Question

A solid cylinder, a solid sphere, and a hollow sphere each of mass $m$ and radius $r$ are released from the top of a smooth inclined plane. Then which of the bodies has minimum acceleration down the plane?
(A) solid cylinder
(B) solid sphere
(C) hollow sphere
(D) all will have the same value

Answer 1

Hint: In this problem, first we have to find the moment of inertia for solid cylinder, solid sphere, and the hollow sphere. By using the relation between the moment of inertia and the acceleration, then which body has minimum acceleration down the plane can be determined.

Formula used:
Moment of inertia for solid sphere,
$I = \dfrac{2}{5}m{r^2}$
Where $I$ is the moment of inertia, $m$ is the mass of the object and $r$ is the radius of the object.
Moment of inertia for solid cylinder,
$I = \dfrac{1}{2}m{r^2}$
Where $I$ is the moment of inertia, $m$ is the mass of the object and $r$ is the radius of the object.
Moment of inertia for a hollow sphere,
$I = \dfrac{2}{3}m{r^2}$
Where $I$ is the moment of inertia, $m$ is the mass of the object and $r$ is the radius of the object.

Complete step by step answer:
Given that,
A solid cylinder, a solid sphere, and a hollow sphere each of mass, $m$
A solid cylinder, a solid sphere, and a hollow sphere each of radius, $r$
It is given that the three objects are having the same mass and same radius, so in calculating the moment of inertia, the $m{r^2}$ value is the same for all the three objects, so eliminate this term in the calculation, then
Moment of inertia for the solid sphere,
$I = \dfrac{2}{5}m{r^2} $
$\Rightarrow I= 0.4m{r^2}\,...............\left( 1 \right)$
Moment of inertia for solid cylinder,
$I = \dfrac{1}{2}m{r^2}$
$\Rightarrow I = 0.5m{r^2}\,..............\left( 2 \right)$
Moment of inertia for hollow sphere,
$I = \dfrac{2}{3}m{r^2} $
$\Rightarrow I= 0.66m{r^2}\,..............\left( 3 \right)$
From the three equations, equation (1), equation (2), and equation (3), the moment of inertia for the hollow sphere is maximum.
Already we know that the moment of inertia is inversely proportional to the acceleration, so the acceleration is minimum for the hollow sphere.

$\therefore$ The moment of inertia for the hollow sphere is maximum. Hence, option (C) is the correct answer.

Note:
In equation (1), equation (2), and equation (3), the mass and radius values are the same, so these terms are eliminated in all the three equations. These values are eliminated because, in this problem the exact value is not required, the required is the maximum value among the three equations. If the mass and radius values are multiplied with these equations, then also the value of equation (3) is high.