Question

The radius of gyration of a rotating metallic disc is independent of which of the following physical quantities.
(A) Position of axis of rotation
(B) Mass of disc
(C) Radius of disc
(D) Centre of mass disc

Answer 1

Hint: We know that Radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated. It is related to the moment of inertia and the total mass of the body. Therefore, the radius of gyration is the distance from the axis of a mass point whose mass is equal to the mass of the whole body and whose moment of inertia is equal to the moment of inertia of the body about the axis.

Complete step-by step answer:
We know that:
$I=m{k}^2$
In the above expression,
I is the moment of Inertia
m is mass
k is the constant
Now we can write that:
$k=\sqrt{{\displaystyle \frac{T}{m}}}$
$\Rightarrow I={\displaystyle \frac{m{R}^2}{2}}$
Now we can evaluate to get:
$k={\displaystyle \frac{R}{\sqrt{2}}}$
So, we can write that:
$r=R_0(1+\alpha \mathrm{\Delta }T)$
Therefore, k is independent of mass. $r=R_0(1+\alpha \mathrm{\Delta }T)$
Hence, the radius of gyration of a rotating metallic disc is independent of which of the following physical quantities is the mass of the disc.

Hence, the correct answer is Option B.

Note: We know that moment of inertia is defined as the ratio of the net angular momentum of a system to its angular velocity around a principal axis, that is. If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. Rotational inertia is important in almost all physics problems that involve mass in rotational motion. It is used to calculate angular momentum and allows us to explain (via conservation of angular momentum) how rotational motion changes when the distribution of mass changes.
It should be known to us that higher moments of inertia indicate that more force has to be applied in order to cause a rotation whereas lower moments of inertia means that only low forces are necessary. Masses that are further away from the axis of rotation have the greatest moment of inertia.