A ring, a solid cylinder and a solid sphere, all having same mass and radius, are placed at the top of an incline and released. The friction coefficents between the objects and the incline are same and not sufficient to allow pure rolling.
(`a`) Which body will reach the bottom first?
(`b`) Which body will have the minimum kinetic energy at the bottom?

Explanation

Case of impure rolling, `f=f_(max)=mu mg costheta`
`mg sintheta-mu mg costheta=ma`
`a=g sintheta-mu g costheta`
`s=ut+(1)/(2)at^(2)=0+(1)/(2)at^(2)`
`t=sqrt((2s)/(a))`
`v=u+at=0+at=at`
(a) Acceleration will be same for all bodies, hence they will reach at the bottom in same time with same linear kinetic energy.
(b) `tau_(0)=fR=Ialpha`
`alpha=(fR)/(I)`
`omega=omega_(0)+at=0+(fR)/(I)t`
`K_(R)=(1)/(2)Iomega^(2)=(1)/(2)xxI((fRt)/(I))^(2)`
` K_(R)alpha(1)/(I)`
The body with maximum moment of inertia will have minimum rotational kinetic energy at the bottom.
Hence the ring will be with minimum kinetic energy at the bottom.