A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length . Which one of the two objects gets to the bottom of the plane first ?

Options

A. Both reach at the same time
B. Depends on their masses
C. Disk
D. Sphere

Answer

D

Explanation

Time taken by the body to reach the bottom when it rolls down on an inclined plane without slipping is given by
`t = sqrt((2l (1 + (k^2)/(R^2)))/(g sin theta))`
Since g is constant and l , R and `sin theta` are same for both
`therefore (t_(d))/(t_(s)) = (sqrt(1 + (k_(d)^(2))/(R^(2))))/(sqrt(1 + (k_(s)^(2))/(R^(2)))) = sqrt((1 + (R^2)/(2R^2))/(1 + (2R^2)/(5R^2))) ( because k_(d) = (R)/(sqrt2) , k_(s) = sqrt((2)/(5)) R)`
`= sqrt((3)/(2) xx (5)/(7)) = sqrt((15)/(14)) implies t_(d) gt t_(s)`
Hence , the sphere gets to the bottom first .