Question 425: Centre of Mass and Rotational Motion

Question

A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is $$\sqrt{\frac{2}{x}}$$. The value of $$x$$ is ___________.

Accepted Answer

Given that the radii of gyration for the ring and the solid sphere are equal, we have:
$$
K_1 = K_2
$$
For the ring, the moment of inertia is:
$$
I_{ring} = mR_1^2 = mK_1^2
$$
Thus, the radius of gyration for the ring is:
$$
K_1 = R_1
$$
For the solid sphere, the moment of inertia is:
$$
I_{sphere} = \frac{2}{5}m'R_2^2 = m'K_2^2
$$
Hence, the radius of gyration for the solid sphere is:
$$
K_2 = \sqrt{\frac{2}{5}}R_2
$$
Since the radii of gyration are equal:
$$
R_1 = \sqrt{\frac{2}{5}}R_2
$$
Therefore, the ratio of the radius of the ring to that of the sphere is:
$$
\frac{R_1}{R_2} = \sqrt{\frac{2}{5}}
$$
So, the value of $$x$$ is:
$$
x = 5
$$

Centre of Mass and Rotational Motion

A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is $\sqrt{\frac{2}{x}}. The value of x$ is ___________.

Choose the correct answer: