Physics 211

•Rolling Motion of a Rigid Body•Vector Product and Torque•Angular Momentum•Rotation of a Rigid Body about a Fixed Axis•Conservation of Angular Momentum

10: Angular Momentum and Torque

Rolling Motion of a Rigid Body

Rotation + Translation

If v

cm r

Pure Rolling Motion

As then one can see that the center of mass moves at v cm

by noting that 2f angular frequency=angular speed

where f frequency of rotation, which has unitsrps=revolutions per second

or cps =cycles per second

Which shows that the distance traveled by the center ofmass in pure rolling motion in one second is

2r f2r2

r

vcm

vtop = vcm+ v

vbottom = vcm- v

Pure rolling motion = no skidding

v = linear speed of rim of wheel just due to the rotational motion; vtop & vbottom are the total linear speeds at the top and bottom

v = vcm thus for pure rolling motionvtop = vcm+ vcm= 2vcm

vbottom = vcm- vcm= 0!!!!!!

Skidding = no rotational motion about of center of mass

Kinetic Energy of moving wheelKtot Krot Ktrans

Ktot 12I cm

2 12Mvcm

2

N

W

Fs

No work done by non conservative forces in pure rolling motion

tot.mech=

Ktot= - Utot

h

Utot = -mghKtot = Ktot,final - Ktot,initial = Ktot,final (as initially at rest)Ktot = Krot+ Ktrans

As there is no slipping (skidding) the rolling object only experiences STATIC friction with the surface

Static friction can NEVER do any work

At each instant of time the static friction stops translational motion and causes rotation

[The static friction does not have to be the maximum possible static friction (i.e. it can be ]

The static friction produces an instantaneous torque on the portion of the object in contact with the surface

This torque does NO work as it is only in contact with the SAME portion of the object for an infinitesimal time.

)s N

Vector Producta X b aa b b

a b a b sin

is the angle between a and b

X

X

axb

b

a

b

a

axb

Right Hand Rule

Vector Product

a b i j kax ay azbx by bz

b a

a ybz azby i a xbz azbx j axby aybx k

i j i j k1 0 00 1 0

k

j k i j k0 1 00 0 1

i

i k i j k1 0 00 0 1

j

j

k

i

right handed axis system

Using the vector product Torque can be written as rF

r = position vector from an origin to the point of contact of the force F

unit vector in direction of rotational axis is

ˆ r ˆ F =ˆ

F

rˆ r ˆ F ˆ

Choose an origin then draw the position vector

If F is the total force then is the total

torque

Properties of Vector Producta X b absin

aa X b & ba ba a 0

a b b addt

a b dadt

b a dbdt

X

X

X

X

X X X

As 0

ddt

ddt

ddt

pτ r F r

r p

r p v p

Angular Momentum

L rpL depends on the choice of origin

L rp rpt

L rptrmvtrmr mr 2 I L I ̂

rpt

ddt

ddt

τ r p

Lτ r F

.Tot

Tot cm Tot cm Tot External

Tot kk

Tot kk

ddt

Lτ r F r F

τ τ

L L

Forces acting on many particles that are rigidlyfixed with respect to each other or an

extended rigid body

Thus if only internal forces act

tot 0 int

In general for any system made upof many objects that are fixed with respect to each other or an extended rigid object

totint extext

The vector product and the choice of origin

defines the axis of rotation of the rigid body

.

is the angular acceleration of the center of mass

about the axis determined by thechoice of origin

TotTot cm Tot cm Tot External

cm

ddt

I

Lτ r F r F

αα

Conservation of Total Angular Momentum

If a system is isolated

tot

0 dL

tot

dtConservation of Total Angular Momentum

L tot,initial L tot,final

e.g.

I1i 1i ˆ I2i 2 i ˆ I1 f 1 f ˆ I2 f 2 f ˆ