2/4!4)/.!,-/ 4)/.! .$! .'5,!2 -/-%.45- ?tc@ cclil mulr m ... lbm rfcp?qncarqmdpm r?rgml?jkm rgml? pc...

40

Transcript of 2/4!4)/.!,-/ 4)/.! .$! .'5,!2 -/-%.45- ?tc@ cclil mulr m ... lbm rfcp?qncarqmdpm r?rgml?jkm rgml? pc...

�� 2/4!4)/.!,�-/4)/.�!.$�!.'5,!2

-/-%.45-

�=;IF9�� �� 4FC�KCLRGML�MD�?�RMPL?BM�AMLHSPCQ�SN�GK?ECQ�MD�P?U�BCQRPSARGTC�NMUCP��4MPL?BMCQ�@JMU�FMSQCQ�?U?W�?Q�GD�RFCW�UCPC�K?BC�MD�N?NCP�?LB�F?TC�@CCL�ILMUL�RM

NGCPAC�RPCC�RPSLIQ�UGRF�NGCACQ�MD�QRP?U��4FCW�BCQACLB�DPMK�AJMSBQ�GL�DSLLCJ JGIC�QF?NCQ�RF?R�QNGL�TGMJCLRJW��N?PRGASJ?PJW�?R�RFC�@MRRMK�UFCPC�RFCW�?PC�KMQR�L?PPMU��NPMBSAGLE

UGLBQ�?Q�FGEF�?Q�����IK�F���APCBGR��$?NFLC�:?P?Q��5�3��.?RGML?J�/AC?LGA�?LB�!RKMQNFCPGA�!BKGLGQRP?RGML

,C?PLGLE�/@HCARGTCQ� ��� �B;I@5F��779@9F5H=CB

z $CQAPG@C�SLGDMPK�AGPASJ?P�KMRGML�

z %VNJ?GL�LML SLGDMPK�AGPASJ?P�KMRGML�

z #?JASJ?RC�?LESJ?P�?AACJCP?RGML�MD�?L�M@HCAR�

z /@QCPTC�RFC�JGLI�@CRUCCL�JGLC?P�?LB�?LESJ?P�?AACJCP?RGML�

� ��� $=B9A5H=7G�C:�+CH5H=CB5@�&CH=CB

z /@QCPTC�RFC�IGLCK?RGAQ�MD�PMR?RGML?J�KMRGML�

z $CPGTC�PMR?RGML?J�IGLCK?RGA�COS?RGMLQ�

z %T?JS?RC�NPM@JCK�QMJTGLE�QRP?RCEGCQ�DMP�PMR?RGML?J�IGLCK?RGAQ�

� ��� �MB5A=7G�C:�+CH5H=CB5@�&CH=CB��+CH5H=CB5@�"B9FH=5

z 5LBCPQR?LB�RFC�PCJ?RGMLQFGN�@CRUCCL�DMPAC��K?QQ�?LB�?AACJCP?RGML�

z 3RSBW�RFC�RSPLGLE�CDDCAR�MD�DMPAC�

z 3RSBW�RFC�?L?JMEW�@CRUCCL�DMPAC�?LB�RMPOSC��K?QQ�?LB�KMKCLR�MD�GLCPRG?��?LB�JGLC?P�?AACJCP?RGML�?LB�?LESJ?P�?AACJCP?RGML�

� ��� +CH5H=CB5@�$=B9H=7��B9F;M��0CF?�5B8��B9F;M�+9J=G=H98

z $CPGTC�RFC�COS?RGML�DMP�PMR?RGML?J�UMPI�

z #?JASJ?RC�PMR?RGML?J�IGLCRGA�CLCPEW�

z $CKMLQRP?RC�RFC�,?U�MD�#MLQCPT?RGML�MD�%LCPEW�

� ��� �B;I@5F�&CA9BHIA�5B8�"HG��CBG9FJ5H=CB

z 5LBCPQR?LB�RFC�?L?JMEW�@CRUCCL�?LESJ?P�KMKCLRSK�?LB�JGLC?P�KMKCLRSK�

z /@QCPTC�RFC�PCJ?RGMLQFGN�@CRUCCL�RMPOSC�?LB�?LESJ?P�KMKCLRSK�

z !NNJW�RFC�J?U�MD�AMLQCPT?RGML�MD�?LESJ?P�KMKCLRSK�

� ��� �C@@=G=CBG�C:��LH9B898��C8=9G�=B�-KC��=A9BG=CBG

z /@QCPTC�AMJJGQGMLQ�MD�CVRCLBCB�@MBGCQ�GL�RUM�BGKCLQGMLQ�

z %V?KGLC�AMJJGQGML�?R�RFC�NMGLR�MD�NCPASQQGML�

� ��� MFCG7CD=7��::97HG��/97HCF��GD97HG�C:��B;I@5F�&CA9BHIA

z $CQAPG@C�RFC�PGEFR F?LB�PSJC�RM�DGLB�RFC�BGPCARGML�MD�?LESJ?P�TCJMAGRW��KMKCLRSK��?LB�RMPOSC�

z %VNJ?GL�RFC�EWPMQAMNGA�CDDCAR�

z 3RSBW�FMU�%?PRF�?ARQ�JGIC�?�EGE?LRGA�EWPMQAMNC�

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

"BHFC8I7H=CB�HC�+CH5H=CB5@�&CH=CB�5B8��B;I@5F�&CA9BHIA

7FW�BM�RMPL?BMCQ�QNGL�?R�?JJ��!LB�UFW�BM�RMPL?BMQ�QNGL�QM�P?NGBJW��4FC�?LQUCP�GQ�RF?R�?GP�K?QQCQ�RF?R�NPMBSAC�RMPL?BMCQ�?PC�RFCKQCJTCQ�PMR?RGLE�

?LB�UFCL�RFC�P?BGG�MD�RFC�?GP�K?QQCQ�BCAPC?QC��RFCGP�P?RC�MD�PMR?RGML�GLAPC?QCQ��!L�GAC�QI?RCP�GLAPC?QCQ�FCP�QNGL�GL�?L�CV?ARJW�?L?JMEMSQ�K?LLCP�?Q

QCCL�GL �=;IF9�� ����4FC�QI?RCP�QR?PRQ�FCP�PMR?RGML�UGRF�MSRQRPCRAFCB�JGK@Q�?LB�GLAPC?QCQ�FCP�QNGL�@W�NSJJGLE�RFCK�GL�RMU?PB�FCP�@MBW��4FC�Q?KC

NFWQGAQ�BCQAPG@CQ�RFC�CVFGJ?P?RGLE�QNGL�MD�?�QI?RCP�?LB�RFC�UPCLAFGLE�DMPAC�MD�?�RMPL?BM�

#JC?PJW��DMPAC��CLCPEW��?LB�NMUCP�?PC�?QQMAG?RCB�UGRF�PMR?RGML?J�KMRGML��4FCQC�?LB�MRFCP�?QNCARQ�MD�PMR?RGML?J�KMRGML�?PC�AMTCPCB�GL�RFGQ�AF?NRCP��7C

QF?JJ�QCC�RF?R�?JJ�GKNMPR?LR�?QNCARQ�MD�PMR?RGML?J�KMRGML�CGRFCP�F?TC�?JPC?BW�@CCL�BCDGLCB�DMP�JGLC?P�KMRGML�MP�F?TC�CV?AR�?L?JMEQ�GL�JGLC?P�KMRGML�

&GPQR��UC�JMMI�?R�?LESJ?P�?AACJCP?RGMLuRFC�PMR?RGML?J�?L?JME�MD�JGLC?P�?AACJCP?RGML�

�=;IF9�� �� 4FGQ�DGESPC�QI?RCP�GLAPC?QCQ�FCP�P?RC�MD�QNGL�@W�NSJJGLE�FCP�?PKQ�?LB�FCP�CVRCLBCB�JCE�AJMQCP�RM�FCP�?VGQ�MD�PMR?RGML���APCBGR��,SS��7GIGKCBG?�#MKKMLQ

� �� �B;I@5F��779@9F5H=CB

.B=:CFA��=F7I@5F�&CH=CB�5B8� F5J=H5H=CB BGQASQQCB�MLJW�SLGDMPK�AGPASJ?P�KMRGML��UFGAF�GQ�KMRGML�GL�?�AGPAJC�?R�AMLQR?LR�QNCCB�?LB��FCLAC��AMLQR?LR

?LESJ?P�TCJMAGRW��2CA?JJ�RF?R�?LESJ?P�TCJMAGRW ω U?Q�BCDGLCB�?Q�RFC�RGKC�P?RC�MD�AF?LEC�MD�?LEJC θ �

�����ω = Δθ

Δt,

UFCPC θ GQ�RFC�?LEJC�MD�PMR?RGML�?Q�QCCL�GL �=;IF9�� ����4FC�PCJ?RGMLQFGN�@CRUCCL�?LESJ?P�TCJMAGRW ω ?LB�JGLC?P�TCJMAGRW v U?Q�?JQM�BCDGLCB�GL

+CH5H=CB��B;@9�5B8��B;I@5F�/9@C7=HM ?Q

�����v = rω

MP

�����ω = vr ,

UFCPC r GQ�RFC�P?BGSQ�MD�ASPT?RSPC��?JQM�QCCL�GL �=;IF9�� ����!AAMPBGLE�RM�RFC�QGEL�AMLTCLRGML��RFC�AMSLRCP�AJMAIUGQC�BGPCARGML�GQ�AMLQGBCPCB�?Q

NMQGRGTC�BGPCARGML�?LB�AJMAIUGQC�BGPCARGML�?Q�LCE?RGTC

�=;IF9�� �� 4FGQ�DGESPC�QFMUQ�SLGDMPK�AGPASJ?P�KMRGML�?LB�QMKC�MD�GRQ�BCDGLCB�OS?LRGRGCQ�

!LESJ?P�TCJMAGRW�GQ�LMR�AMLQR?LR�UFCL�?�QI?RCP�NSJJQ�GL�FCP�?PKQ��UFCL�?�AFGJB�QR?PRQ�SN�?�KCPPW EM PMSLB�DPMK�PCQR��MP�UFCL�?�AMKNSRCPwQ�F?PB�BGQI

QJMUQ�RM�?�F?JR�UFCL�QUGRAFCB�MDD��)L�?JJ�RFCQC�A?QCQ��RFCPC�GQ�?L 5B;I@5F�5779@9F5H=CB��GL�UFGAF ω AF?LECQ��4FC�D?QRCP�RFC�AF?LEC�MAASPQ��RFC

EPC?RCP�RFC�?LESJ?P�?AACJCP?RGML��!LESJ?P�?AACJCP?RGML α GQ�BCDGLCB�?Q�RFC�P?RC�MD�AF?LEC�MD�?LESJ?P�TCJMAGRW��)L�COS?RGML�DMPK��?LESJ?P�?AACJCP?RGML�GQ

CVNPCQQCB�?Q�DMJJMUQ�

�����α = Δω

Δt,

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

UFCPC Δω GQ�RFC 7<5B;9�=B�5B;I@5F�J9@C7=HM ?LB Δt GQ�RFC�AF?LEC�GL�RGKC��4FC�SLGRQ�MD�?LESJ?P�?AACJCP?RGML�?PC (rad/s)/s ��MP rad/s2��)D ω

GLAPC?QCQ��RFCL α GQ�NMQGRGTC��)D ω BCAPC?QCQ��RFCL α GQ�LCE?RGTC�

�L5AD@9�� �� �5@7I@5H=B;�H<9��B;I@5F��779@9F5H=CB�5B8��979@9F5H=CB�C:�5��=?9�0<99@

3SNNMQC�?�RCCL?ECP�NSRQ�FCP�@GAWAJC�ML�GRQ�@?AI�?LB�QR?PRQ�RFC�PC?P�UFCCJ�QNGLLGLE�DPMK�PCQR�RM�?�DGL?J�?LESJ?P�TCJMAGRW�MD�����PNK�GL������Q���?

#?JASJ?RC�RFC�?LESJ?P�?AACJCP?RGML�GL rad/s2���@�)D�QFC�LMU�QJ?KQ�ML�RFC�@P?ICQ��A?SQGLE�?L�?LESJ?P�?AACJCP?RGML�MD – 87.3 rad/s2

��FMU�JMLE

BMCQ�GR�R?IC�RFC�UFCCJ�RM�QRMN�

,HF5H9;M�:CF��5�

4FC�?LESJ?P�?AACJCP?RGML�A?L�@C�DMSLB�BGPCARJW�DPMK�GRQ�BCDGLGRGML�GL α = ΔωΔt

@CA?SQC�RFC�DGL?J�?LESJ?P�TCJMAGRW�?LB�RGKC�?PC�EGTCL��7C�QCC�RF?R

Δω GQ�����PNK�?LB Δt GQ������Q�

,C@IH=CB�:CF��5�

%LRCPGLE�ILMUL�GLDMPK?RGML�GLRM�RFC�BCDGLGRGML�MD�?LESJ?P�?AACJCP?RGML��UC�ECR

�����α = Δω

Δt

=250 rpm5.00 s

.

"CA?SQC Δω GQ�GL�PCTMJSRGMLQ�NCP�KGLSRC��PNK�?LB�UC�U?LR�RFC�QR?LB?PB�SLGRQ�MD rad/s2DMP�?LESJ?P�?AACJCP?RGML��UC�LCCB�RM�AMLTCPR Δω

DPMK�PNK�RM�P?B�Q�

�����Δω = 250 revmin

⋅ 2π radrev ⋅ 1 min

60 sec

= 26.2rads .

%LRCPGLE�RFGQ�OS?LRGRW�GLRM�RFC�CVNPCQQGML�DMP α ��UC�ECR

�����α = Δω

Δt

= 26.2 rad/s5.00 s

= 5.24 rad/s2.

,HF5H9;M�:CF��6�

)L�RFGQ�N?PR��UC�ILMU�RFC�?LESJ?P�?AACJCP?RGML�?LB�RFC�GLGRG?J�?LESJ?P�TCJMAGRW��7C�A?L�DGLB�RFC�QRMNN?EC�RGKC�@W�SQGLE�RFC�BCDGLGRGML�MD�?LESJ?P

?AACJCP?RGML�?LB�QMJTGLE�DMP Δt ��WGCJBGLE

�����Δt = Δωα .

,C@IH=CB�:CF��6�

(CPC�RFC�?LESJ?P�TCJMAGRW�BCAPC?QCQ�DPMK 26.2 rad/s �����PNK�RM�XCPM��QM�RF?R Δω GQ – 26.2 rad/s ��?LB α GQ�EGTCL�RM�@C – 87.3 rad/s2�

4FSQ�

�����Δt = – 26.2 rad/s– 87.3 rad/s2

= 0.300 s.

�=G7IGG=CB

.MRC�RF?R�RFC�?LESJ?P�?AACJCP?RGML�?Q�RFC�EGPJ�QNGLQ�RFC�UFCCJ�GQ�QK?JJ�?LB�NMQGRGTC��GR�R?ICQ���Q�RM�NPMBSAC�?L�?NNPCAG?@JC�?LESJ?P�TCJMAGRW��7FCL

QFC�FGRQ�RFC�@P?IC��RFC�?LESJ?P�?AACJCP?RGML�GQ�J?PEC�?LB�LCE?RGTC��4FC�?LESJ?P�TCJMAGRW�OSGAIJW�EMCQ�RM�XCPM��)L�@MRF�A?QCQ��RFC�PCJ?RGMLQFGNQ�?PC

?L?JMEMSQ�RM�UF?R�F?NNCLQ�UGRF�JGLC?P�KMRGML��&MP�CV?KNJC��RFCPC�GQ�?�J?PEC�BCACJCP?RGML�UFCL�WMS�AP?QF�GLRM�?�@PGAI�U?JJuRFC�TCJMAGRW�AF?LEC

GQ�J?PEC�GL�?�QFMPR�RGKC�GLRCPT?J�

)D�RFC�@GAWAJC�GL�RFC�NPCACBGLE�CV?KNJC�F?B�@CCL�ML�GRQ�UFCCJQ�GLQRC?B�MD�SNQGBC BMUL��GR�UMSJB�DGPQR�F?TC�?AACJCP?RCB�?JMLE�RFC�EPMSLB�?LB�RFCL

AMKC�RM�?�QRMN��4FGQ�AMLLCARGML�@CRUCCL�AGPASJ?P�KMRGML�?LB�JGLC?P�KMRGML�LCCBQ�RM�@C�CVNJMPCB��&MP�CV?KNJC��GR�UMSJB�@C�SQCDSJ�RM�ILMU�FMU�JGLC?P

?LB�?LESJ?P�?AACJCP?RGML�?PC�PCJ?RCB��)L�AGPASJ?P�KMRGML��JGLC?P�?AACJCP?RGML�GQ G4A:8AG RM�RFC�AGPAJC�?R�RFC�NMGLR�MD�GLRCPCQR��?Q�QCCL�GL �=;IF9�� ���

4FSQ��JGLC?P�?AACJCP?RGML�GQ�A?JJCB H5B;9BH=5@�5779@9F5H=CB at �

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

�=;IF9�� �� )L�AGPASJ?P�KMRGML��JGLC?P�?AACJCP?RGML a ��MAASPQ�?Q�RFC�K?ELGRSBC�MD�RFC�TCJMAGRW�AF?LECQ� a GQ�R?LECLR�RM�RFC�KMRGML��)L�RFC�AMLRCVR�MD�AGPASJ?P�KMRGML��JGLC?P

?AACJCP?RGML�GQ�?JQM�A?JJCB�R?LECLRG?J�?AACJCP?RGML at �

,GLC?P�MP�R?LECLRG?J�?AACJCP?RGML�PCDCPQ�RM�AF?LECQ�GL�RFC�K?ELGRSBC�MD�TCJMAGRW�@SR�LMR�GRQ�BGPCARGML��7C�ILMU�DPMK .B=:CFA��=F7I@5F�&CH=CB�5B8

F5J=H5H=CB RF?R�GL�AGPASJ?P�KMRGML�ACLRPGNCR?J�?AACJCP?RGML� ac ��PCDCPQ�RM�AF?LECQ�GL�RFC�BGPCARGML�MD�RFC�TCJMAGRW�@SR�LMR�GRQ�K?ELGRSBC��!L�M@HCAR

SLBCPEMGLE�AGPASJ?P�KMRGML�CVNCPGCLACQ�ACLRPGNCR?J�?AACJCP?RGML��?Q�QCCL�GL �=;IF9�� ����4FSQ� at ?LB ac ?PC�NCPNCLBGASJ?P�?LB�GLBCNCLBCLR�MD

MLC�?LMRFCP��4?LECLRG?J�?AACJCP?RGML at GQ�BGPCARJW�PCJ?RCB�RM�RFC�?LESJ?P�?AACJCP?RGML α ?LB�GQ�JGLICB�RM�?L�GLAPC?QC�MP�BCAPC?QC�GL�RFC�TCJMAGRW��@SR

LMR�GRQ�BGPCARGML�

�=;IF9�� �� #CLRPGNCR?J�?AACJCP?RGML ac MAASPQ�?Q�RFC�BGPCARGML�MD�TCJMAGRW�AF?LECQ��GR�GQ�NCPNCLBGASJ?P�RM�RFC�AGPASJ?P�KMRGML��#CLRPGNCR?J�?LB�R?LECLRG?J�?AACJCP?RGML�?PC�RFSQ

NCPNCLBGASJ?P�RM�C?AF�MRFCP�

.MU�UC�A?L�DGLB�RFC�CV?AR�PCJ?RGMLQFGN�@CRUCCL�JGLC?P�?AACJCP?RGML at ?LB�?LESJ?P�?AACJCP?RGML α ��"CA?SQC�JGLC?P�?AACJCP?RGML�GQ�NPMNMPRGML?J�RM�?

AF?LEC�GL�RFC�K?ELGRSBC�MD�RFC�TCJMAGRW��GR�GQ�BCDGLCB��?Q�GR�U?Q�GL (B9�=A9BG=CB5@�$=B9A5H=7G�RM�@C

������at = Δv

Δt.

&MP�AGPASJ?P�KMRGML��LMRC�RF?R v = rω ��QM�RF?R

������at =

Δ(rω)Δt

.

4FC�P?BGSQ r GQ�AMLQR?LR�DMP�AGPASJ?P�KMRGML��?LB�QM Δ(rω) = r(Δω) ��4FSQ�

������at = rΔω

Δt.

"W�BCDGLGRGML� α = ΔωΔt

��4FSQ�

������at = rα,

MP

������α =

atr .

4FCQC�COS?RGMLQ�KC?L�RF?R�JGLC?P�?AACJCP?RGML�?LB�?LESJ?P�?AACJCP?RGML�?PC�BGPCARJW�NPMNMPRGML?J��4FC�EPC?RCP�RFC�?LESJ?P�?AACJCP?RGML�GQ��RFC�J?PECP

RFC�JGLC?P��R?LECLRG?J�?AACJCP?RGML�GQ��?LB�TGAC�TCPQ?��&MP�CV?KNJC��RFC�EPC?RCP�RFC�?LESJ?P�?AACJCP?RGML�MD�?�A?PwQ�BPGTC�UFCCJQ��RFC�EPC?RCP�RFC

?AACJCP?RGML�MD�RFC�A?P��4FC�P?BGSQ�?JQM�K?RRCPQ��&MP�CV?KNJC��RFC�QK?JJCP�?�UFCCJ��RFC�QK?JJCP�GRQ�JGLC?P�?AACJCP?RGML�DMP�?�EGTCL�?LESJ?P�?AACJCP?RGML

α �

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

�L5AD@9�� �� �5@7I@5H=B;�H<9��B;I@5F��779@9F5H=CB�C:�5�&CHCF7M7@9�0<99@

!�NMUCPDSJ�KMRMPAWAJC�A?L�?AACJCP?RC�DPMK���RM������K�Q��?@MSR�����IK�F�GL������Q��7F?R�GQ�RFC�?LESJ?P�?AACJCP?RGML�MD�GRQ������ K P?BGSQ

UFCCJQ���3CC �=;IF9�� ���

�=;IF9�� �� 4FC�JGLC?P�?AACJCP?RGML�MD�?�KMRMPAWAJC�GQ�?AAMKN?LGCB�@W�?L�?LESJ?P�?AACJCP?RGML�MD�GRQ�UFCCJQ�

,HF5H9;M

7C�?PC�EGTCL�GLDMPK?RGML�?@MSR�RFC�JGLC?P�TCJMAGRGCQ�MD�RFC�KMRMPAWAJC��4FSQ��UC�A?L�DGLB�GRQ�JGLC?P�?AACJCP?RGML at ��4FCL��RFC�CVNPCQQGML

α =atr A?L�@C�SQCB�RM�DGLB�RFC�?LESJ?P�?AACJCP?RGML�

,C@IH=CB

4FC�JGLC?P�?AACJCP?RGML�GQ

������at = Δv

Δt

= 30.0 m/s4.20 s

= 7.14 m/s2.

7C�?JQM�ILMU�RFC�P?BGSQ�MD�RFC�UFCCJQ��%LRCPGLE�RFC�T?JSCQ�DMP at ?LB r GLRM α =atr ��UC�ECR

������α =

atr

= 7.14 m/s2

0.320 m

= 22.3 rad/s2.

�=G7IGG=CB

5LGRQ�MD�P?BG?LQ�?PC�BGKCLQGMLJCQQ�?LB�?NNC?P�GL�?LW�PCJ?RGMLQFGN�@CRUCCL�?LESJ?P�?LB�JGLC?P�OS?LRGRGCQ�

3M�D?P��UC�F?TC�BCDGLCB�RFPCC�PMR?RGML?J�OS?LRGRGCQu θ, ω ��?LB α ��4FCQC�OS?LRGRGCQ�?PC�?L?JMEMSQ�RM�RFC�RP?LQJ?RGML?J�OS?LRGRGCQ x, v ��?LB a �

-56@9�� �� BGQNJ?WQ�PMR?RGML?J�OS?LRGRGCQ��RFC�?L?JMEMSQ�RP?LQJ?RGML?J�OS?LRGRGCQ��?LB�RFC�PCJ?RGMLQFGNQ�@CRUCCL�RFCK�

4?@JC����� 2MR?RGML?J�?LB�4P?LQJ?RGML?J�1S?LRGRGCQ

+CH5H=CB5@ -F5BG@5H=CB5@ +9@5H=CBG<=D

θ x θ = xr

ω v ω = vr

α a α =atr

-?IGLE�#MLLCARGMLQ��4?IC (MKC�%VNCPGKCLR

3GR�BMUL�UGRF�WMSP�DCCR�ML�RFC�EPMSLB�ML�?�AF?GP�RF?R�PMR?RCQ��,GDR�MLC�MD�WMSP�JCEQ�QSAF�RF?R�GR�GQ�SL@CLR��QRP?GEFRCLCB�MSR��5QGLE�RFC�MRFCP�JCE�

@CEGL�RM�PMR?RC�WMSPQCJD�@W�NSQFGLE�ML�RFC�EPMSLB��3RMN�SQGLE�WMSP�JCE�RM�NSQF�RFC�EPMSLB�@SR�?JJMU�RFC�AF?GP�RM�PMR?RC��&PMK�RFC�MPGEGL�UFCPC�WMS

@CE?L��QICRAF�RFC�?LEJC��?LESJ?P�TCJMAGRW��?LB�?LESJ?P�?AACJCP?RGML�MD�WMSP�JCE�?Q�?�DSLARGML�MD�RGKC�GL�RFC�DMPK�MD�RFPCC�QCN?P?RC�EP?NFQ�

%QRGK?RC�RFC�K?ELGRSBCQ�MD�RFCQC�OS?LRGRGCQ�

�<97?�2CIF�.B89FGH5B8=B;

!LESJ?P�?AACJCP?RGML�GQ�?�TCARMP��F?TGLE�@MRF�K?ELGRSBC�?LB�BGPCARGML��(MU�BM�UC�BCLMRC�GRQ�K?ELGRSBC�?LB�BGPCARGML��)JJSQRP?RC�UGRF�?L�CV?KNJC�

,C@IH=CB

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

4FC�K?ELGRSBC�MD�?LESJ?P�?AACJCP?RGML�GQ α ?LB�GRQ�KMQR�AMKKML�SLGRQ�?PC rad/s2��4FC�BGPCARGML�MD�?LESJ?P�?AACJCP?RGML�?JMLE�?�DGVCB�?VGQ�GQ

BCLMRCB�@W�?���MP�?�t�QGEL��HSQR�?Q�RFC�BGPCARGML�MD�JGLC?P�?AACJCP?RGML�GL�MLC�BGKCLQGML�GQ�BCLMRCB�@W�?���MP�?�t�QGEL��&MP�CV?KNJC��AMLQGBCP�?

EWKL?QR�BMGLE�?�DMPU?PB�DJGN��(CP�?LESJ?P�KMKCLRSK�UMSJB�@C�N?P?JJCJ�RM�RFC�K?R�?LB�RM�FCP�JCDR��4FC�K?ELGRSBC�MD�FCP�?LESJ?P�?AACJCP?RGML

UMSJB�@C�NPMNMPRGML?J�RM�FCP�?LESJ?P�TCJMAGRW��QNGL�P?RC�?LB�FCP�KMKCLR�MD�GLCPRG?�?@MSR�FCP�QNGL�?VGQ�

0F%4�%VNJMP?RGMLQ��,?BW@SE�2CTMJSRGML

*MGL�RFC�J?BW@SE�GL�?L�CVNJMP?RGML�MD�PMR?RGML?J�KMRGML��2MR?RC�RFC�KCPPW EM PMSLB�RM�AF?LEC�GRQ�?LEJC��MP�AFMMQC�?�AMLQR?LR�?LESJ?P�TCJMAGRW�MP

?LESJ?P�?AACJCP?RGML��%VNJMPC�FMU�AGPASJ?P�KMRGML�PCJ?RCQ�RM�RFC�@SE�Q�V�W�NMQGRGML��TCJMAGRW��?LB�?AACJCP?RGML�SQGLE�TCARMPQ�MP�EP?NFQ�

�=;IF9�� �� %58M6I;�+9JC@IH=CB��<HHD���7BL�CF;�7CBH9BH�A����������FCH5H=CB49B�>5F�

� �� $=B9A5H=7G�C:�+CH5H=CB5@�&CH=CB

*SQR�@W�SQGLE�MSP�GLRSGRGML��UC�A?L�@CEGL�RM�QCC�FMU�PMR?RGML?J�OS?LRGRGCQ�JGIC θ � ω ��?LB α ?PC�PCJ?RCB�RM�MLC�?LMRFCP��&MP�CV?KNJC��GD�?�KMRMPAWAJC

UFCCJ�F?Q�?�J?PEC�?LESJ?P�?AACJCP?RGML�DMP�?�D?GPJW�JMLE�RGKC��GR�CLBQ�SN�QNGLLGLE�P?NGBJW�?LB�PMR?RCQ�RFPMSEF�K?LW�PCTMJSRGMLQ��)L�KMPC�RCAFLGA?J

RCPKQ��GD�RFC�UFCCJwQ�?LESJ?P�?AACJCP?RGML α GQ�J?PEC�DMP�?�JMLE�NCPGMB�MD�RGKC t ��RFCL�RFC�DGL?J�?LESJ?P�TCJMAGRW ω ?LB�?LEJC�MD�PMR?RGML θ ?PC�J?PEC�

4FC�UFCCJwQ�PMR?RGML?J�KMRGML�GQ�CV?ARJW�?L?JMEMSQ�RM�RFC�D?AR�RF?R�RFC�KMRMPAWAJCwQ�J?PEC�RP?LQJ?RGML?J�?AACJCP?RGML�NPMBSACQ�?�J?PEC�DGL?J�TCJMAGRW��?LB

RFC�BGQR?LAC�RP?TCJCB�UGJJ�?JQM�@C�J?PEC�

+GLCK?RGAQ�GQ�RFC�BCQAPGNRGML�MD�KMRGML��4FC ?=B9A5H=7G�C:�FCH5H=CB5@�ACH=CB BCQAPG@CQ�RFC�PCJ?RGMLQFGNQ�?KMLE�PMR?RGML�?LEJC��?LESJ?P�TCJMAGRW�

?LESJ?P�?AACJCP?RGML��?LB�RGKC��,CR�SQ�QR?PR�@W�DGLBGLE�?L�COS?RGML�PCJ?RGLE ω � α ��?LB t ��4M�BCRCPKGLC�RFGQ�COS?RGML��UC�PCA?JJ�?�D?KGJG?P�IGLCK?RGA

COS?RGML�DMP�RP?LQJ?RGML?J��MP�QRP?GEFR JGLC��KMRGML�

������v = v0 + at (constant a)

.MRC�RF?R�GL�PMR?RGML?J�KMRGML a = at ��?LB�UC�QF?JJ�SQC�RFC�QWK@MJ a DMP�R?LECLRG?J�MP�JGLC?P�?AACJCP?RGML�DPMK�LMU�ML��!Q�GL�JGLC?P�IGLCK?RGAQ��UC

?QQSKC a GQ�AMLQR?LR��UFGAF�KC?LQ�RF?R�?LESJ?P�?AACJCP?RGML α GQ�?JQM�?�AMLQR?LR��@CA?SQC a = rα ��.MU��JCR�SQ�QS@QRGRSRC v = rω ?LB a = rα

GLRM�RFC�JGLC?P�COS?RGML�?@MTC�

������rω = rω0 + rαt.

4FC�P?BGSQ r A?LACJQ�GL�RFC�COS?RGML��WGCJBGLE

������ω = ω0 + at (constant a),

UFCPC ω0 GQ�RFC�GLGRG?J�?LESJ?P�TCJMAGRW��4FGQ�J?QR�COS?RGML�GQ�? ><A8@4G<6�E8?4G<BAF;<C ?KMLE ω � α ��?LB t uRF?R�GQ��GR�BCQAPG@CQ�RFCGP�PCJ?RGMLQFGN

UGRFMSR�PCDCPCLAC�RM�DMPACQ�MP�K?QQCQ�RF?R�K?W�?DDCAR�PMR?RGML��)R�GQ�?JQM�NPCAGQCJW�?L?JMEMSQ�GL�DMPK�RM�GRQ�RP?LQJ?RGML?J�AMSLRCPN?PR�

-?IGLE�#MLLCARGMLQ

+GLCK?RGAQ�DMP�PMR?RGML?J�KMRGML�GQ�AMKNJCRCJW�?L?JMEMSQ�RM�RP?LQJ?RGML?J�IGLCK?RGAQ��DGPQR�NPCQCLRCB�GL (B9�=A9BG=CB5@�$=B9A5H=7G�

+GLCK?RGAQ�GQ�AMLACPLCB�UGRF�RFC�BCQAPGNRGML�MD�KMRGML�UGRFMSR�PCE?PB�RM�DMPAC�MP�K?QQ��7C�UGJJ�DGLB�RF?R�RP?LQJ?RGML?J�IGLCK?RGA�OS?LRGRGCQ��QSAF

?Q�BGQNJ?ACKCLR��TCJMAGRW��?LB�?AACJCP?RGML�F?TC�BGPCAR�?L?JMEQ�GL�PMR?RGML?J�KMRGML�

3R?PRGLE�UGRF�RFC�DMSP�IGLCK?RGA�COS?RGMLQ�UC�BCTCJMNCB�GL (B9�=A9BG=CB5@�$=B9A5H=7G��UC�A?L�BCPGTC�RFC�DMJJMUGLE�DMSP�PMR?RGML?J�IGLCK?RGA

COS?RGMLQ��NPCQCLRCB�RMECRFCP�UGRF�RFCGP�RP?LQJ?RGML?J�AMSLRCPN?PRQ�

4?@JC����� 2MR?RGML?J�+GLCK?RGA�%OS?RGMLQ

+CH5H=CB5@ -F5BG@5H=CB5@

θ = ω¯t x = v- t

ω = ω0 + αt v = v0 + at �AMLQR?LR α � a

θ = ω0t + 12αt2 x = v0t + 1

2at2 �AMLQR?LR α � a

ω2 = ω02 + 2αθ v2 = v0

2 + 2ax �AMLQR?LR α � a

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

)L�RFCQC�COS?RGMLQ��RFC�QS@QAPGNR���BCLMRCQ�GLGRG?J�T?JSCQ�� θ0 � x0 ��?LB t0 ?PC�GLGRG?J�T?JSCQ��?LB�RFC�?TCP?EC�?LESJ?P�TCJMAGRW ω- ?LB�?TCP?EC

TCJMAGRW v- ?PC�BCDGLCB�?Q�DMJJMUQ�

������ω̄ =

ω0 + ω

2 and v̄ =

v0 + v

2.

4FC�COS?RGMLQ�EGTCL�?@MTC�GL -56@9�� �� A?L�@C�SQCB�RM�QMJTC�?LW�PMR?RGML?J�MP�RP?LQJ?RGML?J�IGLCK?RGAQ�NPM@JCK�GL�UFGAF a ?LB α ?PC�AMLQR?LR�

0PM@JCK 3MJTGLE�3RP?RCEW�DMP�2MR?RGML?J�+GLCK?RGAQ

�� �K4@<A8�G;8�F<GH4G<BA�GB�78G8E@<A8�G;4G�EBG4G<BA4?�><A8@4G<6F��EBG4G<BA4?�@BG<BA��<F�<AIB?I87��2MR?RGML�KSQR�@C�GLTMJTCB��@SR�UGRFMSR�RFC�LCCB

RM�AMLQGBCP�DMPACQ�MP�K?QQCQ�RF?R�?DDCAR�RFC�KMRGML�

�� 78AG<9L�8K46G?L�J;4G�A887F�GB�58�78G8E@<A87�<A�G;8�CEB5?8@��<78AG<9L�G;8�HA>ABJAF���!�QICRAF�MD�RFC�QGRS?RGML�GQ�SQCDSJ�

�� $4>8�4�?<FG�B9�J;4G�<F�:<I8A�BE�64A�58�<A98EE87�9EB@�G;8�CEB5?8@�4F�FG4G87��<78AG<9L�G;8�>ABJAF��

�� *B?I8�G;8�4CCEBCE<4G8�8DH4G<BA�BE�8DH4G<BAF�9BE�G;8�DH4AG<GL�GB�58�78G8E@<A87��G;8�HA>ABJA���)R�A?L�@C�SQCDSJ�RM�RFGLI�GL�RCPKQ�MD�?

RP?LQJ?RGML?J�?L?JME�@CA?SQC�@W�LMU�WMS�?PC�D?KGJG?P�UGRF�QSAF�KMRGML�

�� *H5FG<GHG8�G;8�>ABJA�I4?H8F�4?BA:�J<G;�G;8<E�HA<GF�<AGB�G;8�4CCEBCE<4G8�8DH4G<BA��4A7�B5G4<A�AH@8E<64?�FB?HG<BAF�6B@C?8G8�J<G;�HA<GF��"C

QSPC�RM�SQC�SLGRQ�MD�P?BG?LQ�DMP�?LEJCQ�

�� �;86>�LBHE�4AFJ8E�GB�F88�<9�<G�<F�E84FBA45?8���B8F�LBHE�4AFJ8E�@4>8�F8AF8�

�L5AD@9�� �� �5@7I@5H=B;�H<9��779@9F5H=CB�C:�5��=G<=B;�+99@

!�BCCN QC?�DGQFCPK?L�FMMIQ�?�@GE�DGQF�RF?R�QUGKQ�?U?W�DPMK�RFC�@M?R�NSJJGLE�RFC�DGQFGLE�JGLC�DPMK�FGQ�DGQFGLE�PCCJ��4FC�UFMJC�QWQRCK�GQ�GLGRG?JJW�?R

PCQR�?LB�RFC�DGQFGLE�JGLC�SLUGLBQ�DPMK�RFC�PCCJ�?R�?�P?BGSQ�MD������AK�DPMK�GRQ�?VGQ�MD�PMR?RGML��4FC�PCCJ�GQ�EGTCL�?L�?LESJ?P�?AACJCP?RGML�MD

110 rad/s2DMP������Q�?Q�QCCL�GL �=;IF9�� ���

�?�7F?R�GQ�RFC�DGL?J�?LESJ?P�TCJMAGRW�MD�RFC�PCCJ�

�@�!R�UF?R�QNCCB�GQ�DGQFGLE�JGLC�JC?TGLE�RFC�PCCJ�?DRCP������Q�CJ?NQCQ�

�A�(MU�K?LW�PCTMJSRGMLQ�BMCQ�RFC�PCCJ�K?IC�

�B�(MU�K?LW�KCRCPQ�MD�DGQFGLE�JGLC�AMKC�MDD�RFC�PCCJ�GL�RFGQ�RGKC�

,HF5H9;M

)L�C?AF�N?PR�MD�RFGQ�CV?KNJC��RFC�QRP?RCEW�GQ�RFC�Q?KC�?Q�GR�U?Q�DMP�QMJTGLE�NPM@JCKQ�GL�JGLC?P�IGLCK?RGAQ��)L�N?PRGASJ?P��ILMUL�T?JSCQ�?PC�GBCLRGDGCB

?LB�?�PCJ?RGMLQFGN�GQ�RFCL�QMSEFR�RF?R�A?L�@C�SQCB�RM�QMJTC�DMP�RFC�SLILMUL�

,C@IH=CB�:CF��5�

(CPC α ?LB t ?PC�EGTCL�?LB ω LCCBQ�RM�@C�BCRCPKGLCB��4FC�KMQR�QRP?GEFRDMPU?PB�COS?RGML�RM�SQC�GQ ω = ω0 + αt @CA?SQC�RFC�SLILMUL�GQ

?JPC?BW�ML�MLC�QGBC�?LB�?JJ�MRFCP�RCPKQ�?PC�ILMUL��4F?R�COS?RGML�QR?RCQ�RF?R

������ω = ω0 + αt.

7C�?PC�?JQM�EGTCL�RF?R ω0 = 0 �GR�QR?PRQ�DPMK�PCQR��QM�RF?R

������ω = 0 + ⎛⎝110 rad/s2⎞

⎠(2.00s) = 220 rad/s.

,C@IH=CB�:CF��6�

.MU�RF?R ω GQ�ILMUL��RFC�QNCCB v A?L�KMQR�C?QGJW�@C�DMSLB�SQGLE�RFC�PCJ?RGMLQFGN

������v = rω,

UFCPC�RFC�P?BGSQ r MD�RFC�PCCJ�GQ�EGTCL�RM�@C������AK��RFSQ�

������v = (0.0450 m)(220 rad/s) = 9.90 m/s.

.MRC�?E?GL�RF?R�P?BG?LQ�KSQR�?JU?WQ�@C�SQCB�GL�?LW�A?JASJ?RGML�PCJ?RGLE�JGLC?P�?LB�?LESJ?P�OS?LRGRGCQ��!JQM��@CA?SQC�P?BG?LQ�?PC�BGKCLQGMLJCQQ�

UC�F?TC m×rad = m �

,C@IH=CB�:CF��7�

(CPC��UC�?PC�?QICB�RM�DGLB�RFC�LSK@CP�MD�PCTMJSRGMLQ��"CA?SQC 1 rev = 2π rad ��UC�A?L�DGLB�RFC�LSK@CP�MD�PCTMJSRGMLQ�@W�DGLBGLE θ GL�P?BG?LQ�

7C�?PC�EGTCL α ?LB t ��?LB�UC�ILMU ω 0 GQ�XCPM��QM�RF?R θ A?L�@C�M@R?GLCB�SQGLE θ = ω0t + 12αt2 �

������θ = ω0t + 1

2αt2

= 0 + (0.500)⎛⎝110 rad/s2⎞⎠(2.00 s)2 = 220 rad.

#MLTCPRGLE�P?BG?LQ�RM�PCTMJSRGMLQ�EGTCQ

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

������θ = (220 rad) 1 rev

2π rad= 35.0 rev.

,C@IH=CB�:CF��8�

4FC�LSK@CP�MD�KCRCPQ�MD�DGQFGLE�JGLC�GQ x ��UFGAF�A?L�@C�M@R?GLCB�RFPMSEF�GRQ�PCJ?RGMLQFGN�UGRF θ �

������x = rθ = (0.0450 m)(220 rad) = 9.90 m.

�=G7IGG=CB

4FGQ�CV?KNJC�GJJSQRP?RCQ�RF?R�PCJ?RGMLQFGNQ�?KMLE�PMR?RGML?J�OS?LRGRGCQ�?PC�FGEFJW�?L?JMEMSQ�RM�RFMQC�?KMLE�JGLC?P�OS?LRGRGCQ��7C�?JQM�QCC�GL�RFGQ

CV?KNJC�FMU�JGLC?P�?LB�PMR?RGML?J�OS?LRGRGCQ�?PC�AMLLCARCB��4FC�?LQUCPQ�RM�RFC�OSCQRGMLQ�?PC�PC?JGQRGA��!DRCP�SLUGLBGLE�DMP�RUM�QCAMLBQ��RFC�PCCJ

GQ�DMSLB�RM�QNGL�?R�����P?B�Q��UFGAF�GQ������PNK���.M�UMLBCP�PCCJQ�QMKCRGKCQ�K?IC�FGEF NGRAFCB�QMSLBQ��4FC�?KMSLR�MD�DGQFGLE�JGLC�NJ?WCB�MSR

GQ������K��?@MSR�PGEFR�DMP�UFCL�RFC�@GE�DGQF�@GRCQ�

�=;IF9�� �� &GQFGLE�JGLC�AMKGLE�MDD�?�PMR?RGLE�PCCJ�KMTCQ�JGLC?PJW� �L5AD@9�� �� ?LB �L5AD@9�� �� AMLQGBCP�PCJ?RGMLQFGNQ�@CRUCCL�PMR?RGML?J�?LB�JGLC?P�OS?LRGRGCQ�?QQMAG?RCB

UGRF�?�DGQFGLE�PCCJ�

�L5AD@9�� �� �5@7I@5H=B;�H<9��IF5H=CB�0<9B�H<9��=G<=B;�+99@�,@CKG��CKB�5B8�,HCDG

.MU�JCR�SQ�AMLQGBCP�UF?R�F?NNCLQ�GD�RFC�DGQFCPK?L�?NNJGCQ�?�@P?IC�RM�RFC�QNGLLGLE�PCCJ��?AFGCTGLE�?L�?LESJ?P�?AACJCP?RGML�MD – 300 rad/s2�

(MU�JMLE�BMCQ�GR�R?IC�RFC�PCCJ�RM�AMKC�RM�?�QRMN�

,HF5H9;M

7C�?PC�?QICB�RM�DGLB�RFC�RGKC t DMP�RFC�PCCJ�RM�AMKC�RM�?�QRMN��4FC�GLGRG?J�?LB�DGL?J�AMLBGRGMLQ�?PC�BGDDCPCLR�DPMK�RFMQC�GL�RFC�NPCTGMSQ�NPM@JCK�

UFGAF�GLTMJTCB�RFC�Q?KC�DGQFGLE�PCCJ��.MU�UC�QCC�RF?R�RFC�GLGRG?J�?LESJ?P�TCJMAGRW�GQ ω0 = 220 rad/s ?LB�RFC�DGL?J�?LESJ?P�TCJMAGRW ω GQ�XCPM�

4FC�?LESJ?P�?AACJCP?RGML�GQ�EGTCL�RM�@C α = −300 rad/s2��%V?KGLGLE�RFC�?T?GJ?@JC�COS?RGMLQ��UC�QCC�?JJ�OS?LRGRGCQ�@SR G ?PC�ILMUL�GL

ω = ω0 + αt, K?IGLE�GR�C?QGCQR�RM�SQC�RFGQ�COS?RGML�

,C@IH=CB

4FC�COS?RGML�QR?RCQ

������ω = ω0 + αt.

7C�QMJTC�RFC�COS?RGML�?JEC@P?GA?JJW�DMP G��?LB�RFCL�QS@QRGRSRC�RFC�ILMUL�T?JSCQ�?Q�SQS?J��WGCJBGLE

������t =

ω − ω0α = 0 − 220 rad/s

−300 rad/s2= 0.733 s.

�=G7IGG=CB

.MRC�RF?R�A?PC�KSQR�@C�R?ICL�UGRF�RFC�QGELQ�RF?R�GLBGA?RC�RFC�BGPCARGMLQ�MD�T?PGMSQ�OS?LRGRGCQ��!JQM��LMRC�RF?R�RFC�RGKC�RM�QRMN�RFC�PCCJ�GQ�D?GPJW�QK?JJ

@CA?SQC�RFC�?AACJCP?RGML�GQ�P?RFCP�J?PEC��&GQFGLE�JGLCQ�QMKCRGKCQ�QL?N�@CA?SQC�MD�RFC�?AACJCP?RGMLQ�GLTMJTCB��?LB�DGQFCPKCL�MDRCL�JCR�RFC�DGQF

QUGK�DMP�?�UFGJC�@CDMPC�?NNJWGLE�@P?ICQ�ML�RFC�PCCJ��!�RGPCB�DGQF�UGJJ�@C�QJMUCP��PCOSGPGLE�?�QK?JJCP�?AACJCP?RGML�

�L5AD@9�� �� �5@7I@5H=B;�H<9�,@CK��779@9F5H=CB�C:�-F5=BG�5B8�-<9=F�0<99@G

,?PEC�DPCGEFR�RP?GLQ�?AACJCP?RC�TCPW�QJMUJW��3SNNMQC�MLC�QSAF�RP?GL�?AACJCP?RCQ�DPMK�PCQR��EGTGLE�GRQ������ K P?BGSQ�UFCCJQ�?L�?LESJ?P

?AACJCP?RGML�MD 0.250 rad/s2��!DRCP�RFC�UFCCJQ�F?TC�K?BC�����PCTMJSRGMLQ��?QQSKC�LM�QJGNN?EC���?�(MU�D?P�F?Q�RFC�RP?GL�KMTCB�BMUL�RFC

RP?AI���@�7F?R�?PC�RFC�DGL?J�?LESJ?P�TCJMAGRW�MD�RFC�UFCCJQ�?LB�RFC�JGLC?P�TCJMAGRW�MD�RFC�RP?GL�

,HF5H9;M

)L�N?PR��?��UC�?PC�?QICB�RM�DGLB x ��?LB�GL��@�UC�?PC�?QICB�RM�DGLB ω ?LB v ��7C�?PC�EGTCL�RFC�LSK@CP�MD�PCTMJSRGMLQ θ ��RFC�P?BGSQ�MD�RFC

UFCCJQ r ��?LB�RFC�?LESJ?P�?AACJCP?RGML α �

,C@IH=CB�:CF��5�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

4FC�BGQR?LAC x GQ�TCPW�C?QGJW�DMSLB�DPMK�RFC�PCJ?RGMLQFGN�@CRUCCL�BGQR?LAC�?LB�PMR?RGML�?LEJC�

������θ = xr .

3MJTGLE�RFGQ�COS?RGML�DMP x WGCJBQ

������x = rθ.

"CDMPC�SQGLE�RFGQ�COS?RGML��UC�KSQR�AMLTCPR�RFC�LSK@CP�MD�PCTMJSRGMLQ�GLRM�P?BG?LQ��@CA?SQC�UC�?PC�BC?JGLE�UGRF�?�PCJ?RGMLQFGN�@CRUCCL�JGLC?P

?LB�PMR?RGML?J�OS?LRGRGCQ�

������θ = (200 rev)2π rad

1 rev= 1257 rad.

.MU�UC�A?L�QS@QRGRSRC�RFC�ILMUL�T?JSCQ�GLRM x = rθ RM�DGLB�RFC�BGQR?LAC�RFC�RP?GL�KMTCB�BMUL�RFC�RP?AI�

������x = rθ = (0.350 m)(1257 rad) = 440 m.

,C@IH=CB�:CF��6�

7C�A?LLMR�SQC�?LW�COS?RGML�RF?R�GLAMPNMP?RCQ t RM�DGLB ω ��@CA?SQC�RFC�COS?RGML�UMSJB�F?TC�?R�JC?QR�RUM�SLILMUL�T?JSCQ��4FC�COS?RGML

ω2 = ω02 + 2αθ UGJJ�UMPI��@CA?SQC�UC�ILMU�RFC�T?JSCQ�DMP�?JJ�T?PG?@JCQ�CVACNR ω �

������ω2 = ω02 + 2αθ

4?IGLE�RFC�QOS?PC�PMMR�MD�RFGQ�COS?RGML�?LB�CLRCPGLE�RFC�ILMUL�T?JSCQ�EGTCQ

������ω = ⎡

⎣0 + 2(0.250 rad/s2)(1257 rad)⎤⎦1 / 2

= 25.1 rad/s.

7C�A?L�DGLB�RFC�JGLC?P�TCJMAGRW�MD�RFC�RP?GL� v ��RFPMSEF�GRQ�PCJ?RGMLQFGN�RM ω �

������v = rω = (0.350 m)(25.1 rad/s) = 8.77 m/s.

�=G7IGG=CB

4FC�BGQR?LAC�RP?TCJCB�GQ�D?GPJW�J?PEC�?LB�RFC�DGL?J�TCJMAGRW�GQ�D?GPJW�QJMU��HSQR�SLBCP����IK�F�

4FCPC�GQ�RP?LQJ?RGML?J�KMRGML�CTCL�DMP�QMKCRFGLE�QNGLLGLE�GL�NJ?AC��?Q�RFC�DMJJMUGLE�CV?KNJC�GJJSQRP?RCQ� �=;IF9�� �� QFMUQ�?�DJW�ML�RFC�CBEC�MD�?

PMR?RGLE�KGAPMU?TC�MTCL�NJ?RC��4FC�CV?KNJC�@CJMU�A?JASJ?RCQ�RFC�RMR?J�BGQR?LAC�GR�RP?TCJQ�

�=;IF9�� �� 4FC�GK?EC�QFMUQ�?�KGAPMU?TC�NJ?RC��4FC�DJW�K?ICQ�PCTMJSRGMLQ�UFGJC�RFC�DMMB�GQ�FC?RCB��?JMLE�UGRF�RFC�DJW�

�L5AD@9�� �� �5@7I@5H=B;�H<9��=GH5B79�-F5J9@98�6M�5��@M�CB�H<9��8;9�C:�5�&=7FCK5J9�(J9B�)@5H9

!�NCPQML�BCAGBCQ�RM�SQC�?�KGAPMU?TC�MTCL�RM�PCFC?R�QMKC�JSLAF��)L�RFC�NPMACQQ��?�DJW�?AAGBCLR?JJW�DJGCQ�GLRM�RFC�KGAPMU?TC�?LB�J?LBQ�ML�RFC

MSRCP�CBEC�MD�RFC�PMR?RGLE�NJ?RC�?LB�PCK?GLQ�RFCPC��)D�RFC�NJ?RC�F?Q�?�P?BGSQ�MD������K�?LB�PMR?RCQ�?R�����PNK��A?JASJ?RC�RFC�RMR?J�BGQR?LAC�RP?TCJCB

@W�RFC�DJW�BSPGLE�?���� KGL�AMMIGLE�NCPGMB���)ELMPC�RFC�QR?PR SN�?LB�QJMU BMUL�RGKCQ�

,HF5H9;M

&GPQR��DGLB�RFC�RMR?J�LSK@CP�MD�PCTMJSRGMLQ θ ��?LB�RFCL�RFC�JGLC?P�BGQR?LAC x RP?TCJCB� θ = ω̄ t A?L�@C�SQCB�RM�DGLB θ @CA?SQC ω- GQ�EGTCL�RM�@C

����PNK�

,C@IH=CB

%LRCPGLE�ILMUL�T?JSCQ�GLRM θ = ω̄ t EGTCQ

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

������θ = ω- t = ⎛⎝6.0 rpm⎞

⎠(2.0 min) = 12 rev.

!Q�?JU?WQ��GR�GQ�LCACQQ?PW�RM�AMLTCPR�PCTMJSRGMLQ�RM�P?BG?LQ�@CDMPC�A?JASJ?RGLE�?�JGLC?P�OS?LRGRW�JGIC x DPMK�?L�?LESJ?P�OS?LRGRW�JGIC θ �

������θ = (12 rev)

⎛⎝2π rad1 rev

⎞⎠ = 75.4 rad.

.MU��SQGLE�RFC�PCJ?RGMLQFGN�@CRUCCL x ?LB θ ��UC�A?L�BCRCPKGLC�RFC�BGQR?LAC�RP?TCJCB�

������x = rθ = (0.15 m)(75.4 rad) = 11 m.

�=G7IGG=CB

1SGRC�?�RPGN��GD�GR�QSPTGTCQ��.MRC�RF?R�RFGQ�BGQR?LAC�GQ�RFC�RMR?J�BGQR?LAC�RP?TCJCB�@W�RFC�DJW��$GQNJ?ACKCLR�GQ�?ARS?JJW�XCPM�DMP�AMKNJCRC�PCTMJSRGMLQ

@CA?SQC�RFCW�@PGLE�RFC�DJW�@?AI�RM�GRQ�MPGEGL?J�NMQGRGML��4FC�BGQRGLARGML�@CRUCCL�RMR?J�BGQR?LAC�RP?TCJCB�?LB�BGQNJ?ACKCLR�U?Q�DGPQR�LMRCB�GL (B9

�=A9BG=CB5@�$=B9A5H=7G�

�<97?�2CIF�.B89FGH5B8=B;

2MR?RGML?J�IGLCK?RGAQ�F?Q�K?LW�SQCDSJ�PCJ?RGMLQFGNQ��MDRCL�CVNPCQQCB�GL�COS?RGML�DMPK��!PC�RFCQC�PCJ?RGMLQFGNQ�J?UQ�MD�NFWQGAQ�MP�?PC�RFCW�QGKNJW

BCQAPGNRGTC���(GLR��RFC�Q?KC�OSCQRGML�?NNJGCQ�RM�JGLC?P�IGLCK?RGAQ�

,C@IH=CB

2MR?RGML?J�IGLCK?RGAQ��HSQR�JGIC�JGLC?P�IGLCK?RGAQ�GQ�BCQAPGNRGTC�?LB�BMCQ�LMR�PCNPCQCLR�J?UQ�MD�L?RSPC��7GRF�IGLCK?RGAQ��UC�A?L�BCQAPG@C�K?LW

RFGLEQ�RM�EPC?R�NPCAGQGML�@SR�IGLCK?RGAQ�BMCQ�LMR�AMLQGBCP�A?SQCQ��&MP�CV?KNJC��?�J?PEC�?LESJ?P�?AACJCP?RGML�BCQAPG@CQ�?�TCPW�P?NGB�AF?LEC�GL

?LESJ?P�TCJMAGRW�UGRFMSR�?LW�AMLQGBCP?RGML�MD�GRQ�A?SQC�

� �� �MB5A=7G�C:�+CH5H=CB5@�&CH=CB��+CH5H=CB5@�"B9FH=5

)D�WMS�F?TC�CTCP�QNSL�?�@GIC�UFCCJ�MP�NSQFCB�?�KCPPW EM PMSLB��WMS�ILMU�RF?R�DMPAC�GQ�LCCBCB�RM�AF?LEC�?LESJ?P�TCJMAGRW�?Q�QCCL�GL �=;IF9�� �� �

)L�D?AR��WMSP�GLRSGRGML�GQ�PCJG?@JC�GL�NPCBGARGLE�K?LW�MD�RFC�D?ARMPQ�RF?R�?PC�GLTMJTCB��&MP�CV?KNJC��UC�ILMU�RF?R�?�BMMP�MNCLQ�QJMUJW�GD�UC�NSQF�RMM

AJMQC�RM�GRQ�FGLECQ��&SPRFCPKMPC��UC�ILMU�RF?R�RFC�KMPC�K?QQGTC�RFC�BMMP��RFC�KMPC�QJMUJW�GR�MNCLQ��4FC�DGPQR�CV?KNJC�GKNJGCQ�RF?R�RFC�D?PRFCP�RFC

DMPAC�GQ�?NNJGCB�DPMK�RFC�NGTMR��RFC�EPC?RCP�RFC�?LESJ?P�?AACJCP?RGML��?LMRFCP�GKNJGA?RGML�GQ�RF?R�?LESJ?P�?AACJCP?RGML�GQ�GLTCPQCJW�NPMNMPRGML?J�RM�K?QQ�

4FCQC�PCJ?RGMLQFGNQ�QFMSJB�QCCK�TCPW�QGKGJ?P�RM�RFC�D?KGJG?P�PCJ?RGMLQFGNQ�?KMLE�DMPAC��K?QQ��?LB�?AACJCP?RGML�CK@MBGCB�GL�.CURMLwQ�QCAMLB�J?U�MD

KMRGML��4FCPC�?PC��GL�D?AR��NPCAGQC�PMR?RGML?J�?L?JMEQ�RM�@MRF�DMPAC�?LB�K?QQ�

�=;IF9�� �� &MPAC�GQ�PCOSGPCB�RM�QNGL�RFC�@GIC�UFCCJ��4FC�EPC?RCP�RFC�DMPAC��RFC�EPC?RCP�RFC�?LESJ?P�?AACJCP?RGML�NPMBSACB��4FC�KMPC�K?QQGTC�RFC�UFCCJ��RFC�QK?JJCP�RFC

?LESJ?P�?AACJCP?RGML��)D�WMS�NSQF�ML�?�QNMIC�AJMQCP�RM�RFC�?VJC��RFC�?LESJ?P�?AACJCP?RGML�UGJJ�@C�QK?JJCP�

4M�BCTCJMN�RFC�NPCAGQC�PCJ?RGMLQFGN�?KMLE�DMPAC��K?QQ��P?BGSQ��?LB�?LESJ?P�?AACJCP?RGML��AMLQGBCP�UF?R�F?NNCLQ�GD�UC�CVCPR�?�DMPAC F ML�?�NMGLR

K?QQ m RF?R�GQ�?R�?�BGQR?LAC r DPMK�?�NGTMR�NMGLR��?Q�QFMUL�GL �=;IF9�� �����"CA?SQC�RFC�DMPAC�GQ�NCPNCLBGASJ?P�RM r ��?L�?AACJCP?RGML a = Fm GQ

M@R?GLCB�GL�RFC�BGPCARGML�MD F ��7C�A?L�PC?PP?LEC�RFGQ�COS?RGML�QSAF�RF?R F = ma ?LB�RFCL�JMMI�DMP�U?WQ�RM�PCJ?RC�RFGQ�CVNPCQQGML�RM�CVNPCQQGMLQ�DMP

PMR?RGML?J�OS?LRGRGCQ��7C�LMRC�RF?R a = rα ��?LB�UC�QS@QRGRSRC�RFGQ�CVNPCQQGML�GLRM F = ma ��WGCJBGLE

������F = mrα.

2CA?JJ�RF?R HCFEI9 GQ�RFC�RSPLGLE�CDDCARGTCLCQQ�MD�?�DMPAC��)L�RFGQ�A?QC��@CA?SQC F GQ�NCPNCLBGASJ?P�RM r ��RMPOSC�GQ�QGKNJW τ = Fr ��3M��GD�UC�KSJRGNJW

@MRF�QGBCQ�MD�RFC�COS?RGML�?@MTC�@W r ��UC�ECR�RMPOSC�ML�RFC�JCDR F?LB�QGBC��4F?R�GQ�

������rF = mr2α

MP

������τ = mr2α.

4FGQ�J?QR�COS?RGML�GQ�RFC�PMR?RGML?J�?L?JME�MD�.CURMLwQ�QCAMLB�J?U�� F = ma ��UFCPC�RMPOSC�GQ�?L?JMEMSQ�RM�DMPAC��?LESJ?P�?AACJCP?RGML�GQ�?L?JMEMSQ

RM�RP?LQJ?RGML?J�?AACJCP?RGML��?LB mr2GQ�?L?JMEMSQ�RM�K?QQ��MP�GLCPRG?��4FC�OS?LRGRW mr2

GQ�A?JJCB�RFC FCH5H=CB5@�=B9FH=5 MP ACA9BH�C:�=B9FH=5 MD�?

NMGLR�K?QQ m ?�BGQR?LAC r DPMK�RFC�ACLRCP�MD�PMR?RGML�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

�=;IF9�� ��� !L�M@HCAR�GQ�QSNNMPRCB�@W�?�FMPGXMLR?J�DPGARGMLJCQQ�R?@JC�?LB�GQ�?RR?AFCB�RM�?�NGTMR�NMGLR�@W�?�AMPB�RF?R�QSNNJGCQ�ACLRPGNCR?J�DMPAC��!�DMPAC F GQ�?NNJGCB�RM�RFC

M@HCAR�NCPNCLBGASJ?P�RM�RFC�P?BGSQ r ��A?SQGLE�GR�RM�?AACJCP?RC�?@MSR�RFC�NGTMR�NMGLR��4FC�DMPAC�GQ�ICNR�NCPNCLBGASJ?P�RM r �

-?IGLE�#MLLCARGMLQ��2MR?RGML?J�-MRGML�$WL?KGAQ

$WL?KGAQ�DMP�PMR?RGML?J�KMRGML�GQ�AMKNJCRCJW�?L?JMEMSQ�RM�JGLC?P�MP�RP?LQJ?RGML?J�BWL?KGAQ��$WL?KGAQ�GQ�AMLACPLCB�UGRF�DMPAC�?LB�K?QQ�?LB�RFCGP

CDDCARQ�ML�KMRGML��&MP�PMR?RGML?J�KMRGML��UC�UGJJ�DGLB�BGPCAR�?L?JMEQ�RM�DMPAC�?LB�K?QQ�RF?R�@CF?TC�HSQR�?Q�UC�UMSJB�CVNCAR�DPMK�MSP�C?PJGCP

CVNCPGCLACQ�

2MR?RGML?J�)LCPRG?�?LB�-MKCLR�MD�)LCPRG?

