A metal sphere of radius r and specific heat s is rotated about an axis passing through its centre at a speed of n rotation/s. It is suddenly stopped and `50%` of its energy is used in increasing its temperature. Then, the rise in temperature of the sphere is

Options

A. `(2pi^(2) v^(2) r^(2))/(5s)`
B. `(pi^(2) v^(2))/(10 r^(2) s)`
C. `(7)/(8) pi r^(2) v^(2)s`
D. `(5pi^(2) r^(2) v^(2))/(14s)`

Answer

A

Explanation

Moment of inertia of the sphere, `I= (2)/(5) mr^(2)`
Rotational kinetic energy `=(1)/(2) I omega^(2) = (1)/(2) xx (2)/(5) mr^(2) xx (2pi v)^(2)`
`=(4)/(5) m pi^(2) r^(2) v^(2)`
As half of this energy is converted into heat, so
`Delta Q = (1)/(2) ((4)/(5) m pi^(2) r^(2) v^(2)) = (2)/(5) m pi^(2) r^(2) v^(2)` ...(i)
`:.` Specific heat, `s= (1)/(m) (Delta Q)/(Delta T)`
or `Delta T = (Delta Q)/(ms) = ((2)/(5) m pi^(2) r^(2) v^(2))/(ms) = (2pi^(2) r^(2) v^(2))/(5s)` (Using (i))