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

Options

A. the speed and the of descend will be same.
B. the speed will be same but time of descend will be different.
C. the speed will be different but time of descend will be same.
D. speed and time od descend both are different.

Answer

B

Explanation

(b) Speed of the rolling body at the bottom of inclined plane is.
`v = sqrt((2 gh)/(1 + (k^2)/(R^2)))`
Where `h` is height of the inclined plane, `k` and `R` be radius of gyration and radius of the body respectively.
For solid sphere, `(k^2)/(R^2) = (2)/(5)`
:. `v = sqrt((2 gh)/(1 + (2)/(5))) = sqrt((10)/(7) gh)`
As `h` is same in both the cases, therefore speed will be same in both cases.
Time of descend,
`t = (1)/(sin theta) sqrt((2h)/(g)(1 + (k^2)/(R^2))`
Where `theta` is angle of inclination ?
For solid sphere, `(k^2)/(R^2) = (2)/(5)`
:. `t = (1)/(sin theta) sqrt((14 h)/(5 g))`
As `h` is same but `theta` is different in both the cases, hence time of descend will be different in both the cases.