"CDMPC�UC�A?L�AMLQGBCP�RFC�PMR?RGML�MD�?LWRFGLE�MRFCP�RF?L�?�NMGLR�K?QQ�JGIC�RFC�MLC�GL �=;IF9�� �����UC�KSQR�CVRCLB�RFC�GBC?�MD�PMR?RGML?J�GLCPRG?�RM

?JJ�RWNCQ�MD�M@HCARQ��4M�CVN?LB�MSP�AMLACNR�MD�PMR?RGML?J�GLCPRG?��UC�BCDGLC�RFC ACA9BH�C:�=B9FH=5 I MD�?L�M@HCAR�RM�@C�RFC�QSK�MD mr2DMP�?JJ�RFC

NMGLR�K?QQCQ�MD�UFGAF�GR�GQ�AMKNMQCB��4F?R�GQ� I = ∑ mr2��(CPC I GQ�?L?JMEMSQ�RM m GL�RP?LQJ?RGML?J�KMRGML��"CA?SQC�MD�RFC�BGQR?LAC r ��RFC

KMKCLR�MD�GLCPRG?�DMP�?LW�M@HCAR�BCNCLBQ�ML�RFC�AFMQCL�?VGQ��!ARS?JJW��A?JASJ?RGLE I GQ�@CWMLB�RFC�QAMNC�MD�RFGQ�RCVR�CVACNR�DMP�MLC�QGKNJC�A?QCuRF?R

MD�?�FMMN��UFGAF�F?Q�?JJ�GRQ�K?QQ�?R�RFC�Q?KC�BGQR?LAC�DPMK�GRQ�?VGQ��!�FMMNwQ�KMKCLR�MD�GLCPRG?�?PMSLB�GRQ�?VGQ�GQ�RFCPCDMPC MR2��UFCPC M GQ�GRQ

RMR?J�K?QQ�?LB R GRQ�P?BGSQ���7C�SQC M ?LB R DMP�?L�CLRGPC�M@HCAR�RM�BGQRGLESGQF�RFCK�DPMK m ?LB r DMP�NMGLR�K?QQCQ��)L�?JJ�MRFCP�A?QCQ��UC

KSQR�AMLQSJR �=;IF9�� ��� �LMRC�RF?R�RFC�R?@JC�GQ�NGCAC�MD�?PRUMPI�RF?R�F?Q�QF?NCQ�?Q�UCJJ�?Q�DMPKSJ?C�DMP�DMPKSJ?Q�DMP I RF?R�F?TC�@CCL�BCPGTCB

DPMK�GLRCEP?RGML�MTCP�RFC�AMLRGLSMSQ�@MBW��.MRC�RF?R I F?Q�SLGRQ�MD�K?QQ�KSJRGNJGCB�@W�BGQR?LAC�QOS?PCB�� kg ⋅ m2��?Q�UC�KGEFR�CVNCAR�DPMK�GRQ

BCDGLGRGML�

4FC�ECLCP?J�PCJ?RGMLQFGN�?KMLE�RMPOSC��KMKCLR�MD�GLCPRG?��?LB�?LESJ?P�?AACJCP?RGML�GQ

������net τ = Iα

MP

������α = net τI

,

UFCPC�LCR τ GQ�RFC�RMR?J�RMPOSC�DPMK�?JJ�DMPACQ�PCJ?RGTC�RM�?�AFMQCL�?VGQ��&MP�QGKNJGAGRW��UC�UGJJ�MLJW�AMLQGBCP�RMPOSCQ�CVCPRCB�@W�DMPACQ�GL�RFC�NJ?LC�MD

RFC�PMR?RGML��3SAF�RMPOSCQ�?PC�CGRFCP�NMQGRGTC�MP�LCE?RGTC�?LB�?BB�JGIC�MPBGL?PW�LSK@CPQ��4FC�PCJ?RGMLQFGN�GL τ = Iα, α = net τI

GQ�RFC�PMR?RGML?J

?L?JME�RM�.CURMLwQ�QCAMLB�J?U�?LB�GQ�TCPW�ECLCP?JJW�?NNJGA?@JC��4FGQ�COS?RGML�GQ�?ARS?JJW�T?JGB�DMP 4AL RMPOSC��?NNJGCB�RM 4AL M@HCAR��PCJ?RGTC�RM 4AL

?VGQ�

!Q�UC�KGEFR�CVNCAR��RFC�J?PECP�RFC�RMPOSC�GQ��RFC�J?PECP�RFC�?LESJ?P�?AACJCP?RGML�GQ��&MP�CV?KNJC��RFC�F?PBCP�?�AFGJB�NSQFCQ�ML�?�KCPPW EM PMSLB��RFC

D?QRCP�GR�?AACJCP?RCQ��&SPRFCPKMPC��RFC�KMPC�K?QQGTC�?�KCPPW EM PMSLB��RFC�QJMUCP�GR�?AACJCP?RCQ�DMP�RFC�Q?KC�RMPOSC��4FC�@?QGA�PCJ?RGMLQFGN�@CRUCCL

KMKCLR�MD�GLCPRG?�?LB�?LESJ?P�?AACJCP?RGML�GQ�RF?R�RFC�J?PECP�RFC�KMKCLR�MD�GLCPRG?��RFC�QK?JJCP�GQ�RFC�?LESJ?P�?AACJCP?RGML��"SR�RFCPC�GQ�?L�?BBGRGML?J

RUGQR��4FC�KMKCLR�MD�GLCPRG?�BCNCLBQ�LMR�MLJW�ML�RFC�K?QQ�MD�?L�M@HCAR��@SR�?JQM�ML�GRQ 7<FGE<5HG<BA MD�K?QQ�PCJ?RGTC�RM�RFC�?VGQ�?PMSLB�UFGAF�GR�PMR?RCQ�

&MP�CV?KNJC��GR�UGJJ�@C�KSAF�C?QGCP�RM�?AACJCP?RC�?�KCPPW EM PMSLB�DSJJ�MD�AFGJBPCL�GD�RFCW�QR?LB�AJMQC�RM�GRQ�?VGQ�RF?L�GD�RFCW�?JJ�QR?LB�?R�RFC�MSRCP�CBEC�

4FC�K?QQ�GQ�RFC�Q?KC�GL�@MRF�A?QCQ��@SR�RFC�KMKCLR�MD�GLCPRG?�GQ�KSAF�J?PECP�UFCL�RFC�AFGJBPCL�?PC�?R�RFC�CBEC�

4?IC (MKC�%VNCPGKCLR

#SR�MSR�?�AGPAJC�RF?R�F?Q�?@MSR�?����AK�P?BGSQ�DPMK�QRGDD�A?PB@M?PB��.C?P�RFC�CBEC�MD�RFC�AGPAJC��UPGRC�LSK@CPQ���RM����JGIC�FMSPQ�ML�?�AJMAI�D?AC�

0MQGRGML�RFC�AGPAJC�QM�RF?R�GR�A?L�PMR?RC�DPCCJW�?@MSR�?�FMPGXMLR?J�?VGQ�RFPMSEF�GRQ�ACLRCP��JGIC�?�UFCCJ���9MS�AMSJB�JMMQCJW�L?GJ�RFC�AGPAJC�RM�?�U?JJ�

(MJB�RFC�AGPAJC�QR?RGML?PW�?LB�UGRF�RFC�LSK@CP����NMQGRGMLCB�?R�RFC�RMN��?RR?AF�?�JSKN�MD�@JSC�NSRRW��QRGAIW�K?RCPG?J�SQCB�DMP�DGVGLE�NMQRCPQ�RM

U?JJQ�?R�RFC�LSK@CP����(MU�J?PEC�BMCQ�RFC�JSKN�LCCB�RM�@C�RM�HSQR�PMR?RC�RFC�AGPAJC��$CQAPG@C�FMU�WMS�A?L�AF?LEC�RFC�KMKCLR�MD�GLCPRG?�MD�RFC

AGPAJC��(MU�BMCQ�RFGQ�AF?LEC�?DDCAR�RFC�?KMSLR�MD�@JSC�NSRRW�LCCBCB�?R�RFC�LSK@CP���RM�HSQR�PMR?RC�RFC�AGPAJC��#F?LEC�RFC�AGPAJCwQ�KMKCLR�MD

GLCPRG?�?LB�RFCL�RPW�PMR?RGLE�RFC�AGPAJC�@W�SQGLE�BGDDCPCLR�?KMSLRQ�MD�@JSC�NSRRW��2CNC?R�RFGQ�NPMACQQ�QCTCP?J�RGKCQ�

0PM@JCK 3MJTGLE�3RP?RCEW�DMP�2MR?RGML?J�$WL?KGAQ

�� �K4@<A8�G;8�F<GH4G<BA�GB�78G8E@<A8�G;4G�GBEDH8�4A7�@4FF�4E8�<AIB?I87�<A�G;8�EBG4G<BA��$P?U�?�A?PCDSJ�QICRAF�MD�RFC�QGRS?RGML�

�� �8G8E@<A8�G;8�FLFG8@�B9�<AG8E8FG�

�� �E4J�4�9E88�5B7L�7<4:E4@��4F?R�GQ��BP?U�?LB�J?@CJ�?JJ�CVRCPL?J�DMPACQ�?ARGLE�ML�RFC�QWQRCK�MD�GLRCPCQR�

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

�� �CC?L net τ = Iα, α = net τI

��G;8�EBG4G<BA4?�8DH<I4?8AG�B9�%8JGBAVF�F86BA7�?4J��GB�FB?I8�G;8�CEB5?8@��#?PC�KSQR�@C�R?ICL�RM�SQC�RFC

AMPPCAR�KMKCLR�MD�GLCPRG?�?LB�RM�AMLQGBCP�RFC�RMPOSC�?@MSR�RFC�NMGLR�MD�PMR?RGML�

�� �F�4?J4LF��6;86>�G;8�FB?HG<BA�GB�F88�<9�<G�<F�E84FBA45?8�

-?IGLE�#MLLCARGMLQ

)L�QR?RGAQ��RFC�LCR�RMPOSC�GQ�XCPM��?LB�RFCPC�GQ�LM�?LESJ?P�?AACJCP?RGML��)L�PMR?RGML?J�KMRGML��LCR�RMPOSC�GQ�RFC�A?SQC�MD�?LESJ?P�?AACJCP?RGML�

CV?ARJW�?Q�GL�.CURMLwQ�QCAMLB�J?U�MD�KMRGML�DMP�PMR?RGML�

�=;IF9�� ��� 3MKC�PMR?RGML?J�GLCPRG?Q�

�L5AD@9�� �� �5@7I@5H=B;�H<9��::97H�C:�&5GG��=GHF=6IH=CB�CB�5�&9FFM C+CIB8

#MLQGBCP�RFC�D?RFCP�NSQFGLE�?�NJ?WEPMSLB�KCPPW EM PMSLB�GL �=;IF9�� �����(C�CVCPRQ�?�DMPAC�MD�����.�?R�RFC�CBEC�MD�RFC����� IE�KCPPW EM PMSLB�

UFGAF�F?Q�?������K�P?BGSQ��#?JASJ?RC�RFC�?LESJ?P�?AACJCP?RGML�NPMBSACB��?�UFCL�LM�MLC�GQ�ML�RFC�KCPPW EM PMSLB�?LB��@�UFCL�?L����� IE�AFGJB

QGRQ������K�?U?W�DPMK�RFC�ACLRCP��#MLQGBCP�RFC�KCPPW EM PMSLB�GRQCJD�RM�@C�?�SLGDMPK�BGQI�UGRF�LCEJGEG@JC�PCR?PBGLE�DPGARGML�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

�=;IF9�� ��� !�D?RFCP�NSQFCQ�?�NJ?WEPMSLB�KCPPW EM PMSLB�?R�GRQ�CBEC�?LB�NCPNCLBGASJ?P�RM�GRQ�P?BGSQ�RM�?AFGCTC�K?VGKSK�RMPOSC�

,HF5H9;M

!LESJ?P�?AACJCP?RGML�GQ�EGTCL�BGPCARJW�@W�RFC�CVNPCQQGML α = net τI

������α = τI.

4M�QMJTC�DMP α ��UC�KSQR�DGPQR�A?JASJ?RC�RFC�RMPOSC τ �UFGAF�GQ�RFC�Q?KC�GL�@MRF�A?QCQ�?LB�KMKCLR�MD�GLCPRG? I �UFGAF�GQ�EPC?RCP�GL�RFC�QCAMLB

A?QC��4M�DGLB�RFC�RMPOSC��UC�LMRC�RF?R�RFC�?NNJGCB�DMPAC�GQ�NCPNCLBGASJ?P�RM�RFC�P?BGSQ�?LB�DPGARGML�GQ�LCEJGEG@JC��QM�RF?R

������τ = rF sin θ = (1.50 m)(250 N) = 375 N ⋅ m.

,C@IH=CB�:CF��5�

4FC�KMKCLR�MD�GLCPRG?�MD�?�QMJGB�BGQI�?@MSR�RFGQ�?VGQ�GQ�EGTCL�GL �=;IF9�� ��� RM�@C

������12MR2,

UFCPC M = 50.0 kg ?LB R = 1.50 m ��QM�RF?R

������I = (0.500)(50.0 kg)(1.50 m)2 = 56.25 kg ⋅ m2.

.MU��?DRCP�UC�QS@QRGRSRC�RFC�ILMUL�T?JSCQ��UC�DGLB�RFC�?LESJ?P�?AACJCP?RGML�RM�@C

������α = τ

I= 375 N ⋅ m

56.25 kg ⋅ m2= 6.67rad

s2.

,C@IH=CB�:CF��6�

7C�CVNCAR�RFC�?LESJ?P�?AACJCP?RGML�DMP�RFC�QWQRCK�RM�@C�JCQQ�GL�RFGQ�N?PR��@CA?SQC�RFC�KMKCLR�MD�GLCPRG?�GQ�EPC?RCP�UFCL�RFC�AFGJB�GQ�ML�RFC�KCPPW

EM PMSLB��4M�DGLB�RFC�RMR?J�KMKCLR�MD�GLCPRG? I ��UC�DGPQR�DGLB�RFC�AFGJBwQ�KMKCLR�MD�GLCPRG? Ic @W�AMLQGBCPGLE�RFC�AFGJB�RM�@C�COSGT?JCLR�RM�?�NMGLR

K?QQ�?R�?�BGQR?LAC�MD������K�DPMK�RFC�?VGQ��4FCL�

������Ic = MR2 = (18.0 kg)(1.25 m)2 = 28.13 kg ⋅ m2.

4FC�RMR?J�KMKCLR�MD�GLCPRG?�GQ�RFC�QSK�MD�KMKCLRQ�MD�GLCPRG?�MD�RFC�KCPPW EM PMSLB�?LB�RFC�AFGJB��?@MSR�RFC�Q?KC�?VGQ��4M�HSQRGDW�RFGQ�QSK�RM

WMSPQCJD��CV?KGLC�RFC�BCDGLGRGML�MD I �

������I = 28.13 kg ⋅ m2 + 56.25 kg ⋅ m2 = 84.38 kg ⋅ m2.

3S@QRGRSRGLE�ILMUL�T?JSCQ�GLRM�RFC�COS?RGML�DMP α EGTCQ

������α = τ

I= 375 N ⋅ m

84.38 kg ⋅ m2= 4.44rad

s2.

�=G7IGG=CB

4FC�?LESJ?P�?AACJCP?RGML�GQ�JCQQ�UFCL�RFC�AFGJB�GQ�ML�RFC�KCPPW EM PMSLB�RF?L�UFCL�RFC�KCPPW EM PMSLB�GQ�CKNRW��?Q�CVNCARCB��4FC�?LESJ?P

?AACJCP?RGMLQ�DMSLB�?PC�OSGRC�J?PEC��N?PRJW�BSC�RM�RFC�D?AR�RF?R�DPGARGML�U?Q�AMLQGBCPCB�RM�@C�LCEJGEG@JC��)D��DMP�CV?KNJC��RFC�D?RFCP�ICNR�NSQFGLE

NCPNCLBGASJ?PJW�DMP������Q��FC�UMSJB�EGTC�RFC�KCPPW EM PMSLB�?L�?LESJ?P�TCJMAGRW�MD������P?B�Q�UFCL�GR�GQ�CKNRW�@SR�MLJW������P?B�Q�UFCL�RFC�AFGJB

GQ�ML�GR��)L�RCPKQ�MD�PCTMJSRGMLQ�NCP�QCAMLB��RFCQC�?LESJ?P�TCJMAGRGCQ�?PC������PCT�Q�?LB������PCT�Q��PCQNCARGTCJW��4FC�D?RFCP�UMSJB�CLB�SN�PSLLGLE

?R�?@MSR����IK�F�GL�RFC�DGPQR�A?QC��3SKKCP�/JWKNGAQ��FCPC�FC�AMKCQ��#MLDGPK?RGML�MD�RFCQC�LSK@CPQ�GQ�JCDR�?Q�?L�CVCPAGQC�DMP�RFC�PC?BCP�

�<97?�2CIF�.B89FGH5B8=B;

4MPOSC�GQ�RFC�?L?JME�MD�DMPAC�?LB�KMKCLR�MD�GLCPRG?�GQ�RFC�?L?JME�MD�K?QQ��&MPAC�?LB�K?QQ�?PC�NFWQGA?J�OS?LRGRGCQ�RF?R�BCNCLB�ML�MLJW�MLC

D?ARMP��&MP�CV?KNJC��K?QQ�GQ�PCJ?RCB�QMJCJW�RM�RFC�LSK@CPQ�MD�?RMKQ�MD�T?PGMSQ�RWNCQ�GL�?L�M@HCAR��!PC�RMPOSC�?LB�KMKCLR�MD�GLCPRG?�QGKGJ?PJW

QGKNJC�

,C@IH=CB

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

.M��4MPOSC�BCNCLBQ�ML�RFPCC�D?ARMPQ��DMPAC�K?ELGRSBC��DMPAC�BGPCARGML��?LB�NMGLR�MD�?NNJGA?RGML��-MKCLR�MD�GLCPRG?�BCNCLBQ�ML�@MRF�K?QQ�?LB�GRQ

BGQRPG@SRGML�PCJ?RGTC�RM�RFC�?VGQ�MD�PMR?RGML��3M��UFGJC�RFC�?L?JMEGCQ�?PC�NPCAGQC��RFCQC�PMR?RGML?J�OS?LRGRGCQ�BCNCLB�ML�KMPC�D?ARMPQ�

� �� +CH5H=CB5@�$=B9H=7��B9F;M��0CF?�5B8��B9F;M�+9J=G=H98

)L�RFGQ�KMBSJC��UC�UGJJ�JC?PL�?@MSR�UMPI�?LB�CLCPEW�?QQMAG?RCB�UGRF�PMR?RGML?J�KMRGML� �=;IF9�� ��� QFMUQ�?�UMPICP�SQGLE�?L�CJCARPGA�EPGLBQRMLC

NPMNCJJCB�@W�?�KMRMP��3N?PIQ�?PC�DJWGLE��?LB�LMGQC�?LB�TG@P?RGML�?PC�APC?RCB�?Q�J?WCPQ�MD�QRCCJ�?PC�N?PCB�DPMK�RFC�NMJC��4FC�QRMLC�AMLRGLSCQ�RM�RSPL

CTCL�?DRCP�RFC�KMRMP�GQ�RSPLCB�MDD��@SR�GR�GQ�CTCLRS?JJW�@PMSEFR�RM�?�QRMN�@W�DPGARGML��#JC?PJW��RFC�KMRMP�F?B�RM�UMPI�RM�ECR�RFC�QRMLC�QNGLLGLE��4FGQ�UMPI

UCLR�GLRM�FC?R��JGEFR��QMSLB��TG@P?RGML��?LB�AMLQGBCP?@JC FCH5H=CB5@�?=B9H=7�9B9F;M�

�=;IF9�� ��� 4FC�KMRMP�UMPIQ�GL�QNGLLGLE�RFC�EPGLBQRMLC��EGTGLE�GR�PMR?RGML?J�IGLCRGA�CLCPEW��4F?R�CLCPEW�GQ�RFCL�AMLTCPRCB�RM�FC?R��JGEFR��QMSLB��?LB�TG@P?RGML���APCBGR��5�3��.?TW

NFMRM�@W�-?QQ�#MKKSLGA?RGML�3NCAG?JGQR�3C?K?L�:?AF?PW�$?TGB�"CJJ

7MPI�KSQR�@C�BMLC�RM�PMR?RC�M@HCARQ�QSAF�?Q�EPGLBQRMLCQ�MP�KCPPW EM PMSLBQ��7MPI�U?Q�BCDGLCB�GL .B=:CFA��=F7I@5F�&CH=CB�5B8� F5J=H5H=CB DMP

RP?LQJ?RGML?J�KMRGML��?LB�UC�A?L�@SGJB�ML�RF?R�ILMUJCBEC�UFCL�AMLQGBCPGLE�UMPI�BMLC�GL�PMR?RGML?J�KMRGML��4FC�QGKNJCQR�PMR?RGML?J�QGRS?RGML�GQ�MLC�GL

UFGAF�RFC�LCR�DMPAC�GQ�CVCPRCB�NCPNCLBGASJ?P�RM�RFC�P?BGSQ�MD�?�BGQI��?Q�QFMUL�GL �=;IF9�� ����?LB�PCK?GLQ�NCPNCLBGASJ?P�?Q�RFC�BGQI�QR?PRQ�RM�PMR?RC�

4FC�DMPAC�GQ�N?P?JJCJ�RM�RFC�BGQNJ?ACKCLR��?LB�QM�RFC�LCR�UMPI�BMLC�GQ�RFC�NPMBSAR�MD�RFC�DMPAC�RGKCQ�RFC�?PA�JCLERF�RP?TCJCB�

������net W = (net F)Δs.

4M�ECR�RMPOSC�?LB�MRFCP�PMR?RGML?J�OS?LRGRGCQ�GLRM�RFC�COS?RGML��UC�KSJRGNJW�?LB�BGTGBC�RFC�PGEFR F?LB�QGBC�MD�RFC�COS?RGML�@W r ��?LB�E?RFCP�RCPKQ�

������net W = (r net F)Δsr .

7C�PCAMELGXC�RF?R r net F = net τ ?LB Δs / r = θ ��QM�RF?R

������net W = (net τ)θ.

4FGQ�COS?RGML�GQ�RFC�CVNPCQQGML�DMP�PMR?RGML?J�UMPI��)R�GQ�TCPW�QGKGJ?P�RM�RFC�D?KGJG?P�BCDGLGRGML�MD�RP?LQJ?RGML?J�UMPI�?Q�DMPAC�KSJRGNJGCB�@W�BGQR?LAC��(CPC�

RMPOSC�GQ�?L?JMEMSQ�RM�DMPAC��?LB�?LEJC�GQ�?L?JMEMSQ�RM�BGQR?LAC��4FC�COS?RGML net W = (net τ)θ GQ�T?JGB�GL�ECLCP?J��CTCL�RFMSEF�GR�U?Q�BCPGTCB�DMP

?�QNCAG?J�A?QC�

4M�ECR�?L�CVNPCQQGML�DMP�PMR?RGML?J�IGLCRGA�CLCPEW��UC�KSQR�?E?GL�NCPDMPK�QMKC�?JEC@P?GA�K?LGNSJ?RGMLQ��4FC�DGPQR�QRCN�GQ�RM�LMRC�RF?R net τ = Iα ��QM

RF?R

������net W = Iαθ.

�=;IF9�� ��� 4FC�LCR�DMPAC�ML�RFGQ�BGQI�GQ�ICNR�NCPNCLBGASJ?P�RM�GRQ�P?BGSQ�?Q�RFC�DMPAC�A?SQCQ�RFC�BGQI�RM�PMR?RC��4FC�LCR�UMPI�BMLC�GQ�RFSQ (net F)Δs ��4FC�LCR�UMPI�EMCQ

GLRM�PMR?RGML?J�IGLCRGA�CLCPEW�

-?IGLE�#MLLCARGMLQ

7MPI�?LB�CLCPEW�GL�PMR?RGML?J�KMRGML�?PC�AMKNJCRCJW�?L?JMEMSQ�RM�UMPI�?LB�CLCPEW�GL�RP?LQJ?RGML?J�KMRGML��DGPQR�NPCQCLRCB�GL .B=:CFA��=F7I@5F

&CH=CB�5B8� F5J=H5H=CB�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

.MU��UC�QMJTC�MLC�MD�RFC�PMR?RGML?J�IGLCK?RGAQ�COS?RGMLQ�DMP αθ ��7C�QR?PR�UGRF�RFC�COS?RGML

������ω2 = ω02 + 2αθ.

.CVR��UC�QMJTC�DMP αθ �

������

αθ =ω2 − ω0

2

2.

3S@QRGRSRGLE�RFGQ�GLRM�RFC�COS?RGML�DMP�LCR W ?LB�E?RFCPGLE�RCPKQ�WGCJBQ

������net W = 12Iω2 − 1

2Iω0

2.

4FGQ�COS?RGML�GQ�RFC KCF?9B9F;M�H<9CF9A DMP�PMR?RGML?J�KMRGML�MLJW��!Q�WMS�K?W�PCA?JJ��LCR�UMPI�AF?LECQ�RFC�IGLCRGA�CLCPEW�MD�?�QWQRCK��4FPMSEF

?L�?L?JMEW�UGRF�RP?LQJ?RGML?J�KMRGML��UC�BCDGLC�RFC�RCPK⎛⎝12⎞⎠Iω

2RM�@C FCH5H=CB5@�?=B9H=7�9B9F;M KErot DMP�?L�M@HCAR�UGRF�?�KMKCLR�MD�GLCPRG? I

?LB�?L�?LESJ?P�TCJMAGRW ω �

������KErot = 12Iω2.

4FC�CVNPCQQGML�DMP�PMR?RGML?J�IGLCRGA�CLCPEW�GQ�CV?ARJW�?L?JMEMSQ�RM�RP?LQJ?RGML?J�IGLCRGA�CLCPEW��UGRF I @CGLE�?L?JMEMSQ�RM m ?LB ω RM v �

2MR?RGML?J�IGLCRGA�CLCPEW�F?Q�GKNMPR?LR�CDDCARQ��&JWUFCCJQ��DMP�CV?KNJC��A?L�@C�SQCB�RM�QRMPC�J?PEC�?KMSLRQ�MD�PMR?RGML?J�IGLCRGA�CLCPEW�GL�?�TCFGAJC�

?Q�QCCL�GL �=;IF9�� ����

�=;IF9�� ��� %VNCPGKCLR?J�TCFGAJCQ��QSAF�?Q�RFGQ�@SQ��F?TC�@CCL�AMLQRPSARCB�GL�UFGAF�PMR?RGML?J�IGLCRGA�CLCPEW�GQ�QRMPCB�GL�?�J?PEC�DJWUFCCJ��7FCL�RFC�@SQ�EMCQ�BMUL�?�FGJJ��GRQ

RP?LQKGQQGML�AMLTCPRQ�GRQ�EP?TGR?RGML?J�NMRCLRG?J�CLCPEW�GLRM KErot ��)R�A?L�?JQM�AMLTCPR�RP?LQJ?RGML?J�IGLCRGA�CLCPEW��UFCL�RFC�@SQ�QRMNQ��GLRM KErot ��4FC�DJWUFCCJwQ�CLCPEW

A?L�RFCL�@C�SQCB�RM�?AACJCP?RC��RM�EM�SN�?LMRFCP�FGJJ��MP�RM�ICCN�RFC�@SQ�DPMK�EMGLE�?E?GLQR�DPGARGML�

�L5AD@9�� �� �5@7I@5H=B;�H<9�0CF?�5B8��B9F;M�:CF�,D=BB=B;�5� F=B8GHCB9

#MLQGBCP�?�NCPQML�UFM�QNGLQ�?�J?PEC�EPGLBQRMLC�@W�NJ?AGLE�FCP�F?LB�ML�GRQ�CBEC�?LB�CVCPRGLE�?�DMPAC�RFPMSEF�N?PR�MD�?�PCTMJSRGML�?Q�QFMUL�GL

�=;IF9�� �����)L�RFGQ�CV?KNJC��UC�TCPGDW�RF?R�RFC�UMPI�BMLC�@W�RFC�RMPOSC�QFC�CVCPRQ�COS?JQ�RFC�AF?LEC�GL�PMR?RGML?J�CLCPEW���?�(MU�KSAF�UMPI

GQ�BMLC�GD�QFC�CVCPRQ�?�DMPAC�MD�����.�RFPMSEF�?�PMR?RGML�MD 1.00 rad(57.3º) ��4FC�DMPAC�GQ�ICNR�NCPNCLBGASJ?P�RM�RFC�EPGLBQRMLCwQ������ K�P?BGSQ

?R�RFC�NMGLR�MD�?NNJGA?RGML��?LB�RFC�CDDCARQ�MD�DPGARGML�?PC�LCEJGEG@JC���@�7F?R�GQ�RFC�DGL?J�?LESJ?P�TCJMAGRW�GD�RFC�EPGLBQRMLC�F?Q�?�K?QQ�MD������IE�

�A�7F?R�GQ�RFC�DGL?J�PMR?RGML?J�IGLCRGA�CLCPEW���)R�QFMSJB�COS?J�RFC�UMPI�

,HF5H9;M

4M�DGLB�RFC�UMPI��UC�A?L�SQC�RFC�COS?RGML net W = (net τ)θ ��7C�F?TC�CLMSEF�GLDMPK?RGML�RM�A?JASJ?RC�RFC�RMPOSC�?LB�?PC�EGTCL�RFC�PMR?RGML

?LEJC��)L�RFC�QCAMLB�N?PR��UC�A?L�DGLB�RFC�DGL?J�?LESJ?P�TCJMAGRW�SQGLE�MLC�MD�RFC�IGLCK?RGA�PCJ?RGMLQFGNQ��)L�RFC�J?QR�N?PR��UC�A?L�A?JASJ?RC�RFC

PMR?RGML?J�IGLCRGA�CLCPEW�DPMK�GRQ�CVNPCQQGML�GL KErot = 12Iω2

,C@IH=CB�:CF��5�

4FC�LCR�UMPI�GQ�CVNPCQQCB�GL�RFC�COS?RGML

������net W = (net τ)θ,

UFCPC�LCR τ GQ�RFC�?NNJGCB�DMPAC�KSJRGNJGCB�@W�RFC�P?BGSQ (rF) @CA?SQC�RFCPC�GQ�LM�PCR?PBGLE�DPGARGML��?LB�RFC�DMPAC�GQ�NCPNCLBGASJ?P�RM r ��4FC

?LEJC θ GQ�EGTCL��3S@QRGRSRGLE�RFC�EGTCL�T?JSCQ�GL�RFC�COS?RGML�?@MTC�WGCJBQ

������net W = rFθ = (0.320 m)(200 N)(1.00 rad)

= 64.0 N ⋅ m.

.MRGLE�RF?R 1 N · m = 1 J �

������net W = 64.0 J.

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

�=;IF9�� ��� !�J?PEC�EPGLBQRMLC�GQ�EGTCL�?�QNGL�@W�?�NCPQML�EP?QNGLE�GRQ�MSRCP�CBEC�

,C@IH=CB�:CF��6�

4M�DGLB ω DPMK�RFC�EGTCL�GLDMPK?RGML�PCOSGPCQ�KMPC�RF?L�MLC�QRCN��7C�QR?PR�UGRF�RFC�IGLCK?RGA�PCJ?RGMLQFGN�GL�RFC�COS?RGML

������ω2 = ω02 + 2αθ.

.MRC�RF?R ω0 = 0 @CA?SQC�UC�QR?PR�DPMK�PCQR��4?IGLE�RFC�QOS?PC�PMMR�MD�RFC�PCQSJRGLE�COS?RGML�EGTCQ

������ω = (2αθ)1 / 2.

.MU�UC�LCCB�RM�DGLB α ��/LC�NMQQG@GJGRW�GQ

������α = net τI

,

UFCPC�RFC�RMPOSC�GQ

������net τ = rF = (0.320 m)(200 N) = 64.0 N ⋅ m.

4FC�DMPKSJ?�DMP�RFC�KMKCLR�MD�GLCPRG?�DMP�?�BGQI�GQ�DMSLB�GL �=;IF9�� ����

������I = 1

2MR2 = 0.5⎛

⎝85.0 kg⎞⎠(0.320 m)2 = 4.352 kg ⋅ m2.

3S@QRGRSRGLE�RFC�T?JSCQ�MD�RMPOSC�?LB�KMKCLR�MD�GLCPRG?�GLRM�RFC�CVNPCQQGML�DMP α ��UC�M@R?GL

������α = 64.0 N ⋅ m

4.352 kg ⋅ m2= 14.7rad

s2.

.MU��QS@QRGRSRC�RFGQ�T?JSC�?LB�RFC�EGTCL�T?JSC�DMP θ GLRM�RFC�?@MTC�CVNPCQQGML�DMP ω �

������

ω = (2αθ)1 / 2 =⎡⎣2⎛⎝14.7rad

s2

⎞⎠(1.00 rad)

⎤⎦

1 / 2

= 5.42rads .

,C@IH=CB�:CF��7�

4FC�DGL?J�PMR?RGML?J�IGLCRGA�CLCPEW�GQ

������KErot = 12Iω2.

"MRF I ?LB ω UCPC�DMSLB�?@MTC��4FSQ�

������KErot = (0.5)⎛⎝4.352 kg ⋅ m2⎞⎠(5.42 rad/s)2 = 64.0 J.

�=G7IGG=CB

4FC�DGL?J�PMR?RGML?J�IGLCRGA�CLCPEW�COS?JQ�RFC�UMPI�BMLC�@W�RFC�RMPOSC��UFGAF�AMLDGPKQ�RF?R�RFC�UMPI�BMLC�UCLR�GLRM�PMR?RGML?J�IGLCRGA�CLCPEW��7C

AMSJB��GL�D?AR��F?TC�SQCB�?L�CVNPCQQGML�DMP�CLCPEW�GLQRC?B�MD�?�IGLCK?RGA�PCJ?RGML�RM�QMJTC�N?PR��@��7C�UGJJ�BM�RFGQ�GL�J?RCP�CV?KNJCQ�

(CJGAMNRCP�NGJMRQ�?PC�OSGRC�D?KGJG?P�UGRF�PMR?RGML?J�IGLCRGA�CLCPEW��4FCW�ILMU��DMP�CV?KNJC��RF?R�?�NMGLR�MD�LM�PCRSPL�UGJJ�@C�PC?AFCB�GD�RFCW�?JJMU�RFCGP

@J?BCQ�RM�QJMU�@CJMU�?�APGRGA?J�?LESJ?P�TCJMAGRW�BSPGLE�DJGEFR��4FC�@J?BCQ�JMQC�JGDR��?LB�GR�GQ�GKNMQQG@JC�RM�GKKCBG?RCJW�ECR�RFC�@J?BCQ�QNGLLGLE�D?QR

CLMSEF�RM�PCE?GL�GR��2MR?RGML?J�IGLCRGA�CLCPEW�KSQR�@C�QSNNJGCB�RM�RFC�@J?BCQ�RM�ECR�RFCK�RM�PMR?RC�D?QRCP��?LB�CLMSEF�CLCPEW�A?LLMR�@C�QSNNJGCB�GL

RGKC�RM�?TMGB�?�AP?QF��"CA?SQC�MD�UCGEFR�JGKGR?RGMLQ��FCJGAMNRCP�CLEGLCQ�?PC�RMM�QK?JJ�RM�QSNNJW�@MRF�RFC�CLCPEW�LCCBCB�DMP�JGDR�?LB�RM�PCNJCLGQF�RFC

PMR?RGML?J�IGLCRGA�CLCPEW�MD�RFC�@J?BCQ�MLAC�RFCW�F?TC�QJMUCB�BMUL��4FC�PMR?RGML?J�IGLCRGA�CLCPEW�GQ�NSR�GLRM�RFCK�@CDMPC�R?ICMDD�?LB�KSQR�LMR�@C

?JJMUCB�RM�BPMN�@CJMU�RFGQ�APSAG?J�JCTCJ��/LC�NMQQG@JC�U?W�RM�?TMGB�?�AP?QF�GQ�RM�SQC�RFC�EP?TGR?RGML?J�NMRCLRG?J�CLCPEW�MD�RFC�FCJGAMNRCP�RM�PCNJCLGQF�RFC

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

PMR?RGML?J�IGLCRGA�CLCPEW�MD�RFC�@J?BCQ�@W�JMQGLE�?JRGRSBC�?LB�?JGELGLE�RFC�@J?BCQ�QM�RF?R�RFC�FCJGAMNRCP�GQ�QNSL�SN�GL�RFC�BCQACLR��/D�AMSPQC��GD�RFC

FCJGAMNRCPwQ�?JRGRSBC�GQ�RMM�JMU��RFCL�RFCPC�GQ�GLQSDDGAGCLR�RGKC�DMP�RFC�@J?BC�RM�PCE?GL�JGDR�@CDMPC�PC?AFGLE�RFC�EPMSLB�

0PM@JCK 3MJTGLE�3RP?RCEW�DMP�2MR?RGML?J�%LCPEW

�� �8G8E@<A8�G;4G�8A8E:L�BE�JBE>�<F�<AIB?I87�<A�G;8�EBG4G<BA�

�� �8G8E@<A8�G;8�FLFG8@�B9�<AG8E8FG��!�QICRAF�SQS?JJW�FCJNQ�

�� �A4?LM8�G;8�F<GH4G<BA�GB�78G8E@<A8�G;8�GLC8F�B9�JBE>�4A7�8A8E:L�<AIB?I87�

�� �BE�6?BF87�FLFG8@F��@86;4A<64?�8A8E:L�<F�6BAF8EI87��4F?R�GQ� KEi + PEi = KEf + PEf. .MRC�RF?R KEi ?LB KEf K?W�C?AF�GLAJSBC

RP?LQJ?RGML?J�?LB�PMR?RGML?J�AMLRPG@SRGMLQ�

�� �BE�BC8A�FLFG8@F��KCAF?LGA?J�CLCPEW�K?W�LMR�@C�AMLQCPTCB��?LB�MRFCP�DMPKQ�MD�CLCPEW��PCDCPPCB�RM�NPCTGMSQJW�?Q OE ��QSAF�?Q�FC?R

RP?LQDCP��K?W�CLRCP�MP�JC?TC�RFC�QWQRCK��$CRCPKGLC�UF?R�RFCW�?PC��?LB�A?JASJ?RC�RFCK�?Q�LCACQQ?PW�

�� �?<@<A4G8�G8E@F�J;8E8I8E�CBFF<5?8�GB�F<@C?<9L�G;8�4?:85E4�

�� �;86>�G;8�4AFJ8E�GB�F88�<9�<G�<F�E84FBA45?8�

�L5AD@9�� �� �5@7I@5H=B;�!9@=7CDH9F��B9F;=9G

!�RWNGA?J�QK?JJ�PCQASC�FCJGAMNRCP��QGKGJ?P�RM�RFC�MLC�GL �=;IF9�� �����F?Q�DMSP�@J?BCQ��C?AF�GQ������K�JMLE�?LB�F?Q�?�K?QQ�MD������IE��4FC�@J?BCQ

A?L�@C�?NNPMVGK?RCB�?Q�RFGL�PMBQ�RF?R�PMR?RC�?@MSR�MLC�CLB�MD�?L�?VGQ�NCPNCLBGASJ?P�RM�RFCGP�JCLERF��4FC�FCJGAMNRCP�F?Q�?�RMR?J�JM?BCB�K?QQ�MD

�����IE���?�#?JASJ?RC�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�GL�RFC�@J?BCQ�UFCL�RFCW�PMR?RC�?R�����PNK���@�#?JASJ?RC�RFC�RP?LQJ?RGML?J�IGLCRGA�CLCPEW�MD�RFC

FCJGAMNRCP�UFCL�GR�DJGCQ�?R������K�Q��?LB�AMKN?PC�GR�UGRF�RFC�PMR?RGML?J�CLCPEW�GL�RFC�@J?BCQ���A�4M�UF?R�FCGEFR�AMSJB�RFC�FCJGAMNRCP�@C�P?GQCB�GD�?JJ

MD�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�AMSJB�@C�SQCB�RM�JGDR�GR�

,HF5H9;M

2MR?RGML?J�?LB�RP?LQJ?RGML?J�IGLCRGA�CLCPEGCQ�A?L�@C�A?JASJ?RCB�DPMK�RFCGP�BCDGLGRGMLQ��4FC�J?QR�N?PR�MD�RFC�NPM@JCK�PCJ?RCQ�RM�RFC�GBC?�RF?R�CLCPEW

A?L�AF?LEC�DMPK��GL�RFGQ�A?QC�DPMK�PMR?RGML?J�IGLCRGA�CLCPEW�RM�EP?TGR?RGML?J�NMRCLRG?J�CLCPEW�

,C@IH=CB�:CF��5�

4FC�PMR?RGML?J�IGLCRGA�CLCPEW�GQ

������KErot = 12Iω2.

7C�KSQR�AMLTCPR�RFC�?LESJ?P�TCJMAGRW�RM�P?BG?LQ�NCP�QCAMLB�?LB�A?JASJ?RC�RFC�KMKCLR�MD�GLCPRG?�@CDMPC�UC�A?L�DGLB KErot ��4FC�?LESJ?P�TCJMAGRW

ω GQ

������ω = 300 rev

1.00 min⋅ 2π rad

1 rev⋅ 1.00 min

60.0 s= 31.4rad

s .

4FC�KMKCLR�MD�GLCPRG?�MD�MLC�@J?BC�UGJJ�@C�RF?R�MD�?�RFGL�PMB�PMR?RCB�?@MSR�GRQ�CLB��DMSLB�GL �=;IF9�� �����4FC�RMR?J I GQ�DMSP�RGKCQ�RFGQ�KMKCLR�MD

GLCPRG?��@CA?SQC�RFCPC�?PC�DMSP�@J?BCQ��4FSQ�

������I = 4Mℓ2

3= 4×

⎛⎝50.0 kg⎞

⎠(4.00 m)2

3= 1067 kg ⋅ m2.

%LRCPGLE ω ?LB I GLRM�RFC�CVNPCQQGML�DMP�PMR?RGML?J�IGLCRGA�CLCPEW�EGTCQ

������KErot = 0.5(1067 kg ⋅ m2)(31.4 rad/s)2

= 5.26×105 J

,C@IH=CB�:CF��6�

4P?LQJ?RGML?J�IGLCRGA�CLCPEW�U?Q�BCDGLCB�GL .B=:CFA��=F7I@5F�&CH=CB�5B8� F5J=H5H=CB��%LRCPGLE�RFC�EGTCL�T?JSCQ�MD�K?QQ�?LB�TCJMAGRW��UC�M@R?GL

������KEtrans = 12mv2 = (0.5)⎛⎝1000 kg⎞

⎠(20.0 m/s)2 = 2.00×105 J.

4M�AMKN?PC�IGLCRGA�CLCPEGCQ��UC�R?IC�RFC�P?RGM�MD�RP?LQJ?RGML?J�IGLCRGA�CLCPEW�RM�PMR?RGML?J�IGLCRGA�CLCPEW��4FGQ�P?RGM�GQ

������2.00×105 J5.26×105 J

= 0.380.

,C@IH=CB�:CF��7�

!R�RFC�K?VGKSK�FCGEFR��?JJ�PMR?RGML?J�IGLCRGA�CLCPEW�UGJJ�F?TC�@CCL�AMLTCPRCB�RM�EP?TGR?RGML?J�CLCPEW��4M�DGLB�RFGQ�FCGEFR��UC�COS?RC�RFMQC�RUM

CLCPEGCQ�

������KErot = PEgrav

MP

������12Iω2 = mgh.

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

7C�LMU�QMJTC�DMP h ?LB�QS@QRGRSRC�ILMUL�T?JSCQ�GLRM�RFC�PCQSJRGLE�COS?RGML

������

h =12Iω

2

mg = 5.26×105 J⎛⎝1000 kg⎞

⎠⎛⎝9.80 m/s2⎞

= 53.7 m.

�=G7IGG=CB

4FC�P?RGM�MD�RP?LQJ?RGML?J�CLCPEW�RM�PMR?RGML?J�IGLCRGA�CLCPEW�GQ�MLJW��������4FGQ�P?RGM�RCJJQ�SQ�RF?R�KMQR�MD�RFC�IGLCRGA�CLCPEW�MD�RFC�FCJGAMNRCP�GQ�GL

GRQ�QNGLLGLE�@J?BCQuQMKCRFGLE�WMS�NPM@?@JW�UMSJB�LMR�QSQNCAR��4FC������K�FCGEFR�RM�UFGAF�RFC�FCJGAMNRCP�AMSJB�@C�P?GQCB�UGRF�RFC�PMR?RGML?J

IGLCRGA�CLCPEW�GQ�?JQM�GKNPCQQGTC��?E?GL�CKNF?QGXGLE�RFC�?KMSLR�MD�PMR?RGML?J�IGLCRGA�CLCPEW�GL�RFC�@J?BCQ�

�=;IF9�� ��� 4FC�DGPQR�GK?EC�QFMUQ�FMU�FCJGAMNRCPQ�QRMPC�J?PEC�?KMSLRQ�MD�PMR?RGML?J�IGLCRGA�CLCPEW�GL�RFCGP�@J?BCQ��4FGQ�CLCPEW�KSQR�@C�NSR�GLRM�RFC�@J?BCQ�@CDMPC�R?ICMDD�?LB

K?GLR?GLCB�SLRGJ�RFC�CLB�MD�RFC�DJGEFR��4FC�CLEGLCQ�BM�LMR�F?TC�CLMSEF�NMUCP�RM�QGKSJR?LCMSQJW�NPMTGBC�JGDR�?LB�NSR�QGELGDGA?LR�PMR?RGML?J�CLCPEW�GLRM�RFC�@J?BCQ��4FC�QCAMLB

GK?EC�QFMUQ�?�FCJGAMNRCP�DPMK�RFC�!SAIJ?LB�7CQRN?A�2CQASC�(CJGAMNRCP�3CPTGAC��/TCP��������JGTCQ�F?TC�@CCL�Q?TCB�QGLAC�GRQ�MNCP?RGMLQ�@CEGLLGLE�GL�������(CPC��?�U?RCP

PCQASC�MNCP?RGML�GQ�QFMUL���APCBGR������%KCPECLAW��&JGAIP

-?IGLE�#MLLCARGMLQ

#MLQCPT?RGML�MD�CLCPEW�GLAJSBCQ�PMR?RGML?J�KMRGML��@CA?SQC�PMR?RGML?J�IGLCRGA�CLCPEW�GQ�?LMRFCP�DMPK�MD KE � .B=:CFA��=F7I@5F�&CH=CB�5B8

F5J=H5H=CB F?Q�?�BCR?GJCB�RPC?RKCLR�MD�AMLQCPT?RGML�MD�CLCPEW�

(MU�4FGAI�)Q�RFC�3MSN��/P�7FW�$MLwR�!JJ�/@HCARQ�2MJJ�$MULFGJJ�?R�RFC�3?KC�2?RC�

/LC�MD�RFC�OS?JGRW�AMLRPMJQ�GL�?�RMK?RM�QMSN�D?ARMPW�AMLQGQRQ�MD�PMJJGLE�DGJJCB�A?LQ�BMUL�?�P?KN��)D�RFCW�PMJJ�RMM�D?QR��RFC�QMSN�GQ�RMM�RFGL��7FW�QFMSJB

A?LQ�MD�GBCLRGA?J�QGXC�?LB�K?QQ�PMJJ�BMUL�?L�GLAJGLC�?R�BGDDCPCLR�P?RCQ��!LB�UFW�QFMSJB�RFC�RFGAICQR�QMSN�PMJJ�RFC�QJMUCQR�

4FC�C?QGCQR�U?W�RM�?LQUCP�RFCQC�OSCQRGMLQ�GQ�RM�AMLQGBCP�CLCPEW��3SNNMQC�C?AF�A?L�QR?PRQ�BMUL�RFC�P?KN�DPMK�PCQR��%?AF�A?L�QR?PRGLE�DPMK�PCQR

KC?LQ�C?AF�QR?PRQ�UGRF�RFC�Q?KC�EP?TGR?RGML?J�NMRCLRG?J�CLCPEW PEgrav ��UFGAF�GQ�AMLTCPRCB�CLRGPCJW�RM KE ��NPMTGBCB�C?AF�PMJJQ�UGRFMSR�QJGNNGLE�

KE ��FMUCTCP��A?L�R?IC�RFC�DMPK�MD KEtrans MP KErot ��?LB�RMR?J KE GQ�RFC�QSK�MD�RFC�RUM��)D�?�A?L�PMJJQ�BMUL�?�P?KN��GR�NSRQ�N?PR�MD�GRQ�CLCPEW�GLRM

PMR?RGML��JC?TGLE�JCQQ�DMP�RP?LQJ?RGML��4FSQ��RFC�A?L�EMCQ�QJMUCP�RF?L�GR�UMSJB�GD�GR�QJGB�BMUL��&SPRFCPKMPC��RFC�RFGL�QMSN�BMCQ�LMR�PMR?RC��UFCPC?Q�RFC

RFGAI�QMSN�BMCQ��@CA?SQC�GR�QRGAIQ�RM�RFC�A?L��4FC�RFGAI�QMSN�RFSQ�NSRQ�KMPC�MD�RFC�A?LwQ�MPGEGL?J�EP?TGR?RGML?J�NMRCLRG?J�CLCPEW�GLRM�PMR?RGML�RF?L�RFC

RFGL�QMSN��?LB�RFC�A?L�PMJJQ�KMPC�QJMUJW��?Q�QCCL�GL �=;IF9�� ����

�=;IF9�� ��� 4FPCC�A?LQ�MD�QMSN�UGRF�GBCLRGA?J�K?QQCQ�P?AC�BMUL�?L�GLAJGLC��4FC�DGPQR�A?L�F?Q�?�JMU�DPGARGML�AM?RGLE�?LB�BMCQ�LMR�PMJJ�@SR�HSQR�QJGBCQ�BMUL�RFC�GLAJGLC��)R�UGLQ

@CA?SQC�GR�AMLTCPRQ�GRQ�CLRGPC�0%�GLRM�RP?LQJ?RGML?J�+%��4FC�QCAMLB�?LB�RFGPB�A?LQ�@MRF�PMJJ�BMUL�RFC�GLAJGLC�UGRFMSR�QJGNNGLE��4FC�QCAMLB�A?L�AMLR?GLQ�RFGL�QMSN�?LB�AMKCQ�GL

QCAMLB�@CA?SQC�N?PR�MD�GRQ�GLGRG?J�0%�EMCQ�GLRM�PMR?RGLE�RFC�A?L��@SR�LMR�RFC�RFGL�QMSN��4FC�RFGPB�A?L�AMLR?GLQ�RFGAI�QMSN��)R�AMKCQ�GL�RFGPB�@CA?SQC�RFC�QMSN�PMR?RCQ�?JMLE�UGRF

RFC�A?L��R?IGLE�CTCL�KMPC�MD�RFC�GLGRG?J�0%�DMP�PMR?RGML?J�+%��JC?TGLE�JCQQ�DMP�RP?LQJ?RGML?J�+%�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

!QQSKGLE�LM�JMQQCQ�BSC�RM�DPGARGML��RFCPC�GQ�MLJW�MLC�DMPAC�BMGLE�UMPIuEP?TGRW��4FCPCDMPC�RFC�RMR?J�UMPI�BMLC�GQ�RFC�AF?LEC�GL�IGLCRGA�CLCPEW��!Q�RFC

A?LQ�QR?PR�KMTGLE��RFC�NMRCLRG?J�CLCPEW�GQ�AF?LEGLE�GLRM�IGLCRGA�CLCPEW��#MLQCPT?RGML�MD�CLCPEW�EGTCQ

������PEi = KEf.

-MPC�QNCAGDGA?JJW�

������PEgrav = KEtrans + KErot

MP

������mgh = 1

2mv2 + 1

2Iω2.

3M��RFC�GLGRG?J mgh GQ�BGTGBCB�@CRUCCL�RP?LQJ?RGML?J�IGLCRGA�CLCPEW�?LB�PMR?RGML?J�IGLCRGA�CLCPEW��?LB�RFC�EPC?RCP I GQ��RFC�JCQQ�CLCPEW�EMCQ�GLRM

RP?LQJ?RGML��)D�RFC�A?L�QJGBCQ�BMUL�UGRFMSR�DPGARGML��RFCL ω = 0 ?LB�?JJ�RFC�CLCPEW�EMCQ�GLRM�RP?LQJ?RGML��RFSQ��RFC�A?L�EMCQ�D?QRCP�

4?IC (MKC�%VNCPGKCLR

,MA?RC�QCTCP?J�A?LQ�C?AF�AMLR?GLGLE�BGDDCPCLR�RWNCQ�MD�DMMB��&GPQR��NPCBGAR�UFGAF�A?L�UGJJ�UGL�RFC�P?AC�BMUL�?L�GLAJGLCB�NJ?LC�?LB�CVNJ?GL�UFW�

3CC�GD�WMSP�NPCBGARGML�GQ�AMPPCAR��9MS�AMSJB�?JQM�BM�RFGQ�CVNCPGKCLR�@W�AMJJCARGLE�QCTCP?J�CKNRW�AWJGLBPGA?J�AMLR?GLCPQ�MD�RFC�Q?KC�QGXC�?LB�DGJJGLE

RFCK�UGRF�BGDDCPCLR�K?RCPG?JQ�QSAF�?Q�UCR�MP�BPW�Q?LB�

�L5AD@9�� �� �5@7I@5H=B;�H<9�,D998�C:�5��M@=B89F�+C@@=B;��CKB�5B�"B7@=B9

#?JASJ?RC�RFC�DGL?J�QNCCB�MD�?�QMJGB�AWJGLBCP�RF?R�PMJJQ�BMUL�?����� K FGEF�GLAJGLC��4FC�AWJGLBCP�QR?PRQ�DPMK�PCQR��F?Q�?�K?QQ�MD�������IE��?LB�F?Q�?

P?BGSQ�MD������AK�

,HF5H9;M

7C�A?L�QMJTC�DMP�RFC�DGL?J�TCJMAGRW�SQGLE�AMLQCPT?RGML�MD�CLCPEW��@SR�UC�KSQR�DGPQR�CVNPCQQ�PMR?RGML?J�OS?LRGRGCQ�GL�RCPKQ�MD�RP?LQJ?RGML?J�OS?LRGRGCQ

RM�CLB�SN�UGRF v ?Q�RFC�MLJW�SLILMUL�

,C@IH=CB

#MLQCPT?RGML�MD�CLCPEW�DMP�RFGQ�QGRS?RGML�GQ�UPGRRCL�?Q�BCQAPG@CB�?@MTC�

������mgh = 1

2mv2 + 1

2Iω2.

"CDMPC�UC�A?L�QMJTC�DMP v ��UC�KSQR�ECR�?L�CVNPCQQGML�DMP I DPMK �=;IF9�� �����"CA?SQC v ?LB ω ?PC�PCJ?RCB��LMRC�FCPC�RF?R�RFC�AWJGLBCP�GQ

PMJJGLE�UGRFMSR�QJGNNGLE��UC�KSQR�?JQM�QS@QRGRSRC�RFC�PCJ?RGMLQFGN ω = v / R GLRM�RFC�CVNPCQQGML��4FCQC�QS@QRGRSRGMLQ�WGCJB

������mgh = 1

2mv2 + 1

2⎛⎝12mR2⎞

⎠⎛⎝v2

R2

⎞⎠.

)LRCPCQRGLEJW��RFC�AWJGLBCPwQ�P?BGSQ R ?LB�K?QQ m A?LACJ��WGCJBGLE

������gh = 1

2v2 + 1

4v2 = 3

4v2.

3MJTGLE�?JEC@P?GA?JJW��RFC�COS?RGML�DMP�RFC�DGL?J�TCJMAGRW v EGTCQ

������

v =⎛⎝4gh3

⎞⎠

1 / 2

.

3S@QRGRSRGLE�ILMUL�T?JSCQ�GLRM�RFC�PCQSJRGLE�CVNPCQQGML�WGCJBQ

������

v =⎡

⎣⎢4⎛⎝9.80 m/s2⎞

⎠(2.00 m)

3

⎦⎥

1 / 2

= 5.11 m/s.

�=G7IGG=CB

"CA?SQC m ?LB R A?LACJ��RFC�PCQSJR v =⎛⎝43gh

⎞⎠

1 / 2GQ�T?JGB�DMP�?LW�QMJGB�AWJGLBCP��GKNJWGLE�RF?R�?JJ�QMJGB�AWJGLBCPQ�UGJJ�PMJJ�BMUL�?L�GLAJGLC�?R�RFC

Q?KC�P?RC�GLBCNCLBCLR�MD�RFCGP�K?QQCQ�?LB�QGXCQ���2MJJGLE�AWJGLBCPQ�BMUL�GLAJGLCQ�GQ�UF?R�'?JGJCM�?ARS?JJW�BGB�RM�QFMU�RF?R�M@HCARQ�D?JJ�?R�RFC

Q?KC�P?RC�GLBCNCLBCLR�MD�K?QQ��.MRC�RF?R�GD�RFC�AWJGLBCP�QJGB�UGRFMSR�DPGARGML�BMUL�RFC�GLAJGLC�UGRFMSR�PMJJGLE��RFCL�RFC�CLRGPC�EP?TGR?RGML?J�NMRCLRG?J

CLCPEW�UMSJB�EM�GLRM�RP?LQJ?RGML?J�IGLCRGA�CLCPEW��4FSQ� 12mv2 = mgh ?LB v = (2gh)1 / 2

��UFGAF�GQ�����EPC?RCP�RF?L (4gh / 3)1 / 2��4F?R�GQ�

RFC�AWJGLBCP�UMSJB�EM�D?QRCP�?R�RFC�@MRRMK�

�<97?�2CIF�.B89FGH5B8=B;

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

�B5@C;M�C:�+CH5H=CB5@�5B8�-F5BG@5H=CB5@�$=B9H=7��B9F;M

)Q�PMR?RGML?J�IGLCRGA�CLCPEW�AMKNJCRCJW�?L?JMEMSQ�RM�RP?LQJ?RGML?J�IGLCRGA�CLCPEW��7F?R��GD�?LW��?PC�RFCGP�BGDDCPCLACQ��'GTC�?L�CV?KNJC�MD�C?AF

RWNC�MD�IGLCRGA�CLCPEW�

,C@IH=CB

9CQ��PMR?RGML?J�?LB�RP?LQJ?RGML?J�IGLCRGA�CLCPEW�?PC�CV?AR�?L?JMEQ��4FCW�@MRF�?PC�RFC�CLCPEW�MD�KMRGML�GLTMJTCB�UGRF�RFC�AMMPBGL?RCB��LML

P?LBMK�KMTCKCLR�MD�K?QQ�PCJ?RGTC�RM�QMKC�PCDCPCLAC�DP?KC��4FC�MLJW�BGDDCPCLAC�@CRUCCL�PMR?RGML?J�?LB�RP?LQJ?RGML?J�IGLCRGA�CLCPEW�GQ�RF?R

RP?LQJ?RGML?J�GQ�QRP?GEFR�JGLC�KMRGML�UFGJC�PMR?RGML?J�GQ�LMR��!L�CV?KNJC�MD�@MRF�IGLCRGA�?LB�RP?LQJ?RGML?J�IGLCRGA�CLCPEW�GQ�DMSLB�GL�?�@GIC�RGPC�UFGJC

@CGLE�PGBBCL�BMUL�?�@GIC�N?RF��4FC�PMR?RGML?J�KMRGML�MD�RFC�RGPC�KC?LQ�GR�F?Q�PMR?RGML?J�IGLCRGA�CLCPEW�UFGJC�RFC�KMTCKCLR�MD�RFC�@GIC�?JMLE�RFC

N?RF�KC?LQ�RFC�RGPC�?JQM�F?Q�RP?LQJ?RGML?J�IGLCRGA�CLCPEW��)D�WMS�UCPC�RM�JGDR�RFC�DPMLR�UFCCJ�MD�RFC�@GIC�?LB�QNGL�GR�UFGJC�RFC�@GIC�GQ�QR?RGML?PW��RFCL

RFC�UFCCJ�UMSJB�F?TC�MLJW�PMR?RGML?J�IGLCRGA�CLCPEW�PCJ?RGTC�RM�RFC�%?PRF�

0F%4�%VNJMP?RGMLQ��-W�3MJ?P�3WQRCK

"SGJB�WMSP�MUL�QWQRCK�MD�FC?TCLJW�@MBGCQ�?LB�U?RAF�RFC�EP?TGR?RGML?J�@?JJCR��7GRF�RFGQ�MP@GR�QGKSJ?RMP��WMS�A?L�QCR�GLGRG?J�NMQGRGMLQ��TCJMAGRGCQ��?LB

K?QQCQ�MD�������MP���@MBGCQ��?LB�RFCL�QCC�RFCK�MP@GR�C?AF�MRFCP�

�=;IF9�� �� &M�,C@5F�,MGH9A��<HHD���7BL�CF;�7CBH9BH�A���� �����AMGC@5FGMGH9A49B�>5F�

� �� �B;I@5F�&CA9BHIA�5B8�"HG��CBG9FJ5H=CB

7FW�BMCQ�%?PRF�ICCN�ML�QNGLLGLE��7F?R�QR?PRCB�GR�QNGLLGLE�RM�@CEGL�UGRF��!LB�FMU�BMCQ�?L�GAC�QI?RCP�K?L?EC�RM�QNGL�D?QRCP�?LB�D?QRCP�QGKNJW�@W

NSJJGLE�FCP�?PKQ�GL��7FW�BMCQ�QFC�LMR�F?TC�RM�CVCPR�?�RMPOSC�RM�QNGL�D?QRCP��1SCQRGMLQ�JGIC�RFCQC�F?TC�?LQUCPQ�@?QCB�GL�?LESJ?P�KMKCLRSK��RFC

PMR?RGML?J�?L?JME�RM�JGLC?P�KMKCLRSK�

"W�LMU�RFC�N?RRCPL�GQ�AJC?PuCTCPW�PMR?RGML?J�NFCLMKCLML�F?Q�?�BGPCAR�RP?LQJ?RGML?J�?L?JME��)R�QCCKQ�OSGRC�PC?QML?@JC��RFCL��RM�BCDGLC 5B;I@5F

ACA9BHIA L ?Q

������L = Iω.

4FGQ�COS?RGML�GQ�?L�?L?JME�RM�RFC�BCDGLGRGML�MD�JGLC?P�KMKCLRSK�?Q p = mv ��5LGRQ�DMP�JGLC?P�KMKCLRSK�?PC kg ⋅ m/s UFGJC�SLGRQ�DMP�?LESJ?P

KMKCLRSK�?PC kg ⋅ m2/s ��!Q�UC�UMSJB�CVNCAR��?L�M@HCAR�RF?R�F?Q�?�J?PEC�KMKCLR�MD�GLCPRG? I ��QSAF�?Q�%?PRF��F?Q�?�TCPW�J?PEC�?LESJ?P�KMKCLRSK�

!L�M@HCAR�RF?R�F?Q�?�J?PEC�?LESJ?P�TCJMAGRW ω ��QSAF�?Q�?�ACLRPGDSEC��?JQM�F?Q�?�P?RFCP�J?PEC�?LESJ?P�KMKCLRSK�

-?IGLE�#MLLCARGMLQ

!LESJ?P�KMKCLRSK�GQ�AMKNJCRCJW�?L?JMEMSQ�RM�JGLC?P�KMKCLRSK��DGPQR�NPCQCLRCB�GL .B=:CFA��=F7I@5F�&CH=CB�5B8� F5J=H5H=CB��)R�F?Q�RFC�Q?KC

GKNJGA?RGMLQ�GL�RCPKQ�MD�A?PPWGLE�PMR?RGML�DMPU?PB��?LB�GR�GQ�AMLQCPTCB�UFCL�RFC�LCR�CVRCPL?J�RMPOSC�GQ�XCPM��!LESJ?P�KMKCLRSK��JGIC�JGLC?P

KMKCLRSK��GQ�?JQM�?�NPMNCPRW�MD�RFC�?RMKQ�?LB�QS@?RMKGA�N?PRGAJCQ�

�L5AD@9�� ��� �5@7I@5H=B;��B;I@5F�&CA9BHIA�C:�H<9��5FH<

,HF5H9;M

.M�GLDMPK?RGML�GQ�EGTCL�GL�RFC�QR?RCKCLR�MD�RFC�NPM@JCK��QM�UC�KSQR�JMMI�SN�NCPRGLCLR�B?R?�@CDMPC�UC�A?L�A?JASJ?RC L = Iω ��&GPQR��?AAMPBGLE�RM

�=;IF9�� �����RFC�DMPKSJ?�DMP�RFC�KMKCLR�MD�GLCPRG?�MD�?�QNFCPC�GQ

������I = 2MR2

5

QM�RF?R

������L = Iω = 2MR2ω

5.

%?PRFwQ�K?QQ M GQ 5.979×1024 kg ?LB�GRQ�P?BGSQ R GQ 6.376×106 m ��4FC�%?PRFwQ�?LESJ?P�TCJMAGRW ω GQ��MD�AMSPQC��CV?ARJW�MLC�PCTMJSRGML

NCP�B?W��@SR�UC�KSQR�AMTCPR ω RM�P?BG?LQ�NCP�QCAMLB�RM�BM�RFC�A?JASJ?RGML�GL�3)�SLGRQ�

,C@IH=CB

3S@QRGRSRGLE�ILMUL�GLDMPK?RGML�GLRM�RFC�CVNPCQQGML�DMP L ?LB�AMLTCPRGLE ω RM�P?BG?LQ�NCP�QCAMLB�EGTCQ

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

������L = 0.4⎛

⎝5.979×1024 kg⎞⎠⎛⎝6.376×106 m⎞

2⎛⎝1 rev

d⎞⎠

= 9.72×1037 kg ⋅ m2 ⋅ rev/d.

3S@QRGRSRGLE 2π P?B�DMP 1 PCT�?LB 8.64×104 s DMP���B?W�EGTCQ

������L = ⎛

⎝9.72×1037 kg ⋅ m2⎞⎠⎛⎝

2π rad/rev8.64×104 s/d

⎞⎠(1 rev/d)

= 7.07×1033 kg ⋅ m2/s.

�=G7IGG=CB

4FGQ�LSK@CP�GQ�J?PEC��BCKMLQRP?RGLE�RF?R�%?PRF��?Q�CVNCARCB��F?Q�?�RPCKCLBMSQ�?LESJ?P�KMKCLRSK��4FC�?LQUCP�GQ�?NNPMVGK?RC��@CA?SQC�UC

F?TC�?QQSKCB�?�AMLQR?LR�BCLQGRW�DMP�%?PRF�GL�MPBCP�RM�CQRGK?RC�GRQ�KMKCLR�MD�GLCPRG?�

7FCL�WMS�NSQF�?�KCPPW EM PMSLB��QNGL�?�@GIC�UFCCJ��MP�MNCL�?�BMMP��WMS�CVCPR�?�RMPOSC��)D�RFC�RMPOSC�WMS�CVCPR�GQ�EPC?RCP�RF?L�MNNMQGLE�RMPOSCQ�

RFCL�RFC�PMR?RGML�?AACJCP?RCQ��?LB�?LESJ?P�KMKCLRSK�GLAPC?QCQ��4FC�EPC?RCP�RFC�LCR�RMPOSC��RFC�KMPC�P?NGB�RFC�GLAPC?QC�GL L ��4FC�PCJ?RGMLQFGN

@CRUCCL�RMPOSC�?LB�?LESJ?P�KMKCLRSK�GQ

������net τ = ΔLΔt

.

