Question 540: Centre of Mass and Rotational Motion

Question

Ratio of radius of gyration of a circular disc to that of circular ring each of same mass and radius around their respective axes is

Accepted Answer

The radius of gyration ($k$) is related to the moment of inertia ($I$) and the mass ($M$) of an object by the formula:
$$ k = \sqrt{\frac{I}{M}} $$
For a circular disc:
Moment of inertia ($I_{	ext{disc}}$) about its central axis is given by:
$$   I_{	ext{disc}} = \frac{1}{2} M R^2   $$
Radius of gyration ($k_{	ext{disc}}$) is:
$$   k_{	ext{disc}} = \sqrt{\frac{I_{	ext{disc}}}{M}} = \sqrt{\frac{\frac{1}{2} M R^2}{M}} = \sqrt{\frac{1}{2}} R   $$
For a circular ring:
Moment of inertia ($I_{	ext{ring}}$) about its central axis is given by:
$$   I_{	ext{ring}} = M R^2   $$
Radius of gyration ($k_{	ext{ring}}$) is:
$$   k_{	ext{ring}} = \sqrt{\frac{I_{	ext{ring}}}{M}} = \sqrt{\frac{M R^2}{M}} = R   $$
Ratio of the radii of gyration:
The ratio $k_{	ext{disc}} : k_{	ext{ring}}$ is:
$$   \sqrt{\frac{1}{2}} R : R = \frac{\sqrt{2}}{2} R : R = \frac{\sqrt{2}}{2} : 1 = \frac{1}{\sqrt{2}} : 1 = 1 : \sqrt{2}   $$
Therefore, the ratio of the radius of gyration of a circular disc to that of a circular ring, each with the same mass and radius, is:
Option D
: $1: \sqrt{2}$

Answer

$\sqrt{2}: 1$

Answer

$\sqrt{2}: \sqrt{3}$

Answer

$\sqrt{3} : \sqrt{2}$

Answer

$1: \sqrt{2}$

Centre of Mass and Rotational Motion

Ratio of radius of gyration of a circular disc to that of circular ring each of same mass and radius around their respective axes is

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