Question 386: Centre of Mass and Rotational Motion

Question

A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L / 2$ and radius $R / 3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1 / I_2$ is $\_\_\_\_$ .

Accepted Answer

The moment of inertia of a uniform solid cylinder of mass M and radius R about its central axis is given by :
$$ \mathrm{I}=\frac{1}{2} \mathrm{MR}^2 $$
If area of cross-section of the cylinder is A and length is L then volume is,
$$ V=\pi R^2 L $$
If the density of the cylinder is $ho$ and volume is $V$, then the mass of the cylinder is, $m=ho V$
Putting in the expression for the moment of inertia,
$$ I=\frac{1}{2}\left[ho \pi R^2 Light] 	imes R^2=\frac{\pi ho R^4 L}{2} $$
For the larger cylinder, $ho_1=ho$, the radius is $\mathrm{R}_1=\mathrm{R}$ and the length is $\mathrm{L}_1=\mathrm{L}$
So, the moment of inertia of the larger cylinder is,
$$ I_1=\frac{\pi ho_1 R_1^4 L_1}{2}=\frac{\pi ho R^4 L}{2} $$
For the smaller cylinder, as it is carved from the same material, so it has the same density ( $ho_2=ho$ ). The radius is $\mathrm{R}_2=\mathrm{R} / 3$ and the length is $\mathrm{L}_2=\mathrm{L} / 2$
So, the moment of inertia of the smaller cylinder is,
$$ \mathrm{I}_2=\frac{\pi ho_2 \mathrm{R}_2^4 \mathrm{~L}_2}{2}=\frac{\pi ho\left(\frac{\mathrm{R}}{3}ight)^4\left(\frac{\mathrm{~L}}{2}ight)}{2}=\frac{\pi ho \mathrm{R}^4 \mathrm{~L}}{324} $$
Now, the ratio of the moment of inertia of larger cylinder to the smaller cylinder is,
$$ \frac{I_1}{I_2}=\frac{\left(\frac{\pi ho R^4 L}{2}ight)}{\left(\frac{\pi ho R^4 L}{324}ight)}=\frac{324}{2}=162 $$
Therefore, the value of ratio of moment of inertia is $\frac{I_1}{I_2}=162$.
Hence, the correct answer is 162 .

Centre of Mass and Rotational Motion

A uniform solid cylinder of length L and radius R has moment of inertia about its axis equal to I_1. A small co-centric cylinder of length L / 2 and radius R / 3 carved from this cylinder has moment of inertia about its axis equals to I_2. The ratio I_1 / I_2 is \_\_\_\_ .

Choose the correct answer: