Question 645: Centre of Mass and Rotational Motion

Question

A disc 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?

Accepted Answer

Time taken by the body to reach the bottom
when it rolls down on an inclined plane without
slipping is given by
$$t = \sqrt {{{2l\left( {1 + {{{k^2}} \over {{R^2}}}} ight)} \over {g\sin 	heta }}} $$
Since g is constant and $$l$$, R and sin $$	heta $$ are same for
both
$$ 	herefore $$ $${{{t_d}} \over {{t_s}}} = {{\sqrt {1 + {{k_d^2} \over {{R^2}}}} } \over {\sqrt {1 + {{k_s^2} \over {{R^2}}}} }} = \sqrt {{{1 + {{{R^2}} \over {2{R^2}}}} \over {1 + {{2{R^2}} \over {5{R^2}}}}}} $$
$$ \left(\because {{k_d} = {R \over {\sqrt 2 }},{k_s} = \sqrt {{2 \over 5}} R} ight)$$
$$ \Rightarrow \sqrt {{3 \over 2} 	imes {5 \over 7}}  = \sqrt {{{15} \over {14}}}  \Rightarrow {t_d} > {t_s}$$

Answer

Both reach at the same time

Answer

Depends on their masses

Answer

Disc

Answer

Sphere

Centre of Mass and Rotational Motion

A disc 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?

Choose the correct answer: