When a uniform solid sphere and a disc of the same mass and of the same radius roll down an inclined smooth plane from rest to the same distance, then ratio of the time taken by them is

Options

A. `15^(2):14^(2)`
B. `15:14`
C. `sqrt(14):sqrt(15)`
D. `14:15`

Answer

C

Explanation

For the solid sphere `I=(2)/(5)MR^(2)` (about the diameter)
and for the disc, `I=(MR^(2))/(2)` (about its geometrical axis)
The time taken by a rolling body to reach the bottom is given by
`t=(1)/(sin theta)sqrt((2h)/(g)(1+(K^(2))/(R^(2))))`
`therefore(t_(s))/(t_(d))=sqrt(((1+(K^(2))/(R^(2)))_("sphere"))/((1+(K^(2))/(R^(2)))_("disc")))=sqrt((1+(2)/(5)(R^(2))/(R^(2)))/(1+(1)/(2)(R^(2))/(R^(2))))`
`=sqrt(((7)/(5))/((3)/(2)))=sqrt((7xx2)/(5xx2))=sqrt((14)/(15))`