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

Options

A. the speed and time 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 are descend will be same.
D. speed and time of descend both are different..

Answer

B

Explanation

Speed of the rolling body at the bottm on of inclined plance is
`v= sqrt((2gh)/(1+k^(2)/R^(2))`
Where h is the 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`
`therefore v=sqrt((2gh)/(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/(sintheta) sqrt((2h)/g+k^(2)/R^(2))`
Where `theta` is the angle of inclination.
`therefore t=1/(sintheta) sqrt((14h)/(5g)` `(therefore k^(2)/R^(2) = 2/5)`
As h is same but `theta` is different in both the cases, hence time of descend will be different in both the cases.