Three bodies, a ring, a solid cylinder and a solid sphere roll down the same inclinedplane without slipping They - Physics - System Of Particles And Rotational Motion

Question

Three bodies, a ring, a solid cylinder and a solid sphere roll down
the same inclinedplane without slipping They start from rest The
radius of the bodies are identical Which body has the greatest
rotational kinetic energy while reaching the bottom ofthe inclined
plane?

Govind · Accepted Answer

We know that energy of a rolling body is conserved So,

Potential energy lost by the body = kinetic energy gained

As the body starts from rest, the kinetic energy gained will be
equal to the final kinetic energy of the body

Kinetic energy of a rolling bodyK is given byK=12mv21+k2R2     
1Here, v is the final velocity of the body
k=radius of gyrationEquation the above equation with the potential energy lost
 We havemgh=12mv21+k2R2⇒v2=2gh1+k2R2For a ring, k2=R2vring=2gh1+1=ghFor solid cylinder, k2=R22∴ vdisc=2gh1+12=4gh3For a solid sphere, k2=2R25∴ vsphere=2gh1+25=10gh7Thus, sphere has the greatest and ring has the least velocity of the centre mass at the bottom of the inclined plane

Three bodies, a ring, a solid cylinder and a solid sphere roll down the same inclinedplane without slipping. They start from rest. The radius of the bodies are identical.Which body has the greatest rotational kinetic energy while reaching the bottom ofthe inclined plane?