4FGQ�CVNPCQQGML�GQ�CV?ARJW�?L?JMEMSQ�RM�RFC�PCJ?RGMLQFGN�@CRUCCL�DMPAC�?LB�JGLC?P�KMKCLRSK� F = Δp / Δt ��4FC�COS?RGML net τ = ΔLΔt

GQ�TCPW

DSLB?KCLR?J�?LB�@PM?BJW�?NNJGA?@JC��)R�GQ��GL�D?AR��RFC�PMR?RGML?J�DMPK�MD�.CURMLwQ�QCAMLB�J?U�

�L5AD@9�� ��� �5@7I@5H=B;�H<9�-CFEI9�)IHH=B;��B;I@5F�&CA9BHIA�"BHC�5�%5NM�,IG5B

�=;IF9�� ��� QFMUQ�?�,?XW�3SQ?L�DMMB�RP?W�@CGLE�PMR?RCB�@W�?�NCPQML�GL�OSCQR�MD�QSQRCL?LAC��3SNNMQC�RFC�NCPQML�CVCPRQ�?������.�DMPAC

NCPNCLBGASJ?P�RM�RFC�J?XW�3SQ?LwQ������ K�P?BGSQ�DMP�������Q���?�7F?R�GQ�RFC�DGL?J�?LESJ?P�KMKCLRSK�MD�RFC�J?XW�3SQ?L�GD�GR�QR?PRQ�DPMK�PCQR�

?QQSKGLE�DPGARGML�GQ�LCEJGEG@JC���@�7F?R�GQ�RFC�DGL?J�?LESJ?P�TCJMAGRW�MD�RFC�J?XW�3SQ?L��EGTCL�RF?R�GRQ�K?QQ�GQ������IE�?LB�?QQSKGLE�GRQ�KMKCLR�MD

GLCPRG?�GQ�RF?R�MD�?�BGQI�

�=;IF9�� ��� !�N?PRWEMCP�CVCPRQ�?�RMPOSC�ML�?�J?XW�3SQ?L�RM�K?IC�GR�PMR?RC��4FC�COS?RGML net τ = ΔLΔt

EGTCQ�RFC�PCJ?RGMLQFGN�@CRUCCL�RMPOSC�?LB�RFC�?LESJ?P

KMKCLRSK�NPMBSACB�

,HF5H9;M

7C�A?L�DGLB�RFC�?LESJ?P�KMKCLRSK�@W�QMJTGLE net τ = ΔLΔt

DMP ΔL ��?LB�SQGLE�RFC�EGTCL�GLDMPK?RGML�RM�A?JASJ?RC�RFC�RMPOSC��4FC�DGL?J�?LESJ?P

KMKCLRSK�COS?JQ�RFC�AF?LEC�GL�?LESJ?P�KMKCLRSK��@CA?SQC�RFC�J?XW�3SQ?L�QR?PRQ�DPMK�PCQR��4F?R�GQ� ΔL = L ��4M�DGLB�RFC�DGL?J�TCJMAGRW��UC

KSQR�A?JASJ?RC ω DPMK�RFC�BCDGLGRGML�MD L GL L = Iω �

,C@IH=CB�:CF��5�

3MJTGLE net τ = ΔLΔt

DMP ΔL EGTCQ

������ΔL = (net τ)Δt.

"CA?SQC�RFC�DMPAC�GQ�NCPNCLBGASJ?P�RM r ��UC�QCC�RF?R net τ = rF ��QM�RF?R

������L = rFΔt = (0.260 m)(2.50 N)(0.150 s)

= 9.75×10−2 kg ⋅ m2 / s.

,C@IH=CB�:CF��6�

4FC�DGL?J�?LESJ?P�TCJMAGRW�A?L�@C�A?JASJ?RCB�DPMK�RFC�BCDGLGRGML�MD�?LESJ?P�KMKCLRSK�

������L = Iω.

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

3MJTGLE�DMP ω ?LB�QS@QRGRSRGLE�RFC�DMPKSJ?�DMP�RFC�KMKCLR�MD�GLCPRG?�MD�?�BGQI�GLRM�RFC�PCQSJRGLE�COS?RGML�EGTCQ

������ω = LI

= L12MR2

.

!LB�QS@QRGRSRGLE�ILMUL�T?JSCQ�GLRM�RFC�NPCACBGLE�COS?RGML�WGCJBQ

�������ω =

9.75×10−2 kg ⋅ m2/s(0.500)⎛⎝4.00 kg⎞

⎠(0.260 m)= 0.721 rad/s.

�=G7IGG=CB

.MRC�RF?R�RFC�GKN?PRCB�?LESJ?P�KMKCLRSK�BMCQ�LMR�BCNCLB�ML�?LW�NPMNCPRW�MD�RFC�M@HCAR�@SR�MLJW�ML�RMPOSC�?LB�RGKC��4FC�DGL?J�?LESJ?P�TCJMAGRW

GQ�COSGT?JCLR�RM�MLC�PCTMJSRGML�GL������Q��BCRCPKGL?RGML�MD�RFC�RGKC�NCPGMB�GQ�JCDR�?Q�?L�CVCPAGQC�DMP�RFC�PC?BCP��UFGAF�GQ�?@MSR�PGEFR�DMP�?�J?XW

3SQ?L�

�L5AD@9�� ��� �5@7I@5H=B;�H<9�-CFEI9�=B�5�$=7?

4FC�NCPQML�UFMQC�JCE�GQ�QFMUL�GL �=;IF9�� ��� IGAIQ�FGQ�JCE�@W�CVCPRGLE�?����� .�DMPAC�UGRF�FGQ�SNNCP�JCE�KSQAJC��4FC�CDDCARGTC�NCPNCLBGASJ?P

JCTCP�?PK�GQ������AK��'GTCL�RFC�KMKCLR�MD�GLCPRG?�MD�RFC�JMUCP�JCE�GQ 1.25 kg ⋅ m2���?�DGLB�RFC�?LESJ?P�?AACJCP?RGML�MD�RFC�JCE���@�.CEJCARGLE

RFC�EP?TGR?RGML?J�DMPAC��UF?R�GQ�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�MD�RFC�JCE�?DRCP�GR�F?Q�PMR?RCB�RFPMSEF 57.3º ������P?B�

�=;IF9�� ��� 4FC�KSQAJC�GL�RFC�SNNCP�JCE�EGTCQ�RFC�JMUCP�JCE�?L�?LESJ?P�?AACJCP?RGML�?LB�GKN?PRQ�PMR?RGML?J�IGLCRGA�CLCPEW�RM�GR�@W�CVCPRGLE�?�RMPOSC�?@MSR�RFC�ILCC� F GQ

?�TCARMP�RF?R�GQ�NCPNCLBGASJ?P�RM r ��4FGQ�CV?KNJC�CV?KGLCQ�RFC�QGRS?RGML�

,HF5H9;M

4FC�?LESJ?P�?AACJCP?RGML�A?L�@C�DMSLB�SQGLE�RFC�PMR?RGML?J�?L?JME�RM�.CURMLwQ�QCAMLB�J?U��MP α = net τ / I ��4FC�KMKCLR�MD�GLCPRG? I GQ�EGTCL

?LB�RFC�RMPOSC�A?L�@C�DMSLB�C?QGJW�DPMK�RFC�EGTCL�DMPAC�?LB�NCPNCLBGASJ?P�JCTCP�?PK��/LAC�RFC�?LESJ?P�?AACJCP?RGML α GQ�ILMUL��RFC�DGL?J

?LESJ?P�TCJMAGRW�?LB�PMR?RGML?J�IGLCRGA�CLCPEW�A?L�@C�A?JASJ?RCB�

,C@IH=CB�HC��5�

&PMK�RFC�PMR?RGML?J�?L?JME�RM�.CURMLwQ�QCAMLB�J?U��RFC�?LESJ?P�?AACJCP?RGML α GQ

�������α = net τI

.

"CA?SQC�RFC�DMPAC�?LB�RFC�NCPNCLBGASJ?P�JCTCP�?PK�?PC�EGTCL�?LB�RFC�JCE�GQ�TCPRGA?J�QM�RF?R�GRQ�UCGEFR�BMCQ�LMR�APC?RC�?�RMPOSC��RFC�LCR�RMPOSC�GQ

RFSQ

�������net τ = r⊥ F

= (0.0220 m)(2000 N)

= 44.0 N ⋅ m.

3S@QRGRSRGLE�RFGQ�T?JSC�DMP�RFC�RMPOSC�?LB�RFC�EGTCL�T?JSC�DMP�RFC�KMKCLR�MD�GLCPRG?�GLRM�RFC�CVNPCQQGML�DMP α EGTCQ

�������α = 44.0 N ⋅ m

1.25 kg ⋅ m2= 35.2 rad/s2.

,C@IH=CB�HC��6�

4FC�DGL?J�?LESJ?P�TCJMAGRW�A?L�@C�A?JASJ?RCB�DPMK�RFC�IGLCK?RGA�CVNPCQQGML

�������ω2 = ω02 + 2αθ

MP

�������ω2 = 2αθ

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

@CA?SQC�RFC�GLGRG?J�?LESJ?P�TCJMAGRW�GQ�XCPM��4FC�IGLCRGA�CLCPEW�MD�PMR?RGML�GQ

�������KErot = 12Iω2

QM�GR�GQ�KMQR�AMLTCLGCLR�RM�SQC�RFC�T?JSC�MD ω2HSQR�DMSLB�?LB�RFC�EGTCL�T?JSC�DMP�RFC�KMKCLR�MD�GLCPRG?��4FC�IGLCRGA�CLCPEW�GQ�RFCL

�������KErot = 0.5⎛⎝1.25 kg ⋅ m2⎞

⎠⎛⎝70.4 rad2 / s2⎞

= 44.0 J.

�=G7IGG=CB

4FCQC�T?JSCQ�?PC�PC?QML?@JC�DMP�?�NCPQML�IGAIGLE�FGQ�JCE�QR?PRGLE�DPMK�RFC�NMQGRGML�QFMUL��4FC�UCGEFR�MD�RFC�JCE�A?L�@C�LCEJCARCB�GL�N?PR��?

@CA?SQC�GR�CVCPRQ�LM�RMPOSC�UFCL�RFC�ACLRCP�MD�EP?TGRW�MD�RFC�JMUCP�JCE�GQ�BGPCARJW�@CLC?RF�RFC�NGTMR�GL�RFC�ILCC��)L�N?PR��@��RFC�DMPAC�CVCPRCB�@W

RFC�SNNCP�JCE�GQ�QM�J?PEC�RF?R�GRQ�RMPOSC�GQ�KSAF�EPC?RCP�RF?L�RF?R�APC?RCB�@W�RFC�UCGEFR�MD�RFC�JMUCP�JCE�?Q�GR�PMR?RCQ��4FC�PMR?RGML?J�IGLCRGA�CLCPEW

EGTCL�RM�RFC�JMUCP�JCE�GQ�CLMSEF�RF?R�GR�AMSJB�EGTC�?�@?JJ�?�QGELGDGA?LR�TCJMAGRW�@W�RP?LQDCPPGLE�QMKC�MD�RFGQ�CLCPEW�GL�?�IGAI�

-?IGLE�#MLLCARGMLQ��#MLQCPT?RGML�,?UQ

!LESJ?P�KMKCLRSK��JGIC�CLCPEW�?LB�JGLC?P�KMKCLRSK��GQ�AMLQCPTCB��4FGQ�SLGTCPQ?JJW�?NNJGA?@JC�J?U�GQ�?LMRFCP�QGEL�MD�SLBCPJWGLE�SLGRW�GL

NFWQGA?J�J?UQ��!LESJ?P�KMKCLRSK�GQ�AMLQCPTCB�UFCL�LCR�CVRCPL?J�RMPOSC�GQ�XCPM��HSQR�?Q�JGLC?P�KMKCLRSK�GQ�AMLQCPTCB�UFCL�RFC�LCR�CVRCPL?J

DMPAC�GQ�XCPM�

#MLQCPT?RGML�MD�!LESJ?P�-MKCLRSK

7C�A?L�LMU�SLBCPQR?LB�UFW�%?PRF�ICCNQ�ML�QNGLLGLE��!Q�UC�Q?U�GL�RFC�NPCTGMSQ�CV?KNJC� ΔL = (net τ)Δt ��4FGQ�COS?RGML�KC?LQ�RF?R��RM�AF?LEC

?LESJ?P�KMKCLRSK��?�RMPOSC�KSQR�?AR�MTCP�QMKC�NCPGMB�MD�RGKC��"CA?SQC�%?PRF�F?Q�?�J?PEC�?LESJ?P�KMKCLRSK��?�J?PEC�RMPOSC�?ARGLE�MTCP�?�JMLE

RGKC�GQ�LCCBCB�RM�AF?LEC�GRQ�P?RC�MD�QNGL��3M�UF?R�CVRCPL?J�RMPOSCQ�?PC�RFCPC��4GB?J�DPGARGML�CVCPRQ�RMPOSC�RF?R�GQ�QJMUGLE�%?PRFwQ�PMR?RGML��@SR�RCLQ�MD

KGJJGMLQ�MD�WC?PQ�KSQR�N?QQ�@CDMPC�RFC�AF?LEC�GQ�TCPW�QGELGDGA?LR��2CACLR�PCQC?PAF�GLBGA?RCQ�RFC�JCLERF�MD�RFC�B?W�U?Q����F�QMKC�����KGJJGML�WC?PQ

?EM��/LJW�RFC�RGBCQ�CVCPR�QGELGDGA?LR�PCR?PBGLE�RMPOSCQ�ML�%?PRF��?LB�QM�GR�UGJJ�AMLRGLSC�RM�QNGL��?JRFMSEF�CTCP�KMPC�QJMUJW��DMP�K?LW�@GJJGMLQ�MD�WC?PQ�

7F?R�UC�F?TC�FCPC�GQ��GL�D?AR��?LMRFCP�AMLQCPT?RGML�J?U��)D�RFC�LCR�RMPOSC�GQ M8EB��RFCL�?LESJ?P�KMKCLRSK�GQ�AMLQR?LR�MP 6BAF8EI87��7C�A?L�QCC�RFGQ

PGEMPMSQJW�@W�AMLQGBCPGLE net τ = ΔLΔt

DMP�RFC�QGRS?RGML�GL�UFGAF�RFC�LCR�RMPOSC�GQ�XCPM��)L�RF?R�A?QC�

�������netτ = 0

GKNJWGLE�RF?R

�������ΔLΔt

= 0.

)D�RFC�AF?LEC�GL�?LESJ?P�KMKCLRSK ΔL GQ�XCPM��RFCL�RFC�?LESJ?P�KMKCLRSK�GQ�AMLQR?LR��RFSQ�

�������L = constant (net τ = 0)

MP

�������L = L′(netτ = 0).

4FCQC�CVNPCQQGMLQ�?PC�RFC @5K�C:�7CBG9FJ5H=CB�C:�5B;I@5F�ACA9BHIA��#MLQCPT?RGML�J?UQ�?PC�?Q�QA?PAC�?Q�RFCW�?PC�GKNMPR?LR�

!L�CV?KNJC�MD�AMLQCPT?RGML�MD�?LESJ?P�KMKCLRSK�GQ�QCCL�GL �=;IF9�� �����GL�UFGAF�?L�GAC�QI?RCP�GQ�CVCASRGLE�?�QNGL��4FC�LCR�RMPOSC�ML�FCP�GQ�TCPW

AJMQC�RM�XCPM��@CA?SQC�RFCPC�GQ�PCJ?RGTCJW�JGRRJC�DPGARGML�@CRUCCL�FCP�QI?RCQ�?LB�RFC�GAC�?LB�@CA?SQC�RFC�DPGARGML�GQ�CVCPRCB�TCPW�AJMQC�RM�RFC�NGTMR�NMGLR�

�"MRF F ?LB r ?PC�QK?JJ��?LB�QM τ GQ�LCEJGEG@JW�QK?JJ��#MLQCOSCLRJW��QFC�A?L�QNGL�DMP�OSGRC�QMKC�RGKC��3FC�A?L�BM�QMKCRFGLE�CJQC��RMM��3FC�A?L

GLAPC?QC�FCP�P?RC�MD�QNGL�@W�NSJJGLE�FCP�?PKQ�?LB�JCEQ�GL��7FW�BMCQ�NSJJGLE�FCP�?PKQ�?LB�JCEQ�GL�GLAPC?QC�FCP�P?RC�MD�QNGL��4FC�?LQUCP�GQ�RF?R�FCP

?LESJ?P�KMKCLRSK�GQ�AMLQR?LR��QM�RF?R

�������L = L′.

%VNPCQQGLE�RFGQ�COS?RGML�GL�RCPKQ�MD�RFC�KMKCLR�MD�GLCPRG?�

�������Iω = I′ω′,

UFCPC�RFC�NPGKCB�OS?LRGRGCQ�PCDCP�RM�AMLBGRGMLQ�?DRCP�QFC�F?Q�NSJJCB�GL�FCP�?PKQ�?LB�PCBSACB�FCP�KMKCLR�MD�GLCPRG?��"CA?SQC I′ GQ�QK?JJCP��RFC

?LESJ?P�TCJMAGRW ω′ KSQR�GLAPC?QC�RM�ICCN�RFC�?LESJ?P�KMKCLRSK�AMLQR?LR��4FC�AF?LEC�A?L�@C�BP?K?RGA��?Q�RFC�DMJJMUGLE�CV?KNJC�QFMUQ�

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

�=;IF9�� ��� �?�!L�GAC�QI?RCP�GQ�QNGLLGLE�ML�RFC�RGN�MD�FCP�QI?RC�UGRF�FCP�?PKQ�CVRCLBCB��(CP�?LESJ?P�KMKCLRSK�GQ�AMLQCPTCB�@CA?SQC�RFC�LCR�RMPOSC�ML�FCP�GQ�LCEJGEG@JW

QK?JJ��)L�RFC�LCVR�GK?EC��FCP�P?RC�MD�QNGL�GLAPC?QCQ�EPC?RJW�UFCL�QFC�NSJJQ�GL�FCP�?PKQ��BCAPC?QGLE�FCP�KMKCLR�MD�GLCPRG?��4FC�UMPI�QFC�BMCQ�RM�NSJJ�GL�FCP�?PKQ�PCQSJRQ�GL�?L

GLAPC?QC�GL�PMR?RGML?J�IGLCRGA�CLCPEW�

�L5AD@9�� ��� �5@7I@5H=B;�H<9��B;I@5F�&CA9BHIA�C:�5�,D=BB=B;�,?5H9F

3SNNMQC�?L�GAC�QI?RCP��QSAF�?Q�RFC�MLC�GL �=;IF9�� �����GQ�QNGLLGLE�?R�������PCT��Q�UGRF�FCP�?PKQ�CVRCLBCB��3FC�F?Q�?�KMKCLR�MD�GLCPRG?�MD

2.34 kg ⋅ m2UGRF�FCP�?PKQ�CVRCLBCB�?LB�MD 0.363 kg ⋅ m2

UGRF�FCP�?PKQ�AJMQC�RM�FCP�@MBW���4FCQC�KMKCLRQ�MD�GLCPRG?�?PC�@?QCB�ML

PC?QML?@JC�?QQSKNRGMLQ�?@MSR�?����� IE�QI?RCP���?�7F?R�GQ�FCP�?LESJ?P�TCJMAGRW�GL�PCTMJSRGMLQ�NCP�QCAMLB�?DRCP�QFC�NSJJQ�GL�FCP�?PKQ���@�7F?R

GQ�FCP�PMR?RGML?J�IGLCRGA�CLCPEW�@CDMPC�?LB�?DRCP�QFC�BMCQ�RFGQ�

,HF5H9;M

)L�RFC�DGPQR�N?PR�MD�RFC�NPM@JCK��UC�?PC�JMMIGLE�DMP�RFC�QI?RCPwQ�?LESJ?P�TCJMAGRW ω′ ?DRCP�QFC�F?Q�NSJJCB�GL�FCP�?PKQ��4M�DGLB�RFGQ�OS?LRGRW��UC�SQC

RFC�AMLQCPT?RGML�MD�?LESJ?P�KMKCLRSK�?LB�LMRC�RF?R�RFC�KMKCLRQ�MD�GLCPRG?�?LB�GLGRG?J�?LESJ?P�TCJMAGRW�?PC�EGTCL��4M�DGLB�RFC�GLGRG?J�?LB�DGL?J

IGLCRGA�CLCPEGCQ��UC�SQC�RFC�BCDGLGRGML�MD�PMR?RGML?J�IGLCRGA�CLCPEW�EGTCL�@W

�������KErot = 12Iω2.

,C@IH=CB�:CF��5�

"CA?SQC�RMPOSC�GQ�LCEJGEG@JC��?Q�BGQASQQCB�?@MTC��RFC�AMLQCPT?RGML�MD�?LESJ?P�KMKCLRSK�EGTCL�GL Iω = I′ω′ GQ�?NNJGA?@JC��4FSQ�

�������L = L′

MP

�������Iω = I′ω′

3MJTGLE�DMP ω′ ?LB�QS@QRGRSRGLE�ILMUL�T?JSCQ�GLRM�RFC�PCQSJRGLE�COS?RGML�EGTCQ

�������

ω′ = II′ω =

⎝⎜

2.34 kg ⋅ m2

0.363 kg ⋅ m2

⎠⎟(0.800 rev/s)

= 5.16 rev/s.

,C@IH=CB�:CF��6�

2MR?RGML?J�IGLCRGA�CLCPEW�GQ�EGTCL�@W

�������KErot = 12Iω2.

4FC�GLGRG?J�T?JSC�GQ�DMSLB�@W�QS@QRGRSRGLE�ILMUL�T?JSCQ�GLRM�RFC�COS?RGML�?LB�AMLTCPRGLE�RFC�?LESJ?P�TCJMAGRW�RM�P?B�Q�

�������KErot = (0.5)⎛⎝2.34 kg ⋅ m2⎞⎠⎛⎝(0.800 rev/s)(2π rad/rev)⎞

⎠2

= 29.6 J.

4FC�DGL?J�PMR?RGML?J�IGLCRGA�CLCPEW�GQ

�������KErot ′ = 12I′ω′2.

3S@QRGRSRGLE�ILMUL�T?JSCQ�GLRM�RFGQ�COS?RGML�EGTCQ

�������KErot′ = (0.5)⎛⎝0.363 kg ⋅ m2⎞⎠⎡⎣(5.16 rev/s)(2π rad/rev)⎤

⎦2

= 191 J.

�=G7IGG=CB

)L�@MRF�N?PRQ��RFCPC�GQ�?L�GKNPCQQGTC�GLAPC?QC��&GPQR��RFC�DGL?J�?LESJ?P�TCJMAGRW�GQ�J?PEC��?JRFMSEF�KMQR�UMPJB AJ?QQ�QI?RCPQ�A?L�?AFGCTC�QNGL�P?RCQ

?@MSR�RFGQ�EPC?R��3CAMLB��RFC�DGL?J�IGLCRGA�CLCPEW�GQ�KSAF�EPC?RCP�RF?L�RFC�GLGRG?J�IGLCRGA�CLCPEW��4FC�GLAPC?QC�GL�PMR?RGML?J�IGLCRGA�CLCPEW�AMKCQ

DPMK�UMPI�BMLC�@W�RFC�QI?RCP�GL�NSJJGLE�GL�FCP�?PKQ��4FGQ�UMPI�GQ�GLRCPL?J�UMPI�RF?R�BCNJCRCQ�QMKC�MD�RFC�QI?RCPwQ�DMMB�CLCPEW�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

4FCPC�?PC�QCTCP?J�MRFCP�CV?KNJCQ�MD�M@HCARQ�RF?R�GLAPC?QC�RFCGP�P?RC�MD�QNGL�@CA?SQC�QMKCRFGLE�PCBSACB�RFCGP�KMKCLR�MD�GLCPRG?��4MPL?BMCQ�?PC�MLC

CV?KNJC��3RMPK�QWQRCKQ�RF?R�APC?RC�RMPL?BMCQ�?PC�QJMUJW�PMR?RGLE��7FCL�RFC�P?BGSQ�MD�PMR?RGML�L?PPMUQ��CTCL�GL�?�JMA?J�PCEGML��?LESJ?P�TCJMAGRW

GLAPC?QCQ��QMKCRGKCQ�RM�RFC�DSPGMSQ�JCTCJ�MD�?�RMPL?BM��%?PRF�GQ�?LMRFCP�CV?KNJC��/SP�NJ?LCR�U?Q�@MPL�DPMK�?�FSEC�AJMSB�MD�E?Q�?LB�BSQR��RFC�PMR?RGML

MD�UFGAF�A?KC�DPMK�RSP@SJCLAC�GL�?L�CTCL�J?PECP�AJMSB��'P?TGR?RGML?J�DMPACQ�A?SQCB�RFC�AJMSB�RM�AMLRP?AR��?LB�RFC�PMR?RGML�P?RC�GLAPC?QCB�?Q�?�PCQSJR�

�3CC �=;IF9�� ����

�=;IF9�� ��� 4FC�3MJ?P�3WQRCK�AM?JCQACB�DPMK�?�AJMSB�MD�E?Q�?LB�BSQR�RF?R�U?Q�MPGEGL?JJW�PMR?RGLE��4FC�MP@GR?J�KMRGMLQ�?LB�QNGLQ�MD�RFC�NJ?LCRQ�?PC�GL�RFC�Q?KC�BGPCARGML�?Q

RFC�MPGEGL?J�QNGL�?LB�AMLQCPTC�RFC�?LESJ?P�KMKCLRSK�MD�RFC�N?PCLR�AJMSB�

)L�A?QC�MD�FSK?L�KMRGML��MLC�UMSJB�LMR�CVNCAR�?LESJ?P�KMKCLRSK�RM�@C�AMLQCPTCB�UFCL�?�@MBW�GLRCP?ARQ�UGRF�RFC�CLTGPMLKCLR�?Q�GRQ�DMMR�NSQFCQ

MDD�RFC�EPMSLB��!QRPML?SRQ�DJM?RGLE�GL�QN?AC�?@M?PB�RFC�)LRCPL?RGML?J�3N?AC�3R?RGML�F?TC�LM�?LESJ?P�KMKCLRSK�PCJ?RGTC�RM�RFC�GLQGBC�MD�RFC�QFGN�GD�RFCW

?PC�KMRGMLJCQQ��4FCGP�@MBGCQ�UGJJ�AMLRGLSC�RM�F?TC�RFGQ�XCPM�T?JSC�LM�K?RRCP�FMU�RFCW�RUGQR�?@MSR�?Q�JMLE�?Q�RFCW�BM�LMR�EGTC�RFCKQCJTCQ�?�NSQF�MDD

RFC�QGBC�MD�RFC�TCQQCJ�

�<97?�2CIF�.B89GH5B8=B;

)Q�?LESJ?P�KMKCLRSK�AMKNJCRCJW�?L?JMEMSQ�RM�JGLC?P�KMKCLRSK��7F?R��GD�?LW��?PC�RFCGP�BGDDCPCLACQ�

,C@IH=CB

9CQ��?LESJ?P�?LB�JGLC?P�KMKCLRSKQ�?PC�AMKNJCRCJW�?L?JMEMSQ��7FGJC�RFCW�?PC�CV?AR�?L?JMEQ�RFCW�F?TC�BGDDCPCLR�SLGRQ�?LB�?PC�LMR�BGPCARJW�GLRCP

AMLTCPRG@JC�JGIC�DMPKQ�MD�CLCPEW�?PC�

� �� �C@@=G=CBG�C:��LH9B898��C8=9G�=B�-KC��=A9BG=CBG

"MUJGLE�NGLQ�?PC�QCLR�DJWGLE�?LB�QNGLLGLE�UFCL�FGR�@W�?�@MUJGLE�@?JJu?LESJ?P�KMKCLRSK�?Q�UCJJ�?Q�JGLC?P�KMKCLRSK�?LB�CLCPEW�F?TC�@CCL

GKN?PRCB�RM�RFC�NGLQ���3CC �=;IF9�� �����-?LW�AMJJGQGMLQ�GLTMJTC�?LESJ?P�KMKCLRSK��#?PQ��DMP�CV?KNJC��K?W�QNGL�?LB�AMJJGBC�ML�GAC�MP�?�UCR

QSPD?AC��"?QC@?JJ�NGRAFCPQ�RFPMU�ASPTCQ�@W�NSRRGLE�QNGL�ML�RFC�@?QC@?JJ��!�RCLLGQ�NJ?WCP�A?L�NSR�?�JMR�MD�RMN�QNGL�ML�RFC�RCLLGQ�@?JJ�UFGAF�A?SQCQ�GR�RM

BGTC�BMUL�MLRM�RFC�AMSPR�MLAC�GR�APMQQCQ�RFC�LCR��7C�LMU�R?IC�?�@PGCD�JMMI�?R�UF?R�F?NNCLQ�UFCL�M@HCARQ�RF?R�A?L�PMR?RC�AMJJGBC�

#MLQGBCP�RFC�PCJ?RGTCJW�QGKNJC�AMJJGQGML�QFMUL�GL �=;IF9�� �����GL�UFGAF�?�BGQI�QRPGICQ�?LB�?BFCPCQ�RM�?L�GLGRG?JJW�KMRGMLJCQQ�QRGAI�L?GJCB�?R�MLC�CLB�RM

?�DPGARGMLJCQQ�QSPD?AC��!DRCP�RFC�AMJJGQGML��RFC�RUM�PMR?RC�?@MSR�RFC�L?GJ��4FCPC�GQ�?L�SL@?J?LACB�CVRCPL?J�DMPAC�ML�RFC�QWQRCK�?R�RFC�L?GJ��4FGQ�DMPAC

CVCPRQ�LM�RMPOSC�@CA?SQC�GRQ�JCTCP�?PK r GQ�XCPM��!LESJ?P�KMKCLRSK�GQ�RFCPCDMPC�AMLQCPTCB�GL�RFC�AMJJGQGML��+GLCRGA�CLCPEW�GQ�LMR�AMLQCPTCB�

@CA?SQC�RFC�AMJJGQGML�GQ�GLCJ?QRGA��)R�GQ�NMQQG@JC�RF?R�KMKCLRSK�GQ�LMR�AMLQCPTCB�CGRFCP�@CA?SQC�RFC�DMPAC�?R�RFC�L?GJ�K?W�F?TC�?�AMKNMLCLR�GL�RFC

BGPCARGML�MD�RFC�BGQIwQ�GLGRG?J�TCJMAGRW��,CR�SQ�CV?KGLC�?�A?QC�MD�PMR?RGML�GL�?�AMJJGQGML�GL �L5AD@9�� ����

�=;IF9�� ��� 4FC�@MUJGLE�@?JJ�A?SQCQ�RFC�NGLQ�RM�DJW��QMKC�MD�RFCK�QNGLLGLE�TGMJCLRJW���APCBGR��4GLMS�"?M��&JGAIP

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

�=;IF9�� ��� �?�!�BGQI�QJGBCQ�RMU?PB�?�KMRGMLJCQQ�QRGAI�ML�?�DPGARGMLJCQQ�QSPD?AC���@�4FC�BGQI�FGRQ�RFC�QRGAI�?R�MLC�CLB�?LB�?BFCPCQ�RM�GR��?LB�RFCW�PMR?RC�RMECRFCP��NGTMRGLE

?PMSLB�RFC�L?GJ��!LESJ?P�KMKCLRSK�GQ�AMLQCPTCB�DMP�RFGQ�GLCJ?QRGA�AMJJGQGML�@CA?SQC�RFC�QSPD?AC�GQ�DPGARGMLJCQQ�?LB�RFC�SL@?J?LACB�CVRCPL?J�DMPAC�?R�RFC�L?GJ�CVCPRQ�LM�RMPOSC�

�L5AD@9�� ��� +CH5H=CB�=B�5��C@@=G=CB

3SNNMQC�RFC�BGQI�GL �=;IF9�� ��� F?Q�?�K?QQ�MD������E�?LB�?L�GLGRG?J�TCJMAGRW�MD������K�Q�UFCL�GR�QRPGICQ�RFC�QRGAI�RF?R�GQ������K�JMLE�?LB������IE�

�?�7F?R�GQ�RFC�?LESJ?P�TCJMAGRW�MD�RFC�RUM�?DRCP�RFC�AMJJGQGML�

�@�7F?R�GQ�RFC�IGLCRGA�CLCPEW�@CDMPC�?LB�?DRCP�RFC�AMJJGQGML�

�A�7F?R�GQ�RFC�RMR?J�JGLC?P�KMKCLRSK�@CDMPC�?LB�?DRCP�RFC�AMJJGQGML�

,HF5H9;M�:CF��5�

7C�A?L�?LQUCP�RFC�DGPQR�OSCQRGML�SQGLE�AMLQCPT?RGML�MD�?LESJ?P�KMKCLRSK�?Q�LMRCB��"CA?SQC�?LESJ?P�KMKCLRSK�GQ Iω ��UC�A?L�QMJTC�DMP

?LESJ?P�TCJMAGRW�

,C@IH=CB�:CF��5�

#MLQCPT?RGML�MD�?LESJ?P�KMKCLRSK�QR?RCQ

�������L = L′,

UFCPC�NPGKCB�OS?LRGRGCQ�QR?LB�DMP�AMLBGRGMLQ�?DRCP�RFC�AMJJGQGML�?LB�@MRF�KMKCLR?�?PC�A?JASJ?RCB�PCJ?RGTC�RM�RFC�NGTMR�NMGLR��4FC�GLGRG?J�?LESJ?P

KMKCLRSK�MD�RFC�QWQRCK�MD�QRGAI BGQI�GQ�RF?R�MD�RFC�BGQI�HSQR�@CDMPC�GR�QRPGICQ�RFC�QRGAI��4F?R�GQ�

�������L = Iω,

UFCPC I GQ�RFC�KMKCLR�MD�GLCPRG?�MD�RFC�BGQI�?LB ω GQ�GRQ�?LESJ?P�TCJMAGRW�?PMSLB�RFC�NGTMR�NMGLR��.MU� I = mr2�R?IGLE�RFC�BGQI�RM�@C

?NNPMVGK?RCJW�?�NMGLR�K?QQ�?LB ω = v / r ��QM�RF?R

�������L = mr2vr = mvr.

!DRCP�RFC�AMJJGQGML�

�������L′ = I ω .

)R�GQ ω′ RF?R�UC�UGQF�RM�DGLB��#MLQCPT?RGML�MD�?LESJ?P�KMKCLRSK�EGTCQ

�������I′ω′ = mvr.

2C?PP?LEGLE�RFC�COS?RGML�WGCJBQ

�������ω′ = mvrI

,

UFCPC I′ GQ�RFC�KMKCLR�MD�GLCPRG?�MD�RFC�QRGAI�?LB�BGQI�QRSAI�RMECRFCP��UFGAF�GQ�RFC�QSK�MD�RFCGP�GLBGTGBS?J�KMKCLRQ�MD�GLCPRG?�?@MSR�RFC�L?GJ�

�=;IF9�� ��� EGTCQ�RFC�DMPKSJ?�DMP�?�PMB�PMR?RGLE�?PMSLB�MLC�CLB�RM�@C I = Mr2 / 3 ��4FSQ�

�������I = mr2 + Mr2

3=

⎛⎝m + M

3⎞⎠r

2.

%LRCPGLE�ILMUL�T?JSCQ�GL�RFGQ�COS?RGML�WGCJBQ�

�������I′ = ⎛⎝0.0500 kg + 0.667 kg⎞

⎠(1.20 m)2 = 1.032 kg ⋅ m2.

4FC�T?JSC�MD I′ GQ�LMU�CLRCPCB�GLRM�RFC�CVNPCQQGML�DMP ω′ ��UFGAF�WGCJBQ

�������ω′ = mvr

I′=

⎛⎝0.0500 kg⎞

⎠(30.0 m/s)(1.20 m)

1.032 kg ⋅ m2

= 1.744 rad/s ≈ 1.74 rad/s.

,HF5H9;M�:CF��6�

4FC�IGLCRGA�CLCPEW�@CDMPC�RFC�AMJJGQGML�GQ�RFC�GLAMKGLE�BGQIwQ�RP?LQJ?RGML?J�IGLCRGA�CLCPEW��?LB�?DRCP�RFC�AMJJGQGML��GR�GQ�RFC�PMR?RGML?J�IGLCRGA�CLCPEW

MD�RFC�RUM�QRSAI�RMECRFCP�

,C@IH=CB�:CF��6�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

&GPQR��UC�A?JASJ?RC�RFC�RP?LQJ?RGML?J�IGLCRGA�CLCPEW�@W�CLRCPGLE�EGTCL�T?JSCQ�DMP�RFC�K?QQ�?LB�QNCCB�MD�RFC�GLAMKGLE�BGQI�

�������KE = 12mv2 = (0.500)⎛⎝0.0500 kg⎞

⎠(30.0 m/s)2 = 22.5 J

!DRCP�RFC�AMJJGQGML��RFC�PMR?RGML?J�IGLCRGA�CLCPEW�A?L�@C�DMSLB�@CA?SQC�UC�LMU�ILMU�RFC�DGL?J�?LESJ?P�TCJMAGRW�?LB�RFC�DGL?J�KMKCLR�MD�GLCPRG?�

4FSQ��CLRCPGLE�RFC�T?JSCQ�GLRM�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�COS?RGML�EGTCQ

�������KE′ = 1

2I′ω′2 = (0.5)⎛⎝1.032 kg ⋅ m2⎞

⎠⎛⎝1.744rad

s⎞⎠

2

= 1.57 J.

,HF5H9;M�:CF��7�

4FC�JGLC?P�KMKCLRSK�@CDMPC�RFC�AMJJGQGML�GQ�RF?R�MD�RFC�BGQI��!DRCP�RFC�AMJJGQGML��GR�GQ�RFC�QSK�MD�RFC�BGQIwQ�KMKCLRSK�?LB�RF?R�MD�RFC�ACLRCP�MD

K?QQ�MD�RFC�QRGAI�

,C@IH=CB�C:��7�

"CDMPC�RFC�AMJJGQGML��RFCL��JGLC?P�KMKCLRSK�GQ

�������p = mv = ⎛⎝0.0500 kg⎞

⎠(30.0 m/s) = 1.50 kg ⋅ m/s.

!DRCP�RFC�AMJJGQGML��RFC�BGQI�?LB�RFC�QRGAIwQ�ACLRCP�MD�K?QQ�KMTC�GL�RFC�Q?KC�BGPCARGML��4FC�RMR?J�JGLC?P�KMKCLRSK�GQ�RF?R�MD�RFC�BGQI�KMTGLE�?R�?

LCU�TCJMAGRW v′ = rω′ NJSQ�RF?R�MD�RFC�QRGAIwQ�ACLRCP�MD�K?QQ�

UFGAF�KMTCQ�?R�F?JD�RFGQ�QNCCB�@CA?SQC vCM =⎛⎝r2⎞⎠ω′ = v′

2��4FSQ�

�������p′ = mv′ + MvCM = mv′ + Mv′

2.

'?RFCPGLE�QGKGJ?P�RCPKQ�GL�RFC�COS?RGML�WGCJBQ�

�������p′ =

⎛⎝m + M

2⎞⎠v′

QM�RF?R

�������p′ =

⎛⎝m + M

2⎞⎠rω′.

3S@QRGRSRGLE�ILMUL�T?JSCQ�GLRM�RFC�COS?RGML�

�������p′ = ⎛⎝1.050 kg⎞

⎠(1.20 m)(1.744 rad/s) = 2.20 kg ⋅ m/s.

�=G7IGG=CB

&GPQR�LMRC�RF?R�RFC�IGLCRGA�CLCPEW�GQ�JCQQ�?DRCP�RFC�AMJJGQGML��?Q�NPCBGARCB��@CA?SQC�RFC�AMJJGQGML�GQ�GLCJ?QRGA��-MPC�QSPNPGQGLE�GQ�RF?R�RFC�KMKCLRSK

?DRCP�RFC�AMJJGQGML�GQ�?ARS?JJW�EPC?RCP�RF?L�@CDMPC�RFC�AMJJGQGML��4FGQ�PCQSJR�A?L�@C�SLBCPQRMMB�GD�WMS�AMLQGBCP�FMU�RFC�L?GJ�?DDCARQ�RFC�QRGAI�?LB�TGAC

TCPQ?��!NN?PCLRJW��RFC�QRGAI�NSQFCQ�@?AIU?PB�ML�RFC�L?GJ�UFCL�DGPQR�QRPSAI�@W�RFC�BGQI��4FC�L?GJwQ�PC?ARGML��AMLQGQRCLR�UGRF�.CURMLwQ�RFGPB�J?U�GQ�RM

NSQF�DMPU?PB�ML�RFC�QRGAI��GKN?PRGLE�KMKCLRSK�RM�GR�GL�RFC�Q?KC�BGPCARGML�GL�UFGAF�RFC�BGQI�U?Q�GLGRG?JJW�KMTGLE��RFCPC@W�GLAPC?QGLE�RFC

KMKCLRSK�MD�RFC�QWQRCK�

4FC�?@MTC�CV?KNJC�F?Q�MRFCP�GKNJGA?RGMLQ��&MP�CV?KNJC��UF?R�UMSJB�F?NNCL�GD�RFC�BGQI�FGR�TCPW�AJMQC�RM�RFC�L?GJ��/@TGMSQJW��?�DMPAC�UMSJB�@C�CVCPRCB

ML�RFC�L?GJ�GL�RFC�DMPU?PB�BGPCARGML��3M��UFCL�RFC�QRGAI�GQ�QRPSAI�?R�RFC�CLB�D?PRFCQR�DPMK�RFC�L?GJ��?�@?AIU?PB�DMPAC�GQ�CVCPRCB�ML�RFC�L?GJ��?LB�UFCL�GR�GQ

FGR�?R�RFC�CLB�LC?PCQR�RFC�L?GJ��?�DMPU?PB�DMPAC�GQ�CVCPRCB�ML�RFC�L?GJ��4FSQ��QRPGIGLE�GR�?R�?�ACPR?GL�NMGLR�GL�@CRUCCL�NPMBSACQ�LM�DMPAC�ML�RFC�L?GJ��4FGQ

GLRCPKCBG?RC�NMGLR�GQ�ILMUL�?Q�RFC C8E6HFF<BA�CB<AG�

!L�?L?JMEMSQ�QGRS?RGML�MAASPQ�GL�RCLLGQ�?Q�QCCL�GL �=;IF9�� �����)D�WMS�FGR�?�@?JJ�UGRF�RFC�CLB�MD�WMSP�P?AOSCR��RFC�F?LBJC�GQ�NSJJCB�?U?W�DPMK�WMSP

F?LB��)D�WMS�FGR�?�@?JJ�KSAF�D?PRFCP�BMUL��DMP�CV?KNJC��ML�RFC�QF?DR�MD�RFC�P?AOSCR��RFC�F?LBJC�GQ�NSQFCB�GLRM�WMSP�N?JK��!LB�GD�WMS�FGR�RFC�@?JJ�?R�RFC

P?AOSCRwQ�NCPASQQGML�NMGLR��UF?R�QMKC�NCMNJC�A?JJ�RFC�xQUCCR�QNMRy��RFCL�JGRRJC�MP AB DMPAC�GQ�CVCPRCB�ML�WMSP�F?LB��?LB�RFCPC�GQ�JCQQ�TG@P?RGML�

PCBSAGLE�AF?LACQ�MD�?�RCLLGQ�CJ@MU��4FC�Q?KC�CDDCAR�MAASPQ�DMP�?�@?QC@?JJ�@?R�

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

�=;IF9�� ��� !�BGQI�FGRRGLE�?�QRGAI�GQ�AMKN?PCB�RM�?�RCLLGQ�@?JJ�@CGLE�FGR�@W�?�P?AOSCR���?�7FCL�RFC�@?JJ�QRPGICQ�RFC�P?AOSCR�LC?P�RFC�CLB��?�@?AIU?PB�DMPAC�GQ�CVCPRCB�ML�RFC

F?LB���@�7FCL�RFC�P?AOSCR�GQ�QRPSAI�KSAF�D?PRFCP�BMUL��?�DMPU?PB�DMPAC�GQ�CVCPRCB�ML�RFC�F?LB���A�7FCL�RFC�P?AOSCR�GQ�QRPSAI�?R�RFC�NCPASQQGML�NMGLR��LM�DMPAC�GQ�BCJGTCPCB�RM

RFC�F?LB�

�<97?�2CIF�.B89FGH5B8=B;

)Q�PMR?RGML?J�IGLCRGA�CLCPEW�?�TCARMP��*SQRGDW�WMSP�?LQUCP�

,C@IH=CB

.M��CLCPEW�GQ�?JU?WQ�QA?J?P�UFCRFCP�KMRGML�GQ�GLTMJTCB�MP�LMR��.M�DMPK�MD�CLCPEW�F?Q�?�BGPCARGML�GL�QN?AC�?LB�WMS�A?L�QCC�RF?R�PMR?RGML?J�IGLCRGA

CLCPEW�BMCQ�LMR�BCNCLB�ML�RFC�BGPCARGML�MD�KMRGML�HSQR�?Q�JGLC?P�IGLCRGA�CLCPEW�GQ�GLBCNCLBCLR�MD�RFC�BGPCARGML�MD�KMRGML�

� �� MFCG7CD=7��::97HG��/97HCF��GD97HG�C:��B;I@5F�&CA9BHIA

!LESJ?P�KMKCLRSK�GQ�?�TCARMP�?LB��RFCPCDMPC� ;4F�7<E86G<BA�4F�J8??�4F�@4:A<GH78��4MPOSC�?DDCARQ�@MRF�RFC�BGPCARGML�?LB�RFC�K?ELGRSBC�MD�?LESJ?P

KMKCLRSK��7F?R�GQ�RFC�BGPCARGML�MD�RFC�?LESJ?P�KMKCLRSK�MD�?�PMR?RGLE�M@HCAR�JGIC�RFC�BGQI�GL �=;IF9�� �����4FC�DGESPC�QFMUQ�RFC F=;<H<5B8�FI@9

SQCB�RM�DGLB�RFC�BGPCARGML�MD�@MRF�?LESJ?P�KMKCLRSK�?LB�?LESJ?P�TCJMAGRW��"MRF L ?LB ω ?PC�TCARMPQuC?AF�F?Q�BGPCARGML�?LB�K?ELGRSBC��"MRF�A?L

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

@C�PCNPCQCLRCB�@W�?PPMUQ��4FC�PGEFR F?LB�PSJC�BCDGLCQ�@MRF�RM�@C�NCPNCLBGASJ?P�RM�RFC�NJ?LC�MD�PMR?RGML�GL�RFC�BGPCARGML�QFMUL��"CA?SQC�?LESJ?P

KMKCLRSK�GQ�PCJ?RCB�RM�?LESJ?P�TCJMAGRW�@W L = Iω ��RFC�BGPCARGML�MD L GQ�RFC�Q?KC�?Q�RFC�BGPCARGML�MD ω ��.MRGAC�GL�RFC�DGESPC�RF?R�@MRF�NMGLR�?JMLE

RFC�?VGQ�MD�PMR?RGML�

�=;IF9�� ��� &GESPC��?�QFMUQ�?�BGQI�GQ�PMR?RGLE�AMSLRCPAJMAIUGQC�UFCL�TGCUCB�DPMK�?@MTC��&GESPC��@�QFMUQ�RFC�PGEFR F?LB�PSJC��4FC�BGPCARGML�MD�?LESJ?P�TCJMAGRW ω QGXC�?LB

?LESJ?P�KMKCLRSK L ?PC�BCDGLCB�RM�@C�RFC�BGPCARGML�GL�UFGAF�RFC�RFSK@�MD�WMSP�PGEFR�F?LB�NMGLRQ�UFCL�WMS�ASPJ�WMSP�DGLECPQ�GL�RFC�BGPCARGML�MD�RFC�BGQIwQ�PMR?RGML�?Q�QFMUL�

.MU��PCA?JJ�RF?R�RMPOSC�AF?LECQ�?LESJ?P�KMKCLRSK�?Q�CVNPCQQCB�@W

�������net τ = ΔLΔt

.

4FGQ�COS?RGML�KC?LQ�RF?R�RFC�BGPCARGML�MD ΔL GQ�RFC�Q?KC�?Q�RFC�BGPCARGML�MD�RFC�RMPOSC τ RF?R�APC?RCQ�GR��4FGQ�PCQSJR�GQ�GJJSQRP?RCB�GL �=;IF9�� ����

UFGAF�QFMUQ�RFC�BGPCARGML�MD�RMPOSC�?LB�RFC�?LESJ?P�KMKCLRSK�GR�APC?RCQ�

,CR�SQ�LMU�AMLQGBCP�?�@GAWAJC�UFCCJ�UGRF�?�AMSNJC�MD�F?LBJCQ�?RR?AFCB�RM�GR��?Q�QFMUL�GL �=;IF9�� �� ���4FGQ�BCTGAC�GQ�NMNSJ?P�GL�BCKMLQRP?RGMLQ

?KMLE�NFWQGAGQRQ��@CA?SQC�GR�BMCQ�SLCVNCARCB�RFGLEQ��7GRF�RFC�UFCCJ�PMR?RGLE�?Q�QFMUL��GRQ�?LESJ?P�KMKCLRSK�GQ�RM�RFC�UMK?L�Q�JCDR��3SNNMQC�RFC

NCPQML�FMJBGLE�RFC�UFCCJ�RPGCQ�RM�PMR?RC�GR�?Q�GL�RFC�DGESPC��(CP�L?RSP?J�CVNCAR?RGML�GQ�RF?R�RFC�UFCCJ�UGJJ�PMR?RC�GL�RFC�BGPCARGML�QFC�NSQFCQ�GRu@SR�UF?R

F?NNCLQ�GQ�OSGRC�BGDDCPCLR��4FC�DMPACQ�CVCPRCB�APC?RC�?�RMPOSC�RF?R�GQ�FMPGXMLR?J�RMU?PB�RFC�NCPQML��?Q�QFMUL�GL �=;IF9�� �� �?��4FGQ�RMPOSC�APC?RCQ�?

AF?LEC�GL�?LESJ?P�KMKCLRSK L GL�RFC�Q?KC�BGPCARGML��NCPNCLBGASJ?P�RM�RFC�MPGEGL?J�?LESJ?P�KMKCLRSK L ��RFSQ�AF?LEGLE�RFC�BGPCARGML�MD L @SR

LMR�RFC�K?ELGRSBC�MD L � �=;IF9�� �� QFMUQ�FMU ΔL ?LB L ?BB��EGTGLE�?�LCU�?LESJ?P�KMKCLRSK�UGRF�BGPCARGML�RF?R�GQ�GLAJGLCB�KMPC�RMU?PB�RFC

NCPQML�RF?L�@CDMPC��4FC�?VGQ�MD�RFC�UFCCJ�F?Q�RFSQ�KMTCB C8EC8A7<6H?4E�GB�G;8�9BE68F�8K8EG87�BA�<G��GLQRC?B�MD�GL�RFC�CVNCARCB�BGPCARGML�

�=;IF9�� ��� )L�DGESPC��?��RFC�RMPOSC�GQ�NCPNCLBGASJ?P�RM�RFC�NJ?LC�DMPKCB�@W r ?LB F ?LB�GQ�RFC�BGPCARGML�WMSP�PGEFR�RFSK@�UMSJB�NMGLR�RM�GD�WMS�ASPJCB�WMSP�DGLECPQ�GL�RFC

BGPCARGML�MD F ��&GESPC��@�QFMUQ�RF?R�RFC�BGPCARGML�MD�RFC�RMPOSC�GQ�RFC�Q?KC�?Q�RF?R�MD�RFC�?LESJ?P�KMKCLRSK�GR�NPMBSACQ�

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

5B;I@5F�5779@9F5H=CB�

�=;IF9�� �� )L�DGESPC��?��?�NCPQML�FMJBGLE�RFC�QNGLLGLE�@GIC�UFCCJ�JGDRQ�GR�UGRF�FCP�PGEFR�F?LB�?LB�NSQFCQ�BMUL�UGRF�FCP�JCDR�F?LB�GL�?L�?RRCKNR�RM�PMR?RC�RFC�UFCCJ��4FGQ�?ARGML

APC?RCQ�?�RMPOSC�BGPCARJW�RMU?PB�FCP��4FGQ�RMPOSC�A?SQCQ�?�AF?LEC�GL�?LESJ?P�KMKCLRSK ΔL GL�CV?ARJW�RFC�Q?KC�BGPCARGML��&GESPC��@�QFMUQ�?�TCARMP�BG?EP?K�BCNGARGLE�FMU

ΔL ?LB L ?BB��NPMBSAGLE�?�LCU�?LESJ?P�KMKCLRSK�NMGLRGLE�KMPC�RMU?PB�RFC�NCPQML��4FC�UFCCJ�KMTCQ�RMU?PB�RFC�NCPQML��NCPNCLBGASJ?P�RM�RFC�DMPACQ�QFC�CVCPRQ�ML�GR�

4FGQ�Q?KC�JMEGA�CVNJ?GLQ�RFC�@CF?TGMP�MD�EWPMQAMNCQ� �=;IF9�� ��� QFMUQ�RFC�RUM�DMPACQ�?ARGLE�ML�?�QNGLLGLE�EWPMQAMNC��4FC�RMPOSC�NPMBSACB�GQ

NCPNCLBGASJ?P�RM�RFC�?LESJ?P�KMKCLRSK��RFSQ�RFC�BGPCARGML�MD�RFC�RMPOSC�GQ�AF?LECB��@SR�LMR�GRQ�K?ELGRSBC��4FC�EWPMQAMNC CE868FF8F ?PMSLB�?

TCPRGA?J�?VGQ��QGLAC�RFC�RMPOSC�GQ�?JU?WQ�FMPGXMLR?J�?LB�NCPNCLBGASJ?P�RM L ��)D�RFC�EWPMQAMNC�GQ ABG QNGLLGLE��GR�?AOSGPCQ�?LESJ?P�KMKCLRSK�GL�RFC

BGPCARGML�MD�RFC�RMPOSC�� L = ΔL ��?LB�GR�PMR?RCQ�?PMSLB�?�FMPGXMLR?J�?VGQ��D?JJGLE�MTCP�HSQR�?Q�UC�UMSJB�CVNCAR�

%?PRF�GRQCJD�?ARQ�JGIC�?�EGE?LRGA�EWPMQAMNC��)RQ�?LESJ?P�KMKCLRSK�GQ�?JMLE�GRQ�?VGQ�?LB�NMGLRQ�?R�0MJ?PGQ��RFC�.MPRF�3R?P��"SR�%?PRF�GQ�QJMUJW�NPCACQQGLE

�MLAC�GL�?@MSR��������WC?PQ�BSC�RM�RFC�RMPOSC�MD�RFC�3SL�?LB�RFC�-MML�ML�GRQ�LMLQNFCPGA?J�QF?NC�

�=;IF9�� ��� !Q�QCCL�GL�DGESPC��?��RFC�DMPACQ�ML�?�QNGLLGLE�EWPMQAMNC�?PC�GRQ�UCGEFR�?LB�RFC�QSNNMPRGLE�DMPAC�DPMK�RFC�QR?LB��4FCQC�DMPACQ�APC?RC�?�FMPGXMLR?J�RMPOSC�ML�RFC

EWPMQAMNC��UFGAF�APC?RC�?�AF?LEC�GL�?LESJ?P�KMKCLRSK ΔL RF?R�GQ�?JQM�FMPGXMLR?J��)L�DGESPC��@� ΔL ?LB L ?BB�RM�NPMBSAC�?�LCU�?LESJ?P�KMKCLRSK�UGRF�RFC�Q?KC

K?ELGRSBC��@SR�BGDDCPCLR�BGPCARGML��QM�RF?R�RFC�EWPMQAMNC�NPCACQQCQ�GL�RFC�BGPCARGML�QFMUL�GLQRC?B�MD�D?JJGLE�MTCP�

�<97?�2CIF�.B89FGH5B8=B;

2MR?RGML?J�IGLCRGA�CLCPEW�GQ�?QQMAG?RCB�UGRF�?LESJ?P�KMKCLRSK��$MCQ�RF?R�KC?L�RF?R�PMR?RGML?J�IGLCRGA�CLCPEW�GQ�?�TCARMP�

,C@IH=CB

.M��CLCPEW�GQ�?JU?WQ�?�QA?J?P�UFCRFCP�KMRGML�GQ�GLTMJTCB�MP�LMR��.M�DMPK�MD�CLCPEW�F?Q�?�BGPCARGML�GL�QN?AC�?LB�WMS�A?L�QCC�RF?R�PMR?RGML?J

IGLCRGA�CLCPEW�BMCQ�LMR�BCNCLB�ML�RFC�BGPCARGML�MD�KMRGML�HSQR�?Q�JGLC?P�IGLCRGA�CLCPEW�GQ�GLBCNCLBCLR�MD�RFC�BGPCARGML�MD�KMRGML�

@CGG5FM

RFC�P?RC�MD�AF?LEC�MD�?LESJ?P�TCJMAGRW�UGRF�RGKC

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

5B;I@5F�ACA9BHIA�

7<5B;9�=B�5B;I@5F�J9@C7=HM�

?=B9A5H=7G�C:�FCH5H=CB5@�ACH=CB�

@5K�C:�7CBG9FJ5H=CB�C:�5B;I@5F�ACA9BHIA�

ACA9BH�C:�=B9FH=5�

F=;<H<5B8�FI@9�

FCH5H=CB5@�=B9FH=5�

FCH5H=CB5@�?=B9H=7�9B9F;M�

H5B;9BH=5@�5779@9F5H=CB�

HCFEI9�

KCF?9B9F;M�H<9CF9A�

RFC�NPMBSAR�MD�KMKCLR�MD�GLCPRG?�?LB�?LESJ?P�TCJMAGRW

RFC�BGDDCPCLAC�@CRUCCL�DGL?J�?LB�GLGRG?J�T?JSCQ�MD�?LESJ?P�TCJMAGRW

BCQAPG@CQ�RFC�PCJ?RGMLQFGNQ�?KMLE�PMR?RGML�?LEJC��?LESJ?P�TCJMAGRW��?LESJ?P�?AACJCP?RGML��?LB�RGKC

?LESJ?P�KMKCLRSK�GQ�AMLQCPTCB��G�C���RFC�GLGRG?J�?LESJ?P�KMKCLRSK�GQ�COS?J�RM�RFC�DGL?J�?LESJ?P

KMKCLRSK�UFCL�LM�CVRCPL?J�RMPOSC�GQ�?NNJGCB�RM�RFC�QWQRCK

K?QQ�RGKCQ�RFC�QOS?PC�MD�NCPNCLBGASJ?P�BGQR?LAC�DPMK�RFC�PMR?RGML�?VGQ��DMP�?�NMGLR�K?QQ��GR�GQ I = mr2?LB��@CA?SQC�?LW

M@HCAR�A?L�@C�@SGJR�SN�DPMK�?�AMJJCARGML�MD�NMGLR�K?QQCQ��RFGQ�PCJ?RGMLQFGN�GQ�RFC�@?QGQ�DMP�?JJ�MRFCP�KMKCLRQ�MD�GLCPRG?

BGPCARGML�MD�?LESJ?P�TCJMAGRW�r�?LB�?LESJ?P�KMKCLRSK�,�GL�UFGAF�RFC�RFSK@�MD�WMSP�PGEFR�F?LB�NMGLRQ�UFCL�WMS�ASPJ�WMSP�DGLECPQ

GL�RFC�BGPCARGML�MD�RFC�BGQIwQ�PMR?RGML

PCQGQR?LAC�RM�AF?LEC�MD�PMR?RGML��4FC�KMPC�PMR?RGML?J�GLCPRG?�?L�M@HCAR�F?Q��RFC�F?PBCP�GR�GQ�RM�PMR?RC

RFC�IGLCRGA�CLCPEW�BSC�RM�RFC�PMR?RGML�MD�?L�M@HCAR��4FGQ�GQ�N?PR�MD�GRQ�RMR?J�IGLCRGA�CLCPEW

RFC�?AACJCP?RGML�GL�?�BGPCARGML�R?LECLR�RM�RFC�AGPAJC�?R�RFC�NMGLR�MD�GLRCPCQR�GL�AGPASJ?P�KMRGML

RFC�RSPLGLE�CDDCARGTCLCQQ�MD�?�DMPAC

GD�MLC�MP�KMPC�CVRCPL?J�DMPACQ�?AR�SNML�?�PGEGB�M@HCAR��A?SQGLE�GRQ�IGLCRGA�CLCPEW�RM�AF?LEC�DPMK KE1 RM KE2 ��RFCL�RFC

UMPI W BMLC�@W�RFC�LCR�DMPAC�GQ�COS?J�RM�RFC�AF?LEC�GL�IGLCRGA�CLCPEW

,97H=CB�,IAA5FM

� �� �B;I@5F��779@9F5H=CB

z 5LGDMPK�AGPASJ?P�KMRGML�GQ�RFC�KMRGML�UGRF�?�AMLQR?LR�?LESJ?P�TCJMAGRW ω = ΔθΔt

z )L�LML SLGDMPK�AGPASJ?P�KMRGML��RFC�TCJMAGRW�AF?LECQ�UGRF�RGKC�?LB�RFC�P?RC�MD�AF?LEC�MD�?LESJ?P�TCJMAGRW��G�C��?LESJ?P�?AACJCP?RGML�GQ α = ΔωΔt

z ,GLC?P�MP�R?LECLRG?J�?AACJCP?RGML�PCDCPQ�RM�AF?LECQ�GL�RFC�K?ELGRSBC�MD�TCJMAGRW�@SR�LMR�GRQ�BGPCARGML��EGTCL�?Q at = ΔvΔt

z &MP�AGPASJ?P�KMRGML��LMRC�RF?R v = rω ��QM�RF?R

at =Δ(rω)

Δt.

z 4FC�P?BGSQ�P�GQ�AMLQR?LR�DMP�AGPASJ?P�KMRGML��?LB�QM Δ(rω) = rΔω ��4FSQ�

at = rΔωΔt

.

z "W�BCDGLGRGML� Δω / Δt = α ��4FSQ�

at = rα

MP

α =atr .

� �� $=B9A5H=7G�C:�+CH5H=CB5@�&CH=CBz +GLCK?RGAQ�GQ�RFC�BCQAPGNRGML�MD�KMRGML�

z 4FC�IGLCK?RGAQ�MD�PMR?RGML?J�KMRGML�BCQAPG@CQ�RFC�PCJ?RGMLQFGNQ�?KMLE�PMR?RGML�?LEJC��?LESJ?P�TCJMAGRW��?LESJ?P�?AACJCP?RGML��?LB�RGKC�

z 3R?PRGLE�UGRF�RFC�DMSP�IGLCK?RGA�COS?RGMLQ�UC�BCTCJMNCB�GL�RFC (B9�=A9BG=CB5@�$=B9A5H=7G��UC�A?L�BCPGTC�RFC�DMSP�PMR?RGML?J�IGLCK?RGA

COS?RGMLQ��NPCQCLRCB�RMECRFCP�UGRF�RFCGP�RP?LQJ?RGML?J�AMSLRCPN?PRQ�QCCL�GL -56@9�� ���

z )L�RFCQC�COS?RGMLQ��RFC�QS@QAPGNR���BCLMRCQ�GLGRG?J�T?JSCQ�� x0 ?LB t0 ?PC�GLGRG?J�T?JSCQ��?LB�RFC�?TCP?EC�?LESJ?P�TCJMAGRW ω- ?LB�?TCP?EC

TCJMAGRW v- ?PC�BCDGLCB�?Q�DMJJMUQ�

ω̄ =ω0 + ω

2 and v̄ =

v0 + v

2.

� �� �MB5A=7G�C:�+CH5H=CB5@�&CH=CB��+CH5H=CB5@�"B9FH=5z 4FC�D?PRFCP�RFC�DMPAC�GQ�?NNJGCB�DPMK�RFC�NGTMR��RFC�EPC?RCP�GQ�RFC�?LESJ?P�?AACJCP?RGML��?LESJ?P�?AACJCP?RGML�GQ�GLTCPQCJW�NPMNMPRGML?J�RM�K?QQ�

z )D�UC�CVCPR�?�DMPAC F ML�?�NMGLR�K?QQ m RF?R�GQ�?R�?�BGQR?LAC r DPMK�?�NGTMR�NMGLR�?LB�@CA?SQC�RFC�DMPAC�GQ�NCPNCLBGASJ?P�RM r ��?L

?AACJCP?RGML a = F/m GQ�M@R?GLCB�GL�RFC�BGPCARGML�MD F ��7C�A?L�PC?PP?LEC�RFGQ�COS?RGML�QSAF�RF?R

F = ma,

?LB�RFCL�JMMI�DMP�U?WQ�RM�PCJ?RC�RFGQ�CVNPCQQGML�RM�CVNPCQQGMLQ�DMP�PMR?RGML?J�OS?LRGRGCQ��7C�LMRC�RF?R a = rα ��?LB�UC�QS@QRGRSRC�RFGQ�CVNPCQQGML

GLRM F=ma ��WGCJBGLE

F=mrα

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

z 4MPOSC�GQ�RFC�RSPLGLE�CDDCARGTCLCQQ�MD�?�DMPAC��)L�RFGQ�A?QC��@CA?SQC F GQ�NCPNCLBGASJ?P�RM r ��RMPOSC�GQ�QGKNJW τ = rF ��)D�UC�KSJRGNJW�@MRF�QGBCQ

MD�RFC�COS?RGML�?@MTC�@W r ��UC�ECR�RMPOSC�ML�RFC�JCDR F?LB�QGBC��4F?R�GQ�

rF = mr2α

MP

τ = mr2α.z 4FC�KMKCLR�MD�GLCPRG? I MD�?L�M@HCAR�GQ�RFC�QSK�MD MR2

DMP�?JJ�RFC�NMGLR�K?QQCQ�MD�UFGAF�GR�GQ�AMKNMQCB��4F?R�GQ�

I = ∑ mr2.z 4FC�ECLCP?J�PCJ?RGMLQFGN�?KMLE�RMPOSC��KMKCLR�MD�GLCPRG?��?LB�?LESJ?P�?AACJCP?RGML�GQ

τ = Iα

MP

α = net τI

� �� +CH5H=CB5@�$=B9H=7��B9F;M��0CF?�5B8��B9F;M�+9J=G=H98z 4FC�PMR?RGML?J�IGLCRGA�CLCPEW KErot DMP�?L�M@HCAR�UGRF�?�KMKCLR�MD�GLCPRG? I ?LB�?L�?LESJ?P�TCJMAGRW ω GQ�EGTCL�@W

KErot = 12Iω2.

z (CJGAMNRCPQ�QRMPC�J?PEC�?KMSLRQ�MD�PMR?RGML?J�IGLCRGA�CLCPEW�GL�RFCGP�@J?BCQ��4FGQ�CLCPEW�KSQR�@C�NSR�GLRM�RFC�@J?BCQ�@CDMPC�R?ICMDD�?LB

K?GLR?GLCB�SLRGJ�RFC�CLB�MD�RFC�DJGEFR��4FC�CLEGLCQ�BM�LMR�F?TC�CLMSEF�NMUCP�RM�QGKSJR?LCMSQJW�NPMTGBC�JGDR�?LB�NSR�QGELGDGA?LR�PMR?RGML?J�CLCPEW

GLRM�RFC�@J?BCQ�

z 7MPI�?LB�CLCPEW�GL�PMR?RGML?J�KMRGML�?PC�AMKNJCRCJW�?L?JMEMSQ�RM�UMPI�?LB�CLCPEW�GL�RP?LQJ?RGML?J�KMRGML�

z 4FC�COS?RGML�DMP�RFC KCF?9B9F;M�H<9CF9A DMP�PMR?RGML?J�KMRGML�GQ�

net W = 12Iω2 − 1

2Iω0

2.

