A disc and a solid 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. Sphere
B. Both reach at the same time
C. Depends on their masses
D. Disc

Answer

A

Explanation

Acceleration of an object rolling down an inclined plane is given by
`a=(g sin theta)/(1+I//mr^(2))`
where, `theta=` angle of inclination of the inclined plane
m = mass of the object
and `I=` moment of inertia about the axis through centre of mass
For disc, `(I)/(mr^(2))-(1//2mr^(2))/(mr^(2))=(1)/(2)`
For solid sphere, `(I)/(mr^(2))=(2//5mr^(2))/(mr^(2))=(2)/(5)`
For hollow sphere, `(I)/(mr^(2))=(2//3mr^(2))/(mr^(2))=(2)/(3)`
`:. a_("disc")=(g sin theta)/(1+(1)/(2))=(2)/(3)g sin theta`
`= 0.66 g sin theta`
`a_("solid sphere")=(g sin theta)/(1+(2)/(5))=(5)/(7)g sin theta=0.71 g sin theta`
`a_("hollow sphere") =(g sin theta)/(1+(2)/(3))=(3)/(5)g sin theta =0.6 g sin theta`
Clearly, `a_("solid sphere")gt a_("disc") gt a_("hollow sphere")`
Type of sphere is not mentioned in the question. Therefore, we will assume the given sphere as solid sphere
`:. a_("solid sphere")=a_("hollow sphere")gt a_("disc")`.