Question

The position of the axis of rotation of a body is changed so that its moment of inertia decreases by $36\%$. Find the percentage change in its radius of gyration.
A) It decreases by $18\%$
B) It increases by $18\%$
C) It decreases by $20\%$
D) It increases by $20\%$

Answer 1

The radius of gyration, $k = \sqrt {\dfrac{I}{M}}$ . Let, the initial moment of inertia is ${I_1} = M{k_1}^2$ and the final moment of inertia is ${I_2} = M{k_2}^2$. Also, ${k_2} = 0.8{k_1}$ and ${k_1} > {k_2}$.

Complete step by step solution:
Moment of inertia of a body about an axis of rotation is defined as the torque acting on the body divided by the corresponding angular acceleration thus generated about the same axis of rotation. So, moment of inertia not only depends on the mass of a body, but also depends on the distance of the particles constituting the body from the axis of rotation, i.e., on the distribution of mass of the body. Now, if the whole mass of a body is assumed to be concentrated at a point such that the moment of inertia of the whole body equals the moment of inertia of that point, then the distance of the point from the axis of rotation is called the radius of gyration. So, if the mass of an extended body is $M$ and its moment of inertia about an axis of rotation is $I$, then the radius of gyration, $k = \sqrt {\dfrac{I}{M}}$ or, $I = M{k^2}$
Let, the initial moment of inertia is ${I_1} = M{k_1}^2$ and the final moment of inertia is ${I_2} = M{k_2}^2$
According to the question, the initial moment of inertia decreases by $36\%$
$\therefore {I_2} = {I_1} - \dfrac{{36{I_1}}}{{100}} = {I_1} - 0.36{I_1} = 0.64{I_1}$
So, ${I_2} = 0.64{I_1}$ or, $M{k_2}^2 = 0.64M{k_1}^2$ or, ${k_2}^2 = 0.64{k_1}^2$ or, ${k_2} = 0.8{k_1}$ [So, ${k_1} > {k_2}$ ]
So, the change in radius of gyration is $\dfrac{{{k_1} - {k_2}}}{{{k_1}}} \times 100 = \dfrac{{{k_1} - 0.8{k_1}}}{{{k_1}}} \times 100 = \dfrac{{0.2{k_1}}}{{{k_1}}} \times 100 = 20$

So, the radius of gyration decreases by $20\%$.

Note: In case of rotational motion, a body is compelled to change its state of motion, when an external torque acts on it. In absence of external torque, the body either remains at rest or executes uniform circular motion. It means that the moment of inertia of a body can be called its rotational inertia. It resists any change in the rotational motion of the body and it is clear that more the moment of inertia of a body about an axis, more the torque necessary to rotate the body about that axis.