A ring (`R`), a disc (`D`), a solid sphere (`S`) and a hollow sphere with thin walls (`H`), all having the same mass but different radii, start together from rest at the top of inclined plane and roll down without slipping. Then

Options

A. all of them will reach the bottom of the incline together
B. the body with the maximum radius will reach bottom first
C. they will reach the bottom in the order `S, D, H` and `R`
D. all of them will have the same kinetic energy at bottom of the incline

Answer

C, D

Explanation

In rolling without slipping no work is done against friction. Hence, loss in gravitational potential energy of a body is equal to its total kinetic energy, i.e., linear plus rotational kinetic energies.
Also `v=omegar`
Total kinetic energy `=1/2mv^(2)+1/2Iomega^(2)`
`=1/2 mv^(2)+1/2(mK^(2))((v^(2))/(r^(2)))`
`=1/2mv^(2)[1+(K^(2))/(r^(2))]`
Where `K=`radius of gyration
As the bodies have the same final total kinetic energy, their final velocities will depend upon the ratio `K/r` Bodies with the smaller value of `K/r` will have the greater `v` and hence will reach the bottom earlier.