Show that the rotational kinetic energy of a ball rolling over a horizontal plane is 2/7 of its total kinetic - Physics -

Question

Show that the rotational kinetic energy of a ball rolling over a
horizontal plane is 2/7 of its total kinetic energy

Somnath Mazumder · Accepted Answer

Let the radius of the ball be r and its mass be m and it is moving rolling without slipping with a velocity v in thehorizontal table
 Let the ball be uniform solid
    The total KE of the ball about the CM is given byKEtot=12mv2+12Iω2 for uniform solid ball I=25mr2=>KEtot=12mv2+1225mr2ω2For rolling without slipping condition v=rωKEtot=12mv2+15mr2(vr)2=710mv2The only rotational KE of the ball is given by KErot=12Iω2=15mv2KErotKEtot= 15mv2710mv2=27

Show that the rotational kinetic energy of a ball rolling over a horizontal plane is 2/7 of its total kinetic energy.