� �� �B;I@5F�&CA9BHIA�5B8�"HG��CBG9FJ5H=CBz %TCPW�PMR?RGML?J�NFCLMKCLML�F?Q�?�BGPCAR�RP?LQJ?RGML?J�?L?JME���JGICUGQC�?LESJ?P�KMKCLRSK L A?L�@C�BCDGLCB�?Q L = Iω.z 4FGQ�COS?RGML�GQ�?L�?L?JME�RM�RFC�BCDGLGRGML�MD�JGLC?P�KMKCLRSK�?Q p = mv ��4FC�PCJ?RGMLQFGN�@CRUCCL�RMPOSC�?LB�?LESJ?P�KMKCLRSK�GQ

net τ = ΔLΔt

.

z !LESJ?P�KMKCLRSK��JGIC�CLCPEW�?LB�JGLC?P�KMKCLRSK��GQ�AMLQCPTCB��4FGQ�SLGTCPQ?JJW�?NNJGA?@JC�J?U�GQ�?LMRFCP�QGEL�MD�SLBCPJWGLE�SLGRW�GL

NFWQGA?J�J?UQ��!LESJ?P�KMKCLRSK�GQ�AMLQCPTCB�UFCL�LCR�CVRCPL?J�RMPOSC�GQ�XCPM��HSQR�?Q�JGLC?P�KMKCLRSK�GQ�AMLQCPTCB�UFCL�RFC�LCR�CVRCPL?J

DMPAC�GQ�XCPM�

� �� �C@@=G=CBG�C:��LH9B898��C8=9G�=B�-KC��=A9BG=CBGz !LESJ?P�KMKCLRSK L GQ�?L?JMEMSQ�RM�JGLC?P�KMKCLRSK�?LB�GQ�EGTCL�@W L = Iω �

z !LESJ?P�KMKCLRSK�GQ�AF?LECB�@W�RMPOSC��DMJJMUGLE�RFC�PCJ?RGMLQFGN net τ = ΔLΔt

.

z !LESJ?P�KMKCLRSK�GQ�AMLQCPTCB�GD�RFC�LCR�RMPOSC�GQ�XCPM L = constant (net τ = 0) MP L = L′ (net τ = 0) ��4FGQ�COS?RGML�GQ�ILMUL�?Q�RFC

J?U�MD�AMLQCPT?RGML�MD�?LESJ?P�KMKCLRSK��UFGAF�K?W�@C�AMLQCPTCB�GL�AMJJGQGMLQ�

� �� MFCG7CD=7��::97HG��/97HCF��GD97HG�C:��B;I@5F�&CA9BHIAz 4MPOSC�GQ�NCPNCLBGASJ?P�RM�RFC�NJ?LC�DMPKCB�@W r ?LB F ?LB�GQ�RFC�BGPCARGML�WMSP�PGEFR�RFSK@�UMSJB�NMGLR�GD�WMS�ASPJCB�RFC�DGLECPQ�MD�WMSP�PGEFR

F?LB�GL�RFC�BGPCARGML�MD F ��4FC�BGPCARGML�MD�RFC�RMPOSC�GQ�RFSQ�RFC�Q?KC�?Q�RF?R�MD�RFC�?LESJ?P�KMKCLRSK�GR�NPMBSACQ�

z 4FC�EWPMQAMNC�NPCACQQCQ�?PMSLB�?�TCPRGA?J�?VGQ��QGLAC�RFC�RMPOSC�GQ�?JU?WQ�FMPGXMLR?J�?LB�NCPNCLBGASJ?P�RM L ��)D�RFC�EWPMQAMNC�GQ�LMR�QNGLLGLE�

GR�?AOSGPCQ�?LESJ?P�KMKCLRSK�GL�RFC�BGPCARGML�MD�RFC�RMPOSC�� L = ΔL ��?LB�GR�PMR?RCQ�?@MSR�?�FMPGXMLR?J�?VGQ��D?JJGLE�MTCP�HSQR�?Q�UC�UMSJB

CVNCAR�

z %?PRF�GRQCJD�?ARQ�JGIC�?�EGE?LRGA�EWPMQAMNC��)RQ�?LESJ?P�KMKCLRSK�GQ�?JMLE�GRQ�?VGQ�?LB�NMGLRQ�?R�0MJ?PGQ��RFC�.MPRF�3R?P�

�CB79DHI5@�*I9GH=CBG

� �� �B;I@5F��779@9F5H=CB

�� !L?JMEGCQ�CVGQR�@CRUCCL�PMR?RGML?J�?LB�RP?LQJ?RGML?J�NFWQGA?J�OS?LRGRGCQ��)BCLRGDW�RFC�PMR?RGML?J�RCPK�?L?JMEMSQ�RM�C?AF�MD�RFC�DMJJMUGLE��?AACJCP?RGML�

DMPAC��K?QQ��UMPI��RP?LQJ?RGML?J�IGLCRGA�CLCPEW��JGLC?P�KMKCLRSK��GKNSJQC�

�� %VNJ?GL�UFW�ACLRPGNCR?J�?AACJCP?RGML�AF?LECQ�RFC�BGPCARGML�MD�TCJMAGRW�GL�AGPASJ?P�KMRGML�@SR�LMR�GRQ�K?ELGRSBC�

�� )L�AGPASJ?P�KMRGML��?�R?LECLRG?J�?AACJCP?RGML�A?L�AF?LEC�RFC�K?ELGRSBC�MD�RFC�TCJMAGRW�@SR�LMR�GRQ�BGPCARGML��%VNJ?GL�WMSP�?LQUCP�

�� 3SNNMQC�?�NGCAC�MD�DMMB�GQ�ML�RFC�CBEC�MD�?�PMR?RGLE�KGAPMU?TC�MTCL�NJ?RC��$MCQ�GR�CVNCPGCLAC�LMLXCPM�R?LECLRG?J�?AACJCP?RGML��ACLRPGNCR?J

?AACJCP?RGML��MP�@MRF�UFCL���?�4FC�NJ?RC�QR?PRQ�RM�QNGL���@�4FC�NJ?RC�PMR?RCQ�?R�AMLQR?LR�?LESJ?P�TCJMAGRW���A�4FC�NJ?RC�QJMUQ�RM�?�F?JR�

� �� �MB5A=7G�C:�+CH5H=CB5@�&CH=CB��+CH5H=CB5@�"B9FH=5

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

�� 4FC�KMKCLR�MD�GLCPRG?�MD�?�JMLE�PMB�QNSL�?PMSLB�?L�?VGQ�RFPMSEF�MLC�CLB�NCPNCLBGASJ?P�RM�GRQ�JCLERF�GQ ML2 /3 ��7FW�GQ�RFGQ�KMKCLR�MD�GLCPRG?

EPC?RCP�RF?L�GR�UMSJB�@C�GD�WMS�QNSL�?�NMGLR�K?QQ M ?R�RFC�JMA?RGML�MD�RFC�ACLRCP�MD�K?QQ�MD�RFC�PMB��?R L / 2 ���4F?R�UMSJB�@C ML2 /4 �

��7FW�GQ�RFC�KMKCLR�MD�GLCPRG?�MD�?�FMMN�RF?R�F?Q�?�K?QQ M ?LB�?�P?BGSQ R EPC?RCP�RF?L�RFC�KMKCLR�MD�GLCPRG?�MD�?�BGQI�RF?R�F?Q�RFC�Q?KC�K?QQ

?LB�P?BGSQ��7FW�GQ�RFC�KMKCLR�MD�GLCPRG?�MD�?�QNFCPGA?J�QFCJJ�RF?R�F?Q�?�K?QQ M ?LB�?�P?BGSQ R EPC?RCP�RF?L�RF?R�MD�?�QMJGB�QNFCPC�RF?R�F?Q�RFC

Q?KC�K?QQ�?LB�P?BGSQ�

�� 'GTC�?L�CV?KNJC�GL�UFGAF�?�QK?JJ�DMPAC�CVCPRQ�?�J?PEC�RMPOSC��'GTC�?LMRFCP�CV?KNJC�GL�UFGAF�?�J?PEC�DMPAC�CVCPRQ�?�QK?JJ�RMPOSC�

��7FGJC�PCBSAGLE�RFC�K?QQ�MD�?�P?AGLE�@GIC��RFC�EPC?RCQR�@CLCDGR�GQ�PC?JGXCB�DPMK�PCBSAGLE�RFC�K?QQ�MD�RFC�RGPCQ�?LB�UFCCJ�PGKQ��7FW�BMCQ�RFGQ�?JJMU�?

P?ACP�RM�?AFGCTC�EPC?RCP�?AACJCP?RGMLQ�RF?L�UMSJB�?L�GBCLRGA?J�PCBSARGML�GL�RFC�K?QQ�MD�RFC�@GAWAJCwQ�DP?KC�

�=;IF9�� ��� 4FC�GK?EC�QFMUQ�?�QGBC�TGCU�MD�?�P?AGLE�@GAWAJC��#?L�WMS�QCC�CTGBCLAC�GL�RFC�BCQGEL�MD�RFC�UFCCJQ�ML�RFGQ�P?AGLE�@GAWAJC�RF?R�RFCGP�KMKCLR�MD�GLCPRG?�F?Q�@CCL

NSPNMQCJW�PCBSACB���APCBGR��*CQhQ�2MBPGESCX

�� !�@?JJ�QJGBCQ�SN�?�DPGARGMLJCQQ�P?KN��)R�GQ�RFCL�PMJJCB�UGRFMSR�QJGNNGLE�?LB�UGRF�RFC�Q?KC�GLGRG?J�TCJMAGRW�SN�?LMRFCP�DPGARGMLJCQQ�P?KN��UGRF�RFC�Q?KC

QJMNC�?LEJC��)L�UFGAF�A?QC�BMCQ�GR�PC?AF�?�EPC?RCP�FCGEFR��?LB�UFW�

� �� +CH5H=CB5@�$=B9H=7��B9F;M��0CF?�5B8��B9F;M�+9J=G=H98

� � $CQAPG@C�RFC�CLCPEW�RP?LQDMPK?RGMLQ�GLTMJTCB�UFCL�?�WM WM�GQ�RFPMUL�BMULU?PB�?LB�RFCL�AJGK@Q�@?AI�SN�GRQ�QRPGLE�RM�@C�A?SEFR�GL�RFC�SQCPwQ

F?LB�

���7F?R�CLCPEW�RP?LQDMPK?RGMLQ�?PC�GLTMJTCB�UFCL�?�BP?EQRCP�CLEGLC�GQ�PCTTCB��GRQ�AJSRAF�JCR�MSR�P?NGBJW��GRQ�RGPCQ�QNSL��?LB�GR�QR?PRQ�RM�?AACJCP?RC

DMPU?PB��$CQAPG@C�RFC�QMSPAC�?LB�RP?LQDMPK?RGML�MD�CLCPEW�?R�C?AF�QRCN�

��� 4FC�%?PRF�F?Q�KMPC�PMR?RGML?J�IGLCRGA�CLCPEW�LMU�RF?L�BGB�RFC�AJMSB�MD�E?Q�?LB�BSQR�DPMK�UFGAF�GR�DMPKCB��7FCPC�BGB�RFGQ�CLCPEW�AMKC�DPMK�

�=;IF9�� ��� !L�GKKCLQC�AJMSB�MD�PMR?RGLE�E?Q�?LB�BSQR�AMLRP?ARCB�SLBCP�RFC�GLDJSCLAC�MD�EP?TGRW�RM�DMPK�RFC�%?PRF�?LB�GL�RFC�NPMACQQ�PMR?RGML?J�IGLCRGA�CLCPEW�GLAPC?QCB�

�APCBGR��.!3!

� �� �B;I@5F�&CA9BHIA�5B8�"HG��CBG9FJ5H=CB

���7FCL�WMS�QR?PR�RFC�CLEGLC�MD�WMSP�A?P�UGRF�RFC�RP?LQKGQQGML�GL�LCSRP?J��WMS�LMRGAC�RF?R�RFC�A?P�PMAIQ�GL�RFC�MNNMQGRC�QCLQC�MD�RFC�CLEGLCwQ�PMR?RGML�

%VNJ?GL�GL�RCPKQ�MD�AMLQCPT?RGML�MD�?LESJ?P�KMKCLRSK��)Q�RFC�?LESJ?P�KMKCLRSK�MD�RFC�A?P�AMLQCPTCB�DMP�JMLE��DMP�KMPC�RF?L�?�DCU�QCAMLBQ�

��� 3SNNMQC�?�AFGJB�U?JIQ�DPMK�RFC�MSRCP�CBEC�MD�?�PMR?RGLE�KCPPW EM�PMSLB�RM�RFC�GLQGBC��$MCQ�RFC�?LESJ?P�TCJMAGRW�MD�RFC�KCPPW EM PMSLB�GLAPC?QC�

BCAPC?QC��MP�PCK?GL�RFC�Q?KC��%VNJ?GL�WMSP�?LQUCP�

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

�=;IF9�� ��� !�AFGJB�K?W�HSKN�MDD�?�KCPPW EM PMSLB�GL�?�T?PGCRW�MD�BGPCARGMLQ�

��� 3SNNMQC�?�AFGJB�ECRQ�MDD�?�PMR?RGLE�KCPPW EM PMSLB��$MCQ�RFC�?LESJ?P�TCJMAGRW�MD�RFC�KCPPW EM PMSLB�GLAPC?QC��BCAPC?QC��MP�PCK?GL�RFC�Q?KC�GD�

�?�(C�HSKNQ�MDD�P?BG?JJW���@�(C�HSKNQ�@?AIU?PB�RM�J?LB�KMRGMLJCQQ���A�(C�HSKNQ�QRP?GEFR�SN�?LB�F?LEQ�MLRM�?L�MTCPFC?B�RPCC�@P?LAF���B�(C

HSKNQ�MDD�DMPU?PB��R?LECLRG?J�RM�RFC�CBEC��%VNJ?GL�WMSP�?LQUCPQ���2CDCP�RM �=;IF9�� ����

��� (CJGAMNRCPQ�F?TC�?�QK?JJ�NPMNCJJCP�ML�RFCGP�R?GJ�RM�ICCN�RFCK�DPMK�PMR?RGLE�GL�RFC�MNNMQGRC�BGPCARGML�MD�RFCGP�K?GL�JGDRGLE�@J?BCQ��%VNJ?GL�GL�RCPKQ�MD

.CURMLwQ�RFGPB�J?U�UFW�RFC�FCJGAMNRCP�@MBW�PMR?RCQ�GL�RFC�MNNMQGRC�BGPCARGML�RM�RFC�@J?BCQ�

���7FCLCTCP�?�FCJGAMNRCP�F?Q�RUM�QCRQ�MD�JGDRGLE�@J?BCQ��RFCW�PMR?RC�GL�MNNMQGRC�BGPCARGMLQ��?LB�RFCPC�UGJJ�@C�LM�R?GJ�NPMNCJJCP��%VNJ?GL�UFW�GR�GQ�@CQR�RM

F?TC�RFC�@J?BCQ�PMR?RC�GL�MNNMQGRC�BGPCARGMLQ�

��� $CQAPG@C�FMU�UMPI�GQ�BMLC�@W�?�QI?RCP�NSJJGLE�GL�FCP�?PKQ�BSPGLE�?�QNGL��)L�N?PRGASJ?P��GBCLRGDW�RFC�DMPAC�QFC�CVCPRQ�ML�C?AF�?PK�RM�NSJJ�GR�GL�?LB�RFC

BGQR?LAC�C?AF�KMTCQ��LMRGLE�RF?R�?�AMKNMLCLR�MD�RFC�DMPAC�GQ�GL�RFC�BGPCARGML�KMTCB��7FW�GQ�?LESJ?P�KMKCLRSK�LMR�GLAPC?QCB�@W�RFGQ�?ARGML�

���7FCL�RFCPC�GQ�?�EJM@?J�FC?RGLE�RPCLB�ML�%?PRF��RFC�?RKMQNFCPC�CVN?LBQ�?LB�RFC�JCLERF�MD�RFC�B?W�GLAPC?QCQ�TCPW�QJGEFRJW��%VNJ?GL�UFW�RFC�JCLERF

MD�?�B?W�GLAPC?QCQ�

� � .C?PJW�?JJ�AMLTCLRGML?J�NGQRML�CLEGLCQ�F?TC�DJWUFCCJQ�ML�RFCK�RM�QKMMRF�MSR�CLEGLC�TG@P?RGMLQ�A?SQCB�@W�RFC�RFPSQR�MD�GLBGTGBS?J�NGQRML�DGPGLEQ�

7FW�BMCQ�RFC�DJWUFCCJ�F?TC�RFGQ�CDDCAR�

��� *CR�RSP@GLCQ�QNGL�P?NGBJW��4FCW�?PC�BCQGELCB�RM�DJW�?N?PR�GD�QMKCRFGLE�K?ICQ�RFCK�QCGXC�QSBBCLJW��P?RFCP�RF?L�RP?LQDCP�?LESJ?P�KMKCLRSK�RM�RFC

NJ?LCwQ�UGLE��NMQQG@JW�RC?PGLE�GR�MDD��%VNJ?GL�FMU�DJWGLE�?N?PR�AMLQCPTCQ�?LESJ?P�KMKCLRSK�UGRFMSR�RP?LQDCPPGLE�GR�RM�RFC�UGLE�

��� !L�?QRPML?SR�RGEFRCLQ�?�@MJR�ML�?�Q?RCJJGRC�GL�MP@GR��(C�PMR?RCQ�GL�?�BGPCARGML�MNNMQGRC�RM�RF?R�MD�RFC�@MJR��?LB�RFC�Q?RCJJGRC�PMR?RCQ�GL�RFC�Q?KC

BGPCARGML�?Q�RFC�@MJR��%VNJ?GL�UFW��)D�?�F?LBFMJB�GQ�?T?GJ?@JC�ML�RFC�Q?RCJJGRC��A?L�RFGQ�AMSLRCP PMR?RGML�@C�NPCTCLRCB��%VNJ?GL�WMSP�?LQUCP�

��� #MKNCRGRGTC�BGTCPQ�NSJJ�RFCGP�JGK@Q�GL�?LB�ASPJ�SN�RFCGP�@MBGCQ�UFCL�RFCW�BM�DJGNQ��*SQR�@CDMPC�CLRCPGLE�RFC�U?RCP��RFCW�DSJJW�CVRCLB�RFCGP�JGK@Q�RM

CLRCP�QRP?GEFR�BMUL��%VNJ?GL�RFC�CDDCAR�MD�@MRF�?ARGMLQ�ML�RFCGP�?LESJ?P�TCJMAGRGCQ��!JQM�CVNJ?GL�RFC�CDDCAR�ML�RFCGP�?LESJ?P�KMKCLR?�

�=;IF9�� ��� 4FC�BGTCP�QNGLQ�P?NGBJW�UFCL�ASPJCB�SN�?LB�QJMUQ�UFCL�QFC�CVRCLBQ�FCP�JGK@Q�@CDMPC�CLRCPGLE�RFC�U?RCP�

��� $P?U�?�DPCC�@MBW�BG?EP?K�RM�QFMU�FMU�?�BGTCP�E?GLQ�?LESJ?P�KMKCLRSK�UFCL�JC?TGLE�RFC�BGTGLE�@M?PB�

��� )L�RCPKQ�MD�?LESJ?P�KMKCLRSK��UF?R�GQ�RFC�?BT?LR?EC�MD�EGTGLE�?�DMMR@?JJ�MP�?�PGDJC�@SJJCR�?�QNGL�UFCL�RFPMUGLE�MP�PCJC?QGLE�GR�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

�=;IF9�� ��� 4FC�GK?EC�QFMUQ�?�TGCU�BMUL�RFC�@?PPCJ�MD�?�A?LLML��CKNF?QGXGLE�GRQ�PGDJGLE��2GDJGLE�GL�RFC�@?PPCJ�MD�?�A?LML�A?SQCQ�RFC�NPMHCARGJC�RM�QNGL�HSQR�?Q�GQ�RFC�A?QC�DMP

PGDJCQ��FCLAC�RFC�L?KC�DMP�RFC�EPMMTCQ�GL�RFC�@?PPCJ���APCBGR��%JQGC�CQO���&JGAIP

� �� �C@@=G=CBG�C:��LH9B898��C8=9G�=B�-KC��=A9BG=CBG

��� $CQAPG@C�RUM�BGDDCPCLR�AMJJGQGMLQuMLC�GL�UFGAF�?LESJ?P�KMKCLRSK�GQ�AMLQCPTCB��?LB�RFC�MRFCP�GL�UFGAF�GR�GQ�LMR��7FGAF�AMLBGRGML�BCRCPKGLCQ

UFCRFCP�MP�LMR�?LESJ?P�KMKCLRSK�GQ�AMLQCPTCB�GL�?�AMJJGQGML�

��� 3SNNMQC�?L�GAC�FMAICW�NSAI�QRPGICQ�?�FMAICW�QRGAI�RF?R�JGCQ�DJ?R�ML�RFC�GAC�?LB�GQ�DPCC�RM�KMTC�GL�?LW�BGPCARGML��7FGAF�OS?LRGRGCQ�?PC�JGICJW�RM�@C

AMLQCPTCB��?LESJ?P�KMKCLRSK��JGLC?P�KMKCLRSK��MP�IGLCRGA�CLCPEW��?QQSKGLE�RFC�NSAI�?LB�QRGAI�?PC�TCPW�PCQGJGCLR�

���7FGJC�BPGTGLE�FGQ�KMRMPAWAJC�?R�FGEFU?W�QNCCB��?�NFWQGAQ�QRSBCLR�LMRGACQ�RF?R�NSJJGLE�@?AI�JGEFRJW�ML�RFC�PGEFR�F?LBJC@?P�RGNQ�RFC�AWAJC�RM�RFC�JCDR

?LB�NPMBSACQ�?�JCDR�RSPL��%VNJ?GL�UFW�RFGQ�F?NNCLQ�

� �� MFCG7CD=7��::97HG��/97HCF��GD97HG�C:��B;I@5F�&CA9BHIA

���7FGJC�BPGTGLE�FGQ�KMRMPAWAJC�?R�FGEFU?W�QNCCB��?�NFWQGAQ�QRSBCLR�LMRGACQ�RF?R�NSJJGLE�@?AI�JGEFRJW�ML�RFC�PGEFR�F?LBJC@?P�RGNQ�RFC�AWAJC�RM�RFC�JCDR

?LB�NPMBSACQ�?�JCDR�RSPL��%VNJ?GL�UFW�RFGQ�F?NNCLQ�

� � 'WPMQAMNCQ�SQCB�GL�ESGB?LAC�QWQRCKQ�RM�GLBGA?RC�BGPCARGMLQ�GL�QN?AC�KSQR�F?TC�?L�?LESJ?P�KMKCLRSK�RF?R�BMCQ�LMR�AF?LEC�GL�BGPCARGML��9CR

RFCW�?PC�MDRCL�QS@HCARCB�RM�J?PEC�DMPACQ�?LB�?AACJCP?RGMLQ��(MU�A?L�RFC�BGPCARGML�MD�RFCGP�?LESJ?P�KMKCLRSK�@C�AMLQR?LR�UFCL�RFCW�?PC�?AACJCP?RCB�

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

)FC6@9AG����L9F7=G9G

� �� �B;I@5F��779@9F5H=CB

�� !R�GRQ�NC?I��?�RMPL?BM�GQ������K�GL�BG?KCRCP�?LB�A?PPGCQ�����IK�F

UGLBQ��7F?R�GQ�GRQ�?LESJ?P�TCJMAGRW�GL�PCTMJSRGMLQ�NCP�QCAMLB�

�� "BH9;F5H98��CB79DHG

!L�SJRP?ACLRPGDSEC�?AACJCP?RCQ�DPMK�PCQR�RM���������PNK�GL������KGL���?

7F?R�GQ�GRQ�?LESJ?P�?AACJCP?RGML�GL rad/s2���@�7F?R�GQ�RFC�R?LECLRG?J

?AACJCP?RGML�MD�?�NMGLR������AK�DPMK�RFC�?VGQ�MD�PMR?RGML���A�7F?R�GQ�RFC

P?BG?J�?AACJCP?RGML�GL m/s2?LB�KSJRGNJCQ�MD g MD�RFGQ�NMGLR�?R�DSJJ�PNK�

�� "BH9;F5H98��CB79DHG

9MS�F?TC�?�EPGLBQRMLC��?�BGQI�RF?R�GQ������IE��F?Q�?������ K�P?BGSQ��?LB

GQ�RSPLGLE�?R������PNK��?LB�WMS�NPCQQ�?�QRCCJ�?VC�?E?GLQR�GR�UGRF�?�P?BG?J

DMPAC�MD������.���?�!QQSKGLE�RFC�IGLCRGA�AMCDDGAGCLR�MD�DPGARGML�@CRUCCL

QRCCJ�?LB�QRMLC�GQ�������A?JASJ?RC�RFC�?LESJ?P�?AACJCP?RGML�MD�RFC

EPGLBQRMLC���@�(MU�K?LW�RSPLQ�UGJJ�RFC�QRMLC�K?IC�@CDMPC�AMKGLE�RM

PCQR�

�� .BF95GCB56@9�+9GI@HG

9MS�?PC�RMJB�RF?R�?�@?QICR@?JJ�NJ?WCP�QNGLQ�RFC�@?JJ�UGRF�?L�?LESJ?P

?AACJCP?RGML�MD 100 rad/s2���?�7F?R�GQ�RFC�@?JJwQ�DGL?J�?LESJ?P�TCJMAGRW�GD

RFC�@?JJ�QR?PRQ�DPMK�PCQR�?LB�RFC�?AACJCP?RGML�J?QRQ������Q���@�7F?R�GQ

SLPC?QML?@JC�?@MSR�RFC�PCQSJR���A�7FGAF�NPCKGQCQ�?PC�SLPC?QML?@JC�MP

GLAMLQGQRCLR�

� �� $=B9A5H=7G�C:�+CH5H=CB5@�&CH=CB

��7GRF�RFC�?GB�MD�?�QRPGLE��?�EWPMQAMNC�GQ�?AACJCP?RCB�DPMK�PCQR�RM����P?B�

Q�GL������Q�

�?�7F?R�GQ�GRQ�?LESJ?P�?AACJCP?RGML�GL�P?B�Q��

�@�(MU�K?LW�PCTMJSRGMLQ�BMCQ�GR�EM�RFPMSEF�GL�RFC�NPMACQQ�

�� 3SNNMQC�?�NGCAC�MD�BSQR�DGLBQ�GRQCJD�ML�?�#$��)D�RFC�QNGL�P?RC�MD�RFC�#$

GQ�����PNK��?LB�RFC�NGCAC�MD�BSQR�GQ�����AK�DPMK�RFC�ACLRCP��UF?R�GQ�RFC

RMR?J�BGQR?LAC�RP?TCJCB�@W�RFC�BSQR�GL���KGLSRCQ���)ELMPC�?AACJCP?RGMLQ

BSC�RM�ECRRGLE�RFC�#$�PMR?RGLE�

�� !�EWPMQAMNC�QJMUQ�DPMK�?L�GLGRG?J�P?RC�MD������P?B�Q�?R�?�P?RC�MD

0.700 rad/s2�

�?�(MU�JMLE�BMCQ�GR�R?IC�RM�AMKC�RM�PCQR�

�@�(MU�K?LW�PCTMJSRGMLQ�BMCQ�GR�K?IC�@CDMPC�QRMNNGLE�

�� $SPGLE�?�TCPW�OSGAI�QRMN��?�A?P�BCACJCP?RCQ�?R 7.00 m/s2�

�?�7F?R�GQ�RFC�?LESJ?P�?AACJCP?RGML�MD�GRQ������ K P?BGSQ�RGPCQ��?QQSKGLE

RFCW�BM�LMR�QJGN�ML�RFC�N?TCKCLR�

�@�(MU�K?LW�PCTMJSRGMLQ�BM�RFC�RGPCQ�K?IC�@CDMPC�AMKGLE�RM�PCQR��EGTCL

RFCGP�GLGRG?J�?LESJ?P�TCJMAGRW�GQ 95.0 rad/s �

�A�(MU�JMLE�BMCQ�RFC�A?P�R?IC�RM�QRMN�AMKNJCRCJW�

�B�7F?R�BGQR?LAC�BMCQ�RFC�A?P�RP?TCJ�GL�RFGQ�RGKC�

�C�7F?R�U?Q�RFC�A?PwQ�GLGRG?J�TCJMAGRW�

�D�$M�RFC�T?JSCQ�M@R?GLCB�QCCK�PC?QML?@JC��AMLQGBCPGLE�RF?R�RFGQ�QRMN

F?NNCLQ�TCPW�OSGAIJW�

�=;IF9�� ��� 9M WMQ�?PC�?KSQGLE�RMWQ�RF?R�BGQNJ?W�QGELGDGA?LR�NFWQGAQ�?LB�?PC

CLEGLCCPCB�RM�CLF?LAC�NCPDMPK?LAC�@?QCB�ML�NFWQGA?J�J?UQ���APCBGR��"CWMLB�.CML�

&JGAIP

�� %TCPWB?W�?NNJGA?RGML��3SNNMQC�?�WM WM�F?Q�?�ACLRCP�QF?DR�RF?R�F?Q�?

������AK�P?BGSQ�?LB�RF?R�GRQ�QRPGLE�GQ�@CGLE�NSJJCB�

�?�)D�RFC�QRPGLE�GQ�QR?RGML?PW�?LB�RFC�WM WM�?AACJCP?RCQ�?U?W�DPMK�GR�?R�?

P?RC�MD 1.50 m/s2��UF?R�GQ�RFC�?LESJ?P�?AACJCP?RGML�MD�RFC�WM WM�

�@�7F?R�GQ�RFC�?LESJ?P�TCJMAGRW�?DRCP�������Q�GD�GR�QR?PRQ�DPMK�PCQR�

�A�4FC�MSRQGBC�P?BGSQ�MD�RFC�WM WM�GQ������AK��7F?R�GQ�RFC�R?LECLRG?J

?AACJCP?RGML�MD�?�NMGLR�ML�GRQ�CBEC�

� �� �MB5A=7G�C:�+CH5H=CB5@�&CH=CB��+CH5H=CB5@�"B9FH=5

� � 4FGQ�NPM@JCK�AMLQGBCPQ�?BBGRGML?J�?QNCARQ�MD�CV?KNJC �5@7I@5H=B;

H<9��::97H�C:�&5GG��=GHF=6IH=CB�CB�5�&9FFM C+CIB8���?�(MU�JMLE

BMCQ�GR�R?IC�RFC�D?RFCP�RM�EGTC�RFC�KCPPW EM PMSLB�?L�?LESJ?P�TCJMAGRW�MD

�����P?B�Q���@�(MU�K?LW�PCTMJSRGMLQ�KSQR�FC�EM�RFPMSEF�RM�ECLCP?RC

RFGQ�TCJMAGRW���A�)D�FC�CVCPRQ�?�QJMUGLE�DMPAC�MD�����.�?R�?�P?BGSQ�MD�����

K��FMU�JMLE�UMSJB�GR�R?IC�FGK�RM�QRMN�RFCK�

��� #?JASJ?RC�RFC�KMKCLR�MD�GLCPRG?�MD�?�QI?RCP�EGTCL�RFC�DMJJMUGLE

GLDMPK?RGML���?�4FC����� IE�QI?RCP�GQ�?NNPMVGK?RCB�?Q�?�AWJGLBCP�RF?R�F?Q

?������ K�P?BGSQ���@�4FC�QI?RCP�UGRF�?PKQ�CVRCLBCB�GQ�?NNPMVGK?RCJW�?

AWJGLBCP�RF?R�GQ������IE��F?Q�?������ K�P?BGSQ��?LB�F?Q�RUM������ K JMLE

?PKQ�UFGAF�?PC������IE�C?AF�?LB�CVRCLB�QRP?GEFR�MSR�DPMK�RFC�AWJGLBCP

JGIC�PMBQ�PMR?RCB�?@MSR�RFCGP�CLBQ�

��� 4FC�RPGACNQ�KSQAJC�GL�RFC�@?AI�MD�RFC�SNNCP�?PK�CVRCLBQ�RFC�DMPC?PK�

4FGQ�KSQAJC�GL�?�NPMDCQQGML?J�@MVCP�CVCPRQ�?�DMPAC�MD 2.00×103 N UGRF

?L�CDDCARGTC�NCPNCLBGASJ?P�JCTCP�?PK�MD������AK��NPMBSAGLE�?L�?LESJ?P

?AACJCP?RGML�MD�RFC�DMPC?PK�MD 120 rad/s2��7F?R�GQ�RFC�KMKCLR�MD�GLCPRG?

