A solid sphere rolls down two different inclined planes of the same height but of different inclinations

Explanation

Let v is the speed of solid sphere rolling down. Using law of conservation of energy we get
`(1)/(2)mv^(2)+(1)/(2)I omega^(2)= mgh`
`(1)/(2) mv^(2)+(1)/(2)((2)/(5)mr^(2)) omega^(2) mgh`
`(1)/(2) mv^(2)+(1)/(5) mv^(2)=mgh` Mbr> `((7)/(10))mv^(2)=mgh`
`v= sqrt((10)/(7)gh)`
As h is same for both the inclined planes. Therefore, it will reach the bottom with same speed in both planes.
However, acceleration of a rolling sphere
`a=(sin theta)/((1+(k^(2))/(R^(2))))`
So, the sphere will take longer time in case of a plane having smaller inclination