A solid wooden sphere rolls down two different inclined planes of the same height but of different inclinations. (a) Will it reach the bottom with same speed in each case ?
(b) Will it take longer to roll down one inclined plane than other ? Explain.

Explanation

Let `v` be the speed of the solid sphere at the bottom of the incline. Applying principle of conservation of energy, we get
`(1)/(2)m v^(2) + (1)/(2)I omega^(2) = mgh`
As `I = (2)/(5)mr^(2) , (1)/(2)m v^(2) + (1)/(2) ((2)/(5)mr^(2))omega^(2) = mgh` as `r omega = v , (1)/(2)mv^(2) + (1)/(5)mv^(2) = mgh`
`v = sqrt((10)/(7)gh)`
As `h` is same in the two cases, `v` must be the same i.e. it will reach the bottom with the same speed.
As `t prop (1)/(sin theta)`
`:.` The sphere will take longer to roll down the plane with smaller inclination.