Experimental Physics EP1 MECHANICS -Rotation. Basics · Experimental Physics -Mechanics -Basics of...

Experimental Physics - Mechanics - Basics of rotations 1

Experimental Physics EP1 MECHANICS

- Rotation. Basics -

Rustem Valiullinhttps://bloch.physgeo.uni-leipzig.de/amr/


Some basics

vdtds =

q

dsqd

0>w

0<w

r

qrdds =

wqº=rv

dtd

# Along line Rotational

1 v = v0+at w = w0+at2 x=x0+v0t+at2/2 q = q0+w0t+at2/2

3 v2=v02+2a(x-x0) w2 = w02+2a(q-q0)

4 x=x0+(v+v0)t/2 q = q0+(w +w0)t/2

5 x=x0+vt-at2/2 q = q0+wt-at2/2

- angular velocity

awqº=÷

øö

çèæ

dtd

dtd

dtd - angular acceleration

constant angular acceleration

ra=a


Some more basics

wqº=rv

dtd

O

v!

||v^v

r!

j 2sinsinr

rvr

v jjw ==

initialfinal rrr !!!-=D ?--=D initialfinal qqq

1

2

3 3 1

2

3 2 2

One of the properties of vectors is not reproduced.abba !!!!+=+

2rvr !!! ´

=w - it is a vector! (right-hand rule)´

13


Kinetic energy of rotation

ò=Þ=body

kk dmrEdmvdE 22212

21 w

÷ø

öçè

æºº åòi

iibody

rmdmrI 22 moment of inertiarotational inertia

221

,2

21

, mvEIE trkrotk =Û= w

2212

22

21 vMvvmE

iiM

mM

iiik

i === åå

( )òò ===RR

rot MRdrrr

Mdmv

Mv

0

420

02

2

0

22

221 wprrpw

2241 RMEk w=

221 MRI =

- the same result!dm

rRv


Moment of inertia

Depends on the rotation axis!a

h

b

2MRIa =òò === 222 rMMdmrMdmrI

Þ=++ 222 )( Rhxy

h R

xhhRr 2222 --=

( ) 2222

2

2 221 hRdxhxhRR

dx

dxrr

hR

hRhR

hR

hR

hR +=--== òò

ò -

---

--

-

--

22 MhMRIb +=

22 MhMRI CM +=CM hThe parallel-axis theorem.


CM

CM

CM

CM

CM

CM

CM

CM

Rotation seen from two reference systems

Rotating of an object around an arbitrary point is accompanied by the respective rotation around its center of mass.


Perpendicular-axis theorem

22 rMdmrI == ò

h Rj

w

2cos2)cos(

22/

0

222 MRdMRdRdMI === òòò

p

jjp

jjj

( ) cd ++=ò jjjjj sincoscos 212

x

y

z

yxz III += The perpendicular-axis theorem.

dmrIz ò= 2

dmyxIz ò += )( 22

x

yz

rxy

Experimental Physics - Mechanics - System of Particles - Rotational Motion II 8

Rolling motion

CMv

CMv

CMv

0=CMv

Rv w=

Rv w=CMCM vRvv 2=+= w

CMv

0=v

( ) 22222

21

21

21

21 MvIMRIIE CMCMyk +=+== www

22

2

21

21 Mv

RvIMgh CM +=

Tran

slatio

nRo

tatio

nRo

lling ( )2

2

/12MRI

ghvCM+

=

ghvsphere 710=

j

s

swj RdtdR

dtdsvCM ===


Ø Rotational motion is similar to one-dimensional

translational motion.

Ø Moment of inertia is an analogue of mass, which

might be considered as resistance to rotation.

Ø For moment of inertia of planar objects

the parallel-axis and perpendicular-axis

theorems can be applied.

ØFor rolling motion velocity of a selected

point is the sum of that of the center of mass

and of the point in the center of mass frame.

To remember!


Torque

iiiiti lmamF a== å å= 2iitii lmFl a

F!

F!

'l

a

rF tF

im

Inet at =Torquelevel arm

M

R

RT

mg

t

IRaImgR === at

2232 MRMRII CM =+=

32ga =

mgmamgT32

-=-=-

mgT31

=


Rolling motion

CMv

CMv

CMv

0=CMv

Rv w=

Rv w=CMCM vRvv 2=+= w

CMv

0=v

( ) 22222

21

21

21

21 MvIMRIIE CMCMyk +=+== www

22

2

21

21 Mv

RvIMgh CM +=

Tran

slatio

nRo

tatio

nRo

lling ( )2

2

/12MRI

ghvCM+

=

ghvsphere 710= qsin7

5 gasphere =

j

s

swj RdtdR

dtdsvCM ===


Angular momentum

r!v! wIL =prmvr

rvmrIL ==== 2w

r!v!

q r̂

prvrmmvrL !!!!´=´== )(sinq

å å === ww IrmLL iii2

dtdL

dtId

dtdIInet ====

)( wwat

If then L = const0=nettConservation of angular momentum

The net external torque = the rate of change of L.


Conservation of angular momentum

iv102 =iv

cmV

2)(22 2

11

21

1

21

,1,

iikfk

vmmm

mmmEm

E+

=+

=

i1w1I

2I 02 =iwfw

1

212

111, 221

ILIE i

iik == w ( ) ( ) ikf

ffk EIII

IIL

IIE ,21

1

21

22

21, 21

21

÷÷ø

öççè

æ+

=+

=+= w

m1096.6 8´=SunRd3.25=SunT

m100.5 3´=nsR

2KmRI =T

KmRL p22=

222

÷÷ø

öççè

æ=Þ=

Sun

nsSunns

ns

ns

Sun

Sun

RRTT

TR

TR

s1012.1 4-´»


f1w1I

2I

f2w

Conservation of angular momentum

i1w1I

2I

02 =iw

iL1!

fL1!

iti LL 1

!!= fifftf LLLLL 2121

!!!!!+-=+= if LL 12 2

!!=Þ

fL2!

10/ 12 »mms1.01 »T

)(! m1m5.0

2

1

»»

RR

1

211

2

2221 2

2)(

TRm

TRmm

=+ s1

4)(211

2221

12 »+

=RmRmmTT

There are only internal forces acting within the system. To turn the wheel aroundwe have to apply force, i.e., torque. This will be balanced by the reaction torque.


O r!

gm!

Motion of a wheel

T!

( ) vdtrdm

dtvdrm

dtvrdm !

!!!

!!

´+´=´

=

´

t! dtpdrarm!

!!!!´=´=t

w

L!

dtLd t!!= precession

jt LdrmgdtdtdL ===

j

Lrmg

dtd

p ==jw

Nuclear Magnetic Resonance

dtLd!

!=t

Experimental Physics - Mechanics - Static Equilibrium 16

Examples of precessional motion


Static and dynamic imbalance

dtLd!

!=tL

! w!

)( vrmL !!!´=

w!

r!

v!

L!

w!

r!

L! w

!

h

hmhmvF 2

2

w==

w!

r!

v!

L! L

!


ØTorque is the product of the tangential component of the

force and the level arm.

ØThe angular momentum is the cross-product of the radius

vector and the linear momentum.

Ø It is fundamental property that the angular momentum is

always conserved.

Ø Precession is a reaction of the angular momentum

to a net torque applied perpendicularly.

Ø A body is dynamically imbalanced when the

angular velocity and momentum are not

parallel to each other.

To remember!

Experimental Physics EP1 MECHANICS -Rotation. Basics · Experimental Physics -Mechanics -Basics of...

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