PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 19:
PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 18:
18
10/12/2012 PHY 113 A Fall 2012 -- Lecture 18 1 PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 18: Chapter 10 – rotational motion 1.Torque 2.Conservation of energy including both translational and rotational motion
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PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 18: Chapter 10 – rotational motion Torque Conservation of energy including both translational and rotational motion. Review of rotational energy associated with a rigid body. - PowerPoint PPT Presentation
Transcript of PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 18:
PHY 113 A Fall 2012 -- Lecture 18 110/12/2012
PHY 113 A General Physics I9-9:50 AM MWF Olin 101
Plan for Lecture 18:
Chapter 10 – rotational motion
1. Torque
2. Conservation of energy including both translational and rotational motion
PHY 113 A Fall 2012 -- Lecture 18 210/12/2012
PHY 113 A Fall 2012 -- Lecture 18 310/12/2012
Review of rotational energy associated with a rigid body
iii
iii
iii
iiirot
rmI
Irm
rmvmK
2
222
22
here w21
21
21
21
:energy Rotational
PHY 113 A Fall 2012 -- Lecture 18 410/12/2012
Note that for a given center of rotation, any solid object has 3 moments of inertia; some times two or more can be equal
ji
k
iclicker exercise:Which moment of inertia is the smallest? (A) i (B) j (C) k
d dm m
IB=2md2 IC=2md2IA=0
PHY 113 A Fall 2012 -- Lecture 18 510/12/2012
222222 24222223 mkgxmIi
iiyy
From Webassign:
PHY 113 A Fall 2012 -- Lecture 18 610/12/2012
22
21
21
:object rolling ofenergy kinetic Total
CM
CMrottotal
MvI
KKK
CMvRdtdR
dtds
dtd
: thatNote
22
222
21
21
21
CM
CM
CMrottotal
vMRI
MvRRI
KKK
CM CM
PHY 113 A Fall 2012 -- Lecture 18 710/12/2012
iclicker exercise:Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first?
AB C
2MRI A
22 5.021 MRMRIB
22 4.052 MRMRIC
PHY 113 A Fall 2012 -- Lecture 18 810/12/2012
How to make objects rotate.
Define torque:
t = r x F
t = rF sin
r
F
αarτFraF
Imm
sinr
F sin
Note: We will define and use the “vector cross product” next time. For now, we focus on the fact that the direction of the torque determines the direction of rotation.
PHY 113 A Fall 2012 -- Lecture 18 910/12/2012
Another example of torque:
PHY 113 A Fall 2012 -- Lecture 18 1010/12/2012
clockwise)(counter :T from Torque
)(clockwise :T from Torque
222
2
111
1
TR
TR
t
t
clockwise)(counter 52
(1)(5)(0.5)(15)
: torqueTotal15N 0.5m,
5N 1m, :Example
1122
22
11
Nm .Nm
TRTR
TRTR
t
PHY 113 A Fall 2012 -- Lecture 18 1110/12/2012
Newton’s second law applied to rotational motion
dtdM
dtdm CM
totali
ii
ii
vFvF
Newton’s second law applied to center-of-mass motion
mi Fi
ri
dtdm
dtdm i
iiiii
iivrFrvF
axis) principalabout rotating(for αωτ
rωrτ
rωvFrτ
IdtdI
dtdm
total
iiii
ii
iii
i
iidmI 2
di
PHY 113 A Fall 2012 -- Lecture 18 1210/12/2012
An example:
A horizontal 800 N merry-go-round is a solid disc of radius 1.50 m and is started from rest by a constant horizontal force of 50 N applied tangentially to the cylinder. Find the kinetic energy of solid cylinder after 3 s.
K = ½ I 2 t I a i at = atIn this case I = ½ m R2 and t = FR
R F
JsN
Nmg
tFgRItFt
IFRIIK
Rg
mgItI
FRtIFR
625.275)3(80050m/s8.9
/21
21
21
21
22
222
2
2222
2
aa
PHY 113 A Fall 2012 -- Lecture 18 1310/12/2012
Re-examination of “Atwood’s” machine
T1
T2
T2T1
IT1-m1g = m1a
T2-m2g = -m2a
t =T2R – T1R = I a = I a/R
R
212
12
212
12
/RIg τ
/g a
RImmmm
RImmmm
PHY 113 A Fall 2012 -- Lecture 18 1410/12/2012
Another example: Two masses connect by a frictionless pulley having moment of inertia I and radius R, are initially separated by h=3m. What is the velocity v=v2= -v1 when the masses are at the same height? m1=2kg; m2=1kg; I=1kg m2 ; R=0.2m .
m1 m2v1 v2
hh/2
sm
RImmmmv
hgmhgm
vvmvmghm
UKUK
RI
ffii
/19.02.0/112
12
/
0
:energy ofon Conservati
2
221
21
21
221
1
2212
2212
121
1 2
PHY 113 A Fall 2012 -- Lecture 18 1510/12/2012
Kinetic energy associated with rotation:
i
iirot rmIIK 2221
Distance to axis of rotation
Rolling: rotcomtot KKK
222
1 1
:slipping no is thereIf
comtot
com
vMR
IMK
Rv
Rolling motion reconsidered:
PHY 113 A Fall 2012 -- Lecture 18 1610/12/2012
Newton’s law for torque:
αωτ IdtdItotal
F
fs
FfMRI
IMRFf
IRfaRIaIRf
MafF
s
s
sCMCMs
CMs
312
21
2
2
cylinder, solid aFor
/11
/
a
Note that rolling motion is caused by the torque of friction:
PHY 113 A Fall 2012 -- Lecture 18 1710/12/2012
Bicycle or automobile wheel:
t
fs
τ/Rf
MRI
I/MRτ/Rf
RIaIfMaf
s
s
CMs
CMs
21
For
1
/αR-τ
2
2
PHY 113 A Fall 2012 -- Lecture 18 1810/12/2012
iclicker exercise:What happens when the bicycle skids?
A. Too much torque is appliedB. Too little torque is appliedC. The coefficient of kinetic friction is too smallD. The coefficient of static friction is too smallE. More than one of these