MD�RFC�@MVCPwQ�DMPC?PK�

��� !�QMAACP�NJ?WCP�CVRCLBQ�FCP�JMUCP�JCE�GL�?�IGAIGLE�KMRGML�@W�CVCPRGLE

?�DMPAC�UGRF�RFC�KSQAJC�?@MTC�RFC�ILCC�GL�RFC�DPMLR�MD�FCP�JCE��3FC

NPMBSACQ�?L�?LESJ?P�?AACJCP?RGML�MD 30.00 rad/s2?LB�FCP�JMUCP�JCE�F?Q

?�KMKCLR�MD�GLCPRG?�MD 0.750 kg ⋅ m2��7F?R�GQ�RFC�DMPAC�CVCPRCB�@W�RFC

KSQAJC�GD�GRQ�CDDCARGTC�NCPNCLBGASJ?P�JCTCP�?PK�GQ������AK�

��� 3SNNMQC�WMS�CVCPR�?�DMPAC�MD�����.�R?LECLRG?J�RM�?������ K P?BGSQ

���� IE�EPGLBQRMLC��?�QMJGB�BGQI�

�?7F?R�RMPOSC�GQ�CVCPRCB���@�7F?R�GQ�RFC�?LESJ?P�?AACJCP?RGML�?QQSKGLE

LCEJGEG@JC�MNNMQGLE�DPGARGML���A�7F?R�GQ�RFC�?LESJ?P�?AACJCP?RGML�GD�RFCPC

GQ�?L�MNNMQGLE�DPGARGML?J�DMPAC�MD������.�CVCPRCB������AK�DPMK�RFC�?VGQ�

��� #MLQGBCP�RFC������IE�KMRMPAWAJC�UFCCJ�QFMUL�GL �=;IF9�� ����

!QQSKC�GR�RM�@C�?NNPMVGK?RCJW�?L�?LLSJ?P�PGLE�UGRF�?L�GLLCP�P?BGSQ�MD

������K�?LB�?L�MSRCP�P?BGSQ�MD�������K��4FC�KMRMPAWAJC�GQ�ML�GRQ�ACLRCP

QR?LB��QM�RF?R�RFC�UFCCJ�A?L�QNGL�DPCCJW���?�)D�RFC�BPGTC�AF?GL�CVCPRQ�?

DMPAC�MD������.�?R�?�P?BGSQ�MD������AK��UF?R�GQ�RFC�?LESJ?P�?AACJCP?RGML�MD

RFC�UFCCJ���@�7F?R�GQ�RFC�R?LECLRG?J�?AACJCP?RGML�MD�?�NMGLR�ML�RFC�MSRCP

CBEC�MD�RFC�RGPC���A�(MU�JMLE��QR?PRGLE�DPMK�PCQR��BMCQ�GR�R?IC�RM�PC?AF�?L

?LESJ?P�TCJMAGRW�MD������P?B�Q�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

�=;IF9�� ��� !�KMRMPAWAJC�UFCCJ�F?Q�?�KMKCLR�MD�GLCPRG?�?NNPMVGK?RCJW�RF?R�MD�?L

?LLSJ?P�PGLE�

��� :MPAF��?L�?PAFCLCKW�MD�3SNCPK?L��BCAGBCQ�RM�QJMU�%?PRFwQ�PMR?RGML

RM�MLAC�NCP������F�@W�CVCPRGLE�?L�MNNMQGLE�DMPAC�?R�?LB�N?P?JJCJ�RM�RFC

COS?RMP��3SNCPK?L�GQ�LMR�GKKCBG?RCJW�AMLACPLCB��@CA?SQC�FC�ILMUQ

:MPAF�A?L�MLJW�CVCPR�?�DMPAC�MD 4.00×107 N �?�JGRRJC�EPC?RCP�RF?L�?

3?RSPL�6�PMAICRwQ�RFPSQR��(MU�JMLE�KSQR�:MPAF�NSQF�UGRF�RFGQ�DMPAC�RM

?AAMKNJGQF�FGQ�EM?J���4FGQ�NCPGMB�EGTCQ�3SNCPK?L�RGKC�RM�BCTMRC�RM

MRFCP�TGJJ?GLQ��%VNJGAGRJW�QFMU�FMU�WMS�DMJJMU�RFC�QRCNQ�DMSLB�GL

)FC6@9A,C@J=B;�,HF5H9;M�:CF�+CH5H=CB5@��MB5A=7G�

��� !L�?SRMKM@GJC�CLEGLC�A?L�NPMBSAC�����.�_�K�MD�RMPOSC��#?JASJ?RC�RFC

?LESJ?P�?AACJCP?RGML�NPMBSACB�GD�������MD�RFGQ�RMPOSC�GQ�?NNJGCB�RM�RFC

BPGTC�QF?DR��?VJC��?LB�PC?P�UFCCJQ�MD�?�A?P��EGTCL�RFC�DMJJMUGLE�GLDMPK?RGML�

4FC�A?P�GQ�QSQNCLBCB�QM�RF?R�RFC�UFCCJQ�A?L�RSPL�DPCCJW��%?AF�UFCCJ�?ARQ

JGIC�?������IE�BGQI�RF?R�F?Q�?�������K�P?BGSQ��4FC�U?JJQ�MD�C?AF�RGPC�?AR

JGIC�?����� IE�?LLSJ?P�PGLE�RF?R�F?Q�GLQGBC�P?BGSQ�MD�������K�?LB�MSRQGBC

P?BGSQ�MD�������K��4FC�RPC?B�MD�C?AF�RGPC�?ARQ�JGIC�?����� IE�FMMN�MD

P?BGSQ�������K��4FC����� IE�?VJC�?ARQ�JGIC�?�PMB�RF?R�F?Q�?����� AK

P?BGSQ��4FC����� IE�BPGTC�QF?DR�?ARQ�JGIC�?�PMB�RF?R�F?Q�?����� AK�P?BGSQ�

��� 3R?PRGLE�UGRF�RFC�DMPKSJ?�DMP�RFC�KMKCLR�MD�GLCPRG?�MD�?�PMB�PMR?RCB

?PMSLB�?L�?VGQ�RFPMSEF�MLC�CLB�NCPNCLBGASJ?P�RM�GRQ�JCLERF⎛⎝I = Mℓ 2 / 3⎞

⎠ ��NPMTC�RF?R�RFC�KMKCLR�MD�GLCPRG?�MD�?�PMB�PMR?RCB�?@MSR

?L�?VGQ�RFPMSEF�GRQ�ACLRCP�NCPNCLBGASJ?P�RM�GRQ�JCLERF�GQ I = Mℓ 2 / 12 �

9MS�UGJJ�DGLB�RFC�EP?NFGAQ�GL �=;IF9�� ��� SQCDSJ�GL�TGQS?JGXGLE�RFCQC

PMR?RGMLQ�

��� .BF95GCB56@9�+9GI@HG

!�EWKL?QR�BMGLE�?�DMPU?PB�DJGN�J?LBQ�ML�RFC�K?R�?LB�CVCPRQ�?���� .�_�K

RMPOSC�RM�QJMU�?LB�RFCL�PCTCPQC�FCP�?LESJ?P�TCJMAGRW��(CP�GLGRG?J�?LESJ?P

TCJMAGRW�GQ������P?B�Q��?LB�FCP�KMKCLR�MD�GLCPRG?�GQ 0.050 kg ⋅ m2���?

7F?R�RGKC�GQ�PCOSGPCB�DMP�FCP�RM�CV?ARJW�PCTCPQC�FCP�QNGL���@�7F?R�GQ

SLPC?QML?@JC�?@MSR�RFC�PCQSJR���A�7FGAF�NPCKGQCQ�?PC�SLPC?QML?@JC�MP

GLAMLQGQRCLR�

� � .BF95GCB56@9�+9GI@HG

!L�?BTCPRGQCKCLR�AJ?GKQ�RF?R�?L���� IE�A?P�GQ�?GBCB�@W�GRQ����� IE

DJWUFCCJ��UFGAF�A?L�?AACJCP?RC�RFC�A?P�DPMK�PCQR�RM�?�QNCCB�MD������K�Q�

4FC�DJWUFCCJ�GQ�?�BGQI�UGRF�?������ K�P?BGSQ���?�#?JASJ?RC�RFC�?LESJ?P

TCJMAGRW�RFC�DJWUFCCJ�KSQR�F?TC�GD�������MD�GRQ�PMR?RGML?J�CLCPEW�GQ�SQCB�RM

ECR�RFC�A?P�SN�RM�QNCCB���@�7F?R�GQ�SLPC?QML?@JC�?@MSR�RFC�PCQSJR���A

7FGAF�NPCKGQC�GQ�SLPC?QML?@JC�MP�UFGAF�NPCKGQCQ�?PC�GLAMLQGQRCLR�

� �� +CH5H=CB5@�$=B9H=7��B9F;M��0CF?�5B8��B9F;M

+9J=G=H98

��� 4FGQ�NPM@JCK�AMLQGBCPQ�CLCPEW�?LB�UMPI�?QNCARQ�MD �L5AD@9

� ��uSQC�B?R?�DPMK�RF?R�CV?KNJC�?Q�LCCBCB���?�#?JASJ?RC�RFC�PMR?RGML?J

IGLCRGA�CLCPEW�GL�RFC�KCPPW EM PMSLB�NJSQ�AFGJB�UFCL�RFCW�F?TC�?L

?LESJ?P�TCJMAGRW�MD������PNK���@�5QGLE�CLCPEW�AMLQGBCP?RGMLQ��DGLB�RFC

LSK@CP�MD�PCTMJSRGMLQ�RFC�D?RFCP�UGJJ�F?TC�RM�NSQF�RM�?AFGCTC�RFGQ�?LESJ?P

TCJMAGRW�QR?PRGLE�DPMK�PCQR���A�!E?GL��SQGLE�CLCPEW�AMLQGBCP?RGMLQ�

A?JASJ?RC�RFC�DMPAC�RFC�D?RFCP�KSQR�CVCPR�RM�QRMN�RFC�KCPPW EM PMSLB�GL

RUM�PCTMJSRGMLQ

���7F?R�GQ�RFC�DGL?J�TCJMAGRW�MD�?�FMMN�RF?R�PMJJQ�UGRFMSR�QJGNNGLE�BMUL�?

���� K FGEF�FGJJ��QR?PRGLE�DPMK�PCQR�

��� �?�#?JASJ?RC�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�MD�%?PRF�ML�GRQ�?VGQ���@

7F?R�GQ�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�MD�%?PRF�GL�GRQ�MP@GR�?PMSLB�RFC�3SL�

��� #?JASJ?RC�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�GL�RFC�KMRMPAWAJC�UFCCJ

��=;IF9�� ����GD�GRQ�?LESJ?P�TCJMAGRW�GQ�����P?B�Q��!QQSKC�-��������IE�

2� ��������K��?LB�2� ��������K�

��� !�@?QC@?JJ�NGRAFCP�RFPMUQ�RFC�@?JJ�GL�?�KMRGML�UFCPC�RFCPC�GQ�PMR?RGML

MD�RFC�DMPC?PK�?@MSR�RFC�CJ@MU�HMGLR�?Q�UCJJ�?Q�MRFCP�KMTCKCLRQ��)D�RFC

JGLC?P�TCJMAGRW�MD�RFC�@?JJ�PCJ?RGTC�RM�RFC�CJ@MU�HMGLR�GQ������K�Q�?R�?

BGQR?LAC�MD�������K�DPMK�RFC�HMGLR�?LB�RFC�KMKCLR�MD�GLCPRG?�MD�RFC

DMPC?PK�GQ 0.500 kg ⋅ m2��UF?R�GQ�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�MD�RFC

DMPC?PK�

���7FGJC�NSLRGLE�?�DMMR@?JJ��?�IGAICP�PMR?RCQ�FGQ�JCE�?@MSR�RFC�FGN�HMGLR�

4FC�KMKCLR�MD�GLCPRG?�MD�RFC�JCE�GQ 3.75 kg ⋅ m2?LB�GRQ�PMR?RGML?J

IGLCRGA�CLCPEW�GQ�����*���?�7F?R�GQ�RFC�?LESJ?P�TCJMAGRW�MD�RFC�JCE���@

7F?R�GQ�RFC�TCJMAGRW�MD�RGN�MD�RFC�NSLRCPwQ�QFMC�GD�GR�GQ������K�DPMK�RFC�FGN

HMGLR���A�%VNJ?GL�FMU�RFC�DMMR@?JJ�A?L�@C�EGTCL�?�TCJMAGRW�EPC?RCP�RF?L�RFC

RGN�MD�RFC�QFMC��LCACQQ?PW�DMP�?�BCACLR�IGAI�BGQR?LAC�

��� !�@SQ�AMLR?GLQ�?������IE�DJWUFCCJ��?�BGQI�RF?R�F?Q�?�������K�P?BGSQ

?LB�F?Q�?�RMR?J�K?QQ�MD��������IE���?�#?JASJ?RC�RFC�?LESJ?P�TCJMAGRW�RFC

DJWUFCCJ�KSQR�F?TC�RM�AMLR?GL�CLMSEF�CLCPEW�RM�R?IC�RFC�@SQ�DPMK�PCQR�RM

?�QNCCB�MD������K�Q��?QQSKGLE�������MD�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�A?L

@C�RP?LQDMPKCB�GLRM�RP?LQJ?RGML?J�CLCPEW���@�(MU�FGEF�?�FGJJ�A?L�RFC�@SQ

AJGK@�UGRF�RFGQ�QRMPCB�CLCPEW�?LB�QRGJJ�F?TC�?�QNCCB�MD������K�Q�?R�RFC�RMN

MD�RFC�FGJJ��%VNJGAGRJW�QFMU�FMU�WMS�DMJJMU�RFC�QRCNQ�GL�RFC )FC6@9A

,C@J=B;�,HF5H9;M�:CF�+CH5H=CB5@��B9F;M�

��� !�@?JJ�UGRF�?L�GLGRG?J�TCJMAGRW�MD������K�Q�PMJJQ�SN�?�FGJJ�UGRFMSR�QJGNNGLE�

4PC?RGLE�RFC�@?JJ�?Q�?�QNFCPGA?J�QFCJJ��A?JASJ?RC�RFC�TCPRGA?J�FCGEFR�GR

PC?AFCQ���@�2CNC?R�RFC�A?JASJ?RGML�DMP�RFC�Q?KC�@?JJ�GD�GR�QJGBCQ�SN�RFC�FGJJ

UGRFMSR�PMJJGLE�

���7FGJC�CVCPAGQGLE�GL�?�DGRLCQQ�ACLRCP��?�K?L�JGCQ�D?AC�BMUL�ML�?�@CLAF

?LB�JGDRQ�?�UCGEFR�UGRF�MLC�JMUCP�JCE�@W�AMLR?ARGLE�RFC�KSQAJCQ�GL�RFC

@?AI�MD�RFC�SNNCP�JCE���?�&GLB�RFC�?LESJ?P�?AACJCP?RGML�NPMBSACB�EGTCL

RFC�K?QQ�JGDRCB�GQ������IE�?R�?�BGQR?LAC�MD������AK�DPMK�RFC�ILCC�HMGLR��RFC

KMKCLR�MD�GLCPRG?�MD�RFC�JMUCP�JCE�GQ 0.900 kg ⋅ m2��RFC�KSQAJC�DMPAC�GQ

�����.��?LB�GRQ�CDDCARGTC�NCPNCLBGASJ?P�JCTCP�?PK�GQ������AK���@�(MU

KSAF�UMPI�GQ�BMLC�GD�RFC�JCE�PMR?RCQ�RFPMSEF�?L�?LEJC�MD 20.0º UGRF�?

AMLQR?LR�DMPAC�CVCPRCB�@W�RFC�KSQAJC�

� � 4M�BCTCJMN�KSQAJC�RMLC��?�UMK?L�JGDRQ�?����� IE�UCGEFR�FCJB�GL�FCP

F?LB��3FC�SQCQ�FCP�@GACNQ�KSQAJC�RM�DJCV�RFC�JMUCP�?PK�RFPMSEF�?L�?LEJC

MD 60.0º ���?�7F?R�GQ�RFC�?LESJ?P�?AACJCP?RGML�GD�RFC�UCGEFR�GQ������AK

DPMK�RFC�CJ@MU�HMGLR��FCP�DMPC?PK�F?Q�?�KMKCLR�MD�GLCPRG?�MD

0.250 kg ⋅ m2��?LB�RFC�LCR�DMPAC�QFC�CVCPRQ�GQ�����.�?R�?L�CDDCARGTC

NCPNCLBGASJ?P�JCTCP�?PK�MD������AK���@�(MU�KSAF�UMPI�BMCQ�QFC�BM�

��� #MLQGBCP�RUM�AWJGLBCPQ�RF?R�QR?PR�BMUL�GBCLRGA?J�GLAJGLCQ�DPMK�PCQR

CVACNR�RF?R�MLC�GQ�DPGARGMLJCQQ��4FSQ�MLC�AWJGLBCP�PMJJQ�UGRFMSR�QJGNNGLE�

UFGJC�RFC�MRFCP�QJGBCQ�DPGARGMLJCQQJW�UGRFMSR�PMJJGLE��4FCW�@MRF�RP?TCJ�?�QFMPR

BGQR?LAC�?R�RFC�@MRRMK�?LB�RFCL�QR?PR�SN�?LMRFCP�GLAJGLC���?�3FMU�RF?R

RFCW�@MRF�PC?AF�RFC�Q?KC�FCGEFR�ML�RFC�MRFCP�GLAJGLC��?LB�RF?R�RFGQ�FCGEFR

GQ�COS?J�RM�RFCGP�MPGEGL?J�FCGEFR���@�&GLB�RFC�P?RGM�MD�RFC�RGKC�RFC�PMJJGLE

AWJGLBCP�R?ICQ�RM�PC?AF�RFC�FCGEFR�ML�RFC�QCAMLB�GLAJGLC�RM�RFC�RGKC�RFC

QJGBGLE�AWJGLBCP�R?ICQ�RM�PC?AF�RFC�FCGEFR�ML�RFC�QCAMLB�GLAJGLC���A

%VNJ?GL�UFW�RFC�RGKC�DMP�RFC�PMJJGLE�KMRGML�GQ�EPC?RCP�RF?L�RF?R�DMP�RFC

QJGBGLE�KMRGML�

���7F?R�GQ�RFC�KMKCLR�MD�GLCPRG?�MD�?L�M@HCAR�RF?R�PMJJQ�UGRFMSR�QJGNNGLE

BMUL�?����� K FGEF�GLAJGLC�QR?PRGLE�DPMK�PCQR��?LB�F?Q�?�DGL?J�TCJMAGRW�MD

�����K�Q��%VNPCQQ�RFC�KMKCLR�MD�GLCPRG?�?Q�?�KSJRGNJC�MD MR2��UFCPC

M GQ�RFC�K?QQ�MD�RFC�M@HCAR�?LB R GQ�GRQ�P?BGSQ�

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

��� 3SNNMQC�?���� IE�KMRMPAWAJC�F?Q�RUM�UFCCJQ�JGIC� H<9�CB9

89G7F=698�=B�)FC6@9A�� ��� ?LB�GQ�FC?BGLE�RMU?PB�?�FGJJ�?R�?�QNCCB�MD

�����K�Q���?�(MU�FGEF�A?L�GR�AM?QR�SN�RFC�FGJJ��GD�WMS�LCEJCAR�DPGARGML���@

(MU�KSAF�CLCPEW�GQ�JMQR�RM�DPGARGML�GD�RFC�KMRMPAWAJC�MLJW�E?GLQ�?L�?JRGRSBC

MD������K�@CDMPC�AMKGLE�RM�PCQR�

��� )L�QMDR@?JJ��RFC�NGRAFCP�RFPMUQ�UGRF�RFC�?PK�DSJJW�CVRCLBCB��QRP?GEFR�?R

RFC�CJ@MU��)L�?�D?QR�NGRAF�RFC�@?JJ�JC?TCQ�RFC�F?LB�UGRF�?�QNCCB�MD����

IK�F���?�&GLB�RFC�PMR?RGML?J�IGLCRGA�CLCPEW�MD�RFC�NGRAFCPwQ�?PK�EGTCL�GRQ

KMKCLR�MD�GLCPRG?�GQ 0.720 kg ⋅ m2?LB�RFC�@?JJ�JC?TCQ�RFC�F?LB�?R�?

BGQR?LAC�MD�������K�DPMK�RFC�NGTMR�?R�RFC�QFMSJBCP���@�7F?R�DMPAC�BGB�RFC

KSQAJCQ�CVCPR�RM�A?SQC�RFC�?PK�RM�PMR?RC�GD�RFCGP�CDDCARGTC�NCPNCLBGASJ?P

JCTCP�?PK�GQ������AK�?LB�RFC�@?JJ�GQ�������IE�

��� �CBGHFI7H�2CIF�(KB�)FC6@9A

#MLQGBCP�RFC�UMPI�BMLC�@W�?�QNGLLGLE�QI?RCP�NSJJGLE�FCP�?PKQ�GL�RM

GLAPC?QC�FCP�P?RC�MD�QNGL��#MLQRPSAR�?�NPM@JCK�GL�UFGAF�WMS�A?JASJ?RC�RFC

UMPI�BMLC�UGRF�?�xDMPAC�KSJRGNJGCB�@W�BGQR?LACy�A?JASJ?RGML�?LB�AMKN?PC�GR

RM�RFC�QI?RCPwQ�GLAPC?QC�GL�IGLCRGA�CLCPEW�

� �� �B;I@5F�&CA9BHIA�5B8�"HG��CBG9FJ5H=CB

��� �?�#?JASJ?RC�RFC�?LESJ?P�KMKCLRSK�MD�RFC�%?PRF�GL�GRQ�MP@GR�?PMSLB

RFC�3SL�

�@�#MKN?PC�RFGQ�?LESJ?P�KMKCLRSK�UGRF�RFC�?LESJ?P�KMKCLRSK�MD

%?PRF�ML�GRQ�?VGQ�

��� �?�7F?R�GQ�RFC�?LESJ?P�KMKCLRSK�MD�RFC�-MML�GL�GRQ�MP@GR�?PMSLB

%?PRF�

�@�(MU�BMCQ�RFGQ�?LESJ?P�KMKCLRSK�AMKN?PC�UGRF�RFC�?LESJ?P

KMKCLRSK�MD�RFC�-MML�ML�GRQ�?VGQ��2CKCK@CP�RF?R�RFC�-MML�ICCNQ�MLC

QGBC�RMU?PB�%?PRF�?R�?JJ�RGKCQ�

�A�$GQASQQ�UFCRFCP�RFC�T?JSCQ�DMSLB�GL�N?PRQ��?�?LB��@�QCCK�AMLQGQRCLR

UGRF�RFC�D?AR�RF?R�RGB?J�CDDCARQ�UGRF�%?PRF�F?TC�A?SQCB�RFC�-MML�RM�PMR?RC

UGRF�MLC�QGBC�?JU?WQ�D?AGLE�%?PRF�

��� 3SNNMQC�WMS�QR?PR�?L�?LRGOSC�A?P�@W�CVCPRGLE�?�DMPAC�MD�����.�ML�GRQ

AP?LI�DMP�������Q��7F?R�?LESJ?P�KMKCLRSK�GQ�EGTCL�RM�RFC�CLEGLC�GD�RFC

F?LBJC�MD�RFC�AP?LI�GQ�������K�DPMK�RFC�NGTMR�?LB�RFC�DMPAC�GQ�CVCPRCB�RM

APC?RC�K?VGKSK�RMPOSC�RFC�CLRGPC�RGKC�

��� !�NJ?WEPMSLB�KCPPW EM PMSLB�F?Q�?�K?QQ�MD�����IE�?LB�?�P?BGSQ�MD

�����K�?LB�GR�GQ�PMR?RGLE�UGRF�?L�?LESJ?P�TCJMAGRW�MD�������PCT�Q��7F?R�GQ

GRQ�?LESJ?P�TCJMAGRW�?DRCP�?����� IE�AFGJB�ECRQ�MLRM�GR�@W�EP?@@GLE�GRQ�MSRCP

CBEC��4FC�AFGJB�GQ�GLGRG?JJW�?R�PCQR�

� � 4FPCC�AFGJBPCL�?PC�PGBGLE�ML�RFC�CBEC�MD�?�KCPPW EM PMSLB�RF?R�GQ����

IE��F?Q�?����� K�P?BGSQ��?LB�GQ�QNGLLGLE�?R������PNK��4FC�AFGJBPCL�F?TC

K?QQCQ�MD�������������?LB������IE��)D�RFC�AFGJB�UFM�F?Q�?�K?QQ�MD������IE

KMTCQ�RM�RFC�ACLRCP�MD�RFC�KCPPW EM PMSLB��UF?R�GQ�RFC�LCU�?LESJ?P

TCJMAGRW�GL�PNK�

��� �?�#?JASJ?RC�RFC�?LESJ?P�KMKCLRSK�MD�?L�GAC�QI?RCP�QNGLLGLE�?R�����

PCT�Q�EGTCL�FGQ�KMKCLR�MD�GLCPRG?�GQ 0.400 kg ⋅ m2���@�(C�PCBSACQ�FGQ

P?RC�MD�QNGL��FGQ�?LESJ?P�TCJMAGRW�@W�CVRCLBGLE�FGQ�?PKQ�?LB�GLAPC?QGLE

FGQ�KMKCLR�MD�GLCPRG?��&GLB�RFC�T?JSC�MD�FGQ�KMKCLR�MD�GLCPRG?�GD�FGQ

?LESJ?P�TCJMAGRW�BCAPC?QCQ�RM������PCT�Q���A�3SNNMQC�GLQRC?B�FC�ICCNQ

FGQ�?PKQ�GL�?LB�?JJMUQ�DPGARGML�MD�RFC�GAC�RM�QJMU�FGK�RM������PCT�Q��7F?R

?TCP?EC�RMPOSC�U?Q�CVCPRCB�GD�RFGQ�R?ICQ������Q�

��� #MLQGBCP�RFC�%?PRF -MML�QWQRCK��#MLQRPSAR�?�NPM@JCK�GL�UFGAF�WMS

A?JASJ?RC�RFC�RMR?J�?LESJ?P�KMKCLRSK�MD�RFC�QWQRCK�GLAJSBGLE�RFC�QNGLQ�MD

RFC�%?PRF�?LB�RFC�-MML�ML�RFCGP�?VCQ�?LB�RFC�MP@GR?J�?LESJ?P�KMKCLRSK

MD�RFC�%?PRF -MML�QWQRCK�GL�GRQ�LC?PJW�KMLRFJW�PMR?RGML��#?JASJ?RC�UF?R

F?NNCLQ�RM�RFC�-MMLwQ�MP@GR?J�P?BGSQ�GD�RFC�%?PRFwQ�PMR?RGML�BCAPC?QCQ�BSC

RM�RGB?J�BP?E��!KMLE�RFC�RFGLEQ�RM�@C�AMLQGBCPCB�?PC�RFC�?KMSLR�@W�UFGAF

RFC�%?PRFwQ�PMR?RGML�QJMUQ�?LB�RFC�D?AR�RF?R�RFC�-MML�UGJJ�AMLRGLSC�RM�F?TC

MLC�QGBC�?JU?WQ�D?AGLE�RFC�%?PRF�

� �� �C@@=G=CBG�C:��LH9B898��C8=9G�=B�-KC��=A9BG=CBG

��� 2CNC?R �L5AD@9�� ��� GL�UFGAF�RFC�BGQI�QRPGICQ�?LB�?BFCPCQ�RM�RFC

QRGAI�������K�DPMK�RFC�L?GJ�

��� 2CNC?R �L5AD@9�� ��� GL�UFGAF�RFC�BGQI�MPGEGL?JJW�QNGLQ�AJMAIUGQC�?R

�����PNK�?LB�F?Q�?�P?BGSQ�MD������AK�

��� 4UGL�QI?RCPQ�?NNPM?AF�MLC�?LMRFCP�?Q�QFMUL�GL �=;IF9�� ��� ?LB

JMAI�F?LBQ���?�#?JASJ?RC�RFCGP�DGL?J�?LESJ?P�TCJMAGRW��EGTCL�C?AF�F?B�?L

GLGRG?J�QNCCB�MD������K�Q�PCJ?RGTC�RM�RFC�GAC��%?AF�F?Q�?�K?QQ�MD������IE�

?LB�C?AF�F?Q�?�ACLRCP�MD�K?QQ�JMA?RCB�������K�DPMK�RFCGP�JMAICB�F?LBQ�

9MS�K?W�?NNPMVGK?RC�RFCGP�KMKCLRQ�MD�GLCPRG?�RM�@C�RF?R�MD�NMGLR�K?QQCQ

?R�RFGQ�P?BGSQ���@�#MKN?PC�RFC�GLGRG?J�IGLCRGA�CLCPEW�?LB�DGL?J�IGLCRGA

CLCPEW�

�=;IF9�� ��� 4UGL�QI?RCPQ�?NNPM?AF�C?AF�MRFCP�UGRF�GBCLRGA?J�QNCCBQ��4FCL��RFC

QI?RCPQ�JMAI�F?LBQ�?LB�QNGL�

��� 3SNNMQC�?������ IE�@?JJ�GQ�RFPMUL�?R������K�Q�RM�?�KMRGMLJCQQ�NCPQML

QR?LBGLE�ML�GAC�UFM�A?RAFCQ�GR�UGRF�?L�MSRQRPCRAFCB�?PK�?Q�QFMUL�GL

�=;IF9�� �� �

�?�#?JASJ?RC�RFC�DGL?J�JGLC?P�TCJMAGRW�MD�RFC�NCPQML��EGTCL�FGQ�K?QQ�GQ�����

IE�

�@�7F?R�GQ�FGQ�?LESJ?P�TCJMAGRW�GD�C?AF�?PK�GQ������IE��9MS�K?W�RPC?R�RFC

@?JJ�?Q�?�NMGLR�K?QQ�?LB�RPC?R�RFC�NCPQML�Q�?PKQ�?Q�SLGDMPK�PMBQ��C?AF

F?Q�?�JCLERF�MD�������K�?LB�RFC�PCQR�MD�FGQ�@MBW�?Q�?�SLGDMPK�AWJGLBCP�MD

P?BGSQ�������K��.CEJCAR�RFC�CDDCAR�MD�RFC�@?JJ�ML�FGQ�ACLRCP�MD�K?QQ�QM

RF?R�FGQ�ACLRCP�MD�K?QQ�PCK?GLQ�GL�FGQ�ECMKCRPGA?J�ACLRCP�

�A�#MKN?PC�RFC�GLGRG?J�?LB�DGL?J�RMR?J�IGLCRGA�CLCPEGCQ�

�=;IF9�� �� 4FC�DGESPC�QFMUQ�RFC�MTCPFC?B�TGCU�MD�?�NCPQML�QR?LBGLE�KMRGMLJCQQ�ML

GAC�?@MSR�RM�A?RAF�?�@?JJ��"MRF�?PKQ�?PC�MSRQRPCRAFCB��!DRCP�A?RAFGLE�RFC�@?JJ��RFC

QI?RCP�PCAMGJQ�?LB�PMR?RCQ�

��� 2CNC?R �L5AD@9�� ��� GL�UFGAF�RFC�QRGAI�GQ�DPCC�RM�F?TC�RP?LQJ?RGML?J

KMRGML�?Q�UCJJ�?Q�PMR?RGML?J�KMRGML�

� �� MFCG7CD=7��::97HG��/97HCF��GD97HG�C:��B;I@5F

&CA9BHIA

��� "BH9;F5H98��CB79DHG

4FC�?VGQ�MD�%?PRF�K?ICQ�?�����\�?LEJC�UGRF�?�BGPCARGML�NCPNCLBGASJ?P�RM

RFC�NJ?LC�MD�%?PRFwQ�MP@GR��!Q�QFMUL�GL �=;IF9�� �����RFGQ�?VGQ�NPCACQQCQ�

K?IGLE�MLC�AMKNJCRC�PMR?RGML�GL��������W�

�?�#?JASJ?RC�RFC�AF?LEC�GL�?LESJ?P�KMKCLRSK�GL�F?JD�RFGQ�RGKC�

�@�7F?R�GQ�RFC�?TCP?EC�RMPOSC�NPMBSAGLE�RFGQ�AF?LEC�GL�?LESJ?P

KMKCLRSK�

�A�)D�RFGQ�RMPOSC�UCPC�APC?RCB�@W�?�QGLEJC�DMPAC��GR�GQ�LMR�?ARGLE�?R�RFC

KMQR�CDDCARGTC�NMGLR�ML�RFC�COS?RMP��UF?R�UMSJB�GRQ�K?ELGRSBC�@C�

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������

�=;IF9�� ��� 4FC�%?PRFwQ�?VGQ�QJMUJW�NPCACQQCQ��?JU?WQ�K?IGLE�?L�?LEJC�MD�����\�UGRF

RFC�BGPCARGML�NCPNCLBGASJ?P�RM�RFC�NJ?LC�MD�%?PRFwQ�MP@GR��4FC�AF?LEC�GL�?LESJ?P

KMKCLRSK�DMP�RFC�RUM�QFMUL�NMQGRGMLQ�GQ�OSGRC�J?PEC��?JRFMSEF�RFC�K?ELGRSBC L GQ

SLAF?LECB�

#(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45- ���

��� #(!04%2����Y�2/4!4)/.!,�-/4)/.�!.$�!.'5,!2�-/-%.45-

4FGQ�AMLRCLR�GQ�?T?GJ?@JC�DMP�DPCC�?R�FRRN���ALV�MPE�AMLRCLR�AMJ���������