Question

The cylinders P and Q are of equal mass and length but made of metals with densities \[{\rho_P}\] and ${\rho_Q}$. If their moment of inertia about an axis passing through centre and normal to the circular face be ${I_P}$ and ${I_P}$ then:
(A) \[{I_p}{\text{ }} = {\text{ }}{I_Q}\]
(B) \[{I_p}{\text{ }} < {\text{ }}{I_Q}\]
(C) \[{I_p}{\text{ }} > {\text{ }}{I_Q}\]
(D) \[{I_p}{\text{ }} \geqslant {\text{ }}{I_Q}\]

Answer 1

Hint: Find the relation between density of a body and its moment of inertia. Try to find the relation between mass and density first in terms of radius of cylinder.

Complete step-by-step solution:
As we know that moment of inertia of any body is given by
\[I = M{R^2}\]
Where I is the moment of inertia of the body
M is the mass of the body
R is the radius of mass from the axis of rotation or the radius of the farthest rotating particle in the body
We need to find a relation between I and density of the material, we know that:
\[
  \rho = mV \\
  V = \pi {r^2}h \\
  \]
Substituting this value in the above equation,
\[
  I = \dfrac{\rho }{V}{r^2} \\
  I = \dfrac{\rho }{{\pi {r^2}h}}{r^2} \\
  I = \dfrac{\rho }{{\pi h}} \\
  \]
Therefore we can infer that density is directly proportional to the moment of inertia of a body, so the body having more mass will have more moment of inertia. As the object P has higher density, it will have higher moment of inertia.

Hence, the correct answer is option C.

Note: Whenever you have to find the relation between 2 such quantities which are not directly related to each other, try to find a relation between other existing quantities to the required quantities.