Physics 211

1. Physics 211 10: Angular Momentum and Torque • 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

2. Rolling Motion of a Rigid Body Rotation + Translation If v = r w cm ß Pure Rolling Motion

3. As then one can see that the center of mass moves at v cm by noting that [ ] w = 2 p f w = angular speed angular frequency = where f = frequency of rotation, which has units rps = revolutions per second or cps = cycles per second Which shows that the distance traveled by the center of mass in pure rolling motion in one second is w 2 p r ´ f = 2 p r ´ = r w 2 p

4. Pure rolling motion = no skidding vtop = vcm+ v vcm vbottom = vcm- v 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

5. v= vcm thus for pure rolling motion vtop = vcm+ vcm= 2vcm vbottom = vcm- vcm= 0!!!!!!

6. Skidding = no rotational motion about of center of mass

7. No work done by non conservative forces in pure rolling motion DEtot.mech= 0 DKtot=-DUtot N Fs h W • DUtot =-mgh • DKtot =Ktot,final - Ktot,initial = Ktot,final (as initially at rest) • Ktot = Krot+ Ktrans

8. 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.

9. Vector Product a b ^ a X a b ^ b X a b = a b sin q X q is the angle between a and b

10. Right Hand Rule b axb b a axb a

12. right handed axis system k j i

13. Using the vector product Torque can be written as t = r ´ F r = position vector from an origin to the point of F contact of the force unit vector in direction of rotational axis is ˆ ˆ ˆ r ´ F = t

14. F r Choose an origin then draw the position vector

15. If F is the total force then t is the total torque

16. Properties of Vector Product a b = ab sin q X a ^ a b & b ^ a b X X a a = 0 X a b = - b a X X d d a d b ( ) b + a a b = X X X dt dt dt

18. Angular Momentum L = r ´ p L depends on the choice of origin pt = L = r ´ p = rpt r ´ L = rpt = rmvt = rmr w = mr w = I w 2 ( ) ˆ Þ L = I w w

20. Forces acting on many particles that are rigidly fixed with respect to each other or an extended rigid body

21. Thus if only internal forces act t = 0 = t tot int In general for any system made up of many objects that are fixed with respect to each other or an extended rigid object t = t + t = t tot int ext ext

22. The vector product and the choice of origin defines the axis of rotation of the rigid body

23. Conservation of Total Angular Momentum If a system is isolated d L t = 0 = tot dt tot Conservation of Total Angular Momentum L = L tot,initial tot,final e . g . ( ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ I w w + I w w = I w w + I w w 1 i 1 i 2 i 2 i 1 f 1 f 2 f 2 f