Rolling Motion

of a Rigid Object

AP Physics C

Rolling Motion: a combination of pure translation and pure rotation.

For pure rolling motion there is “rolling without slipping”, so at

point P vp =0.• All points

instantaneously rotate about the point of contact between the object and the surface (P).

vp’ = 2 vcm

Rolling Motion

CMds dv R Rdt dt

CMCMdv da R Rdt dt

Speed and Acceleration of the CM of a Rolling Object

vcm = ωR

acm = α R

Red Line: Path of a particle on a rolling object (cycloid)

Green line: Path of the center of mass of the rolling object

http://cnx.org/content/m14374/latest/

The Total Kinetic Energy of a Rolling Object is the sum of the rotational and the translational

kinetic energy.

K = ½ ICM ω2 + ½ MvCM2

Note

• Rolling is possible when there is friction between the surface and the rolling object.

• The frictional force provides the torque to rotate the object.

Ex: Accelerated Rolling MotionKi + Ui = Kf + U f

Mgh = ½ ICM ω2 + ½ MvCM2

vcm = ωR

There is no frictional work. Why not?

Does friction cause a displacement at its point of action?

Ex: #52A bowling ball (on a horizontal surface) has a mass M, radius R, and a moment of inertia of (2/5)MR2 . If it starts from rest, how much work must be done on it to set it rolling without slipping at a linear speed v? Express the work in terms of M and v. Hint: use kinetic energy theorem.Ans: (7/10)Mv2

Ex: #54• A uniform solid disk and a uniform hoop are

placed side by side at the top of an incline of height h. If they are released from rest and roll without slipping, which object reaches the bottom first? Verify your answer by calculating their speeds when they reach the bottom in terms of h. Use conservation of energy

• Ans: The disk, vdisk =(4gh/3)1/2 , vring =(gh)1/2

Sample Problem

A solid sphere of mass M and radius R rolls from rest down a ramp of height h and angle . Use Conservation of Energy to find the linear acceleration and the speed at the bottom of the ramp.

710ghv sin)7/5( ga

Sample Problem

• A solid sphere of mass M and radius R rolls from rest down a ramp of length L and angle q. Use Rotational Dynamics to find the linear acceleration and the speed at the bottom of the ramp.

710ghv sin)7/5( ga