Three objects , A ( a solid sphere), B ( a thin circular disc ) and C ( a circular ring), each have the same mass M and radius R. They all spin with the same angular speed `omega` about their own symmetry axes. The amounts of work (W) required to bring them to rest, would satisfy the relation

Options

A. `W_(B) gt W_(A) gt W_(C )`
B. `W_(A) gt W_(B) gt W_(C )`
C. `W_(C ) gt W_(B) gt W_(A)`
D. `W_(A) gt W_(C ) gt W_(B)`

Answer

C

Explanation

The amount of work (W) required to bring the objects = change in kinetic energy (`DeltaE_(k)`)
for the solid sphere,
`W_(A) = (1)/(2) I omega^(2) = (1)/(2) ((2)/(5) MR^(2)) omega^(2) = (1)/(5) MR^(2) omega^(2)`
For the thin circular disc, `W_(B) = (1)/(2) I omega^(2) = (1)/(2) ((2)/(2) MR^(2)) omega^(2) = (1)/(4) MR^(2) omega^(2)`
For the circular ring.
`W_(C) = (1)/(2) I omega^(2) = (1)/(2) ( MR^(2)) omega^(2) = (1)/(2) MR^(2) omega^(2)`
`therefore W_(A) : W_(B) : W_(C ) = (1)/(5) : (1)/(4): (1)/(2) = 4 : 5 : 10 `
Therefore, `W_(C ) gt W_(B) gr W_(A)`