Question

The moment of inertia of a hollow cylinder of mass M and radius R about its axis of rotation is \[M{R^2}\]. The radius of gyration of the cylinder about the axis is
A.\[\dfrac{R}{{\sqrt 2 }}\]
B.\[\sqrt 2 R\]
C.\[R\]
D.\[\dfrac{R}{2}\]

Answer 1

Hint:The radius of gyration of the body about the axis of rotation is the radial distance from the axis of rotation to a point at which if whole of the distributed mass of the body is assumed to be concentrated then the moment of inertia will be same as the actual distribution of the mass about axis of rotation.

Formula used:
The moment of inertia of an object is,
\[I = M{K^2}\]
Here \[M\] is the mass of the object and \[K\] is the radius of gyration.

Complete step by step solution:
The mass of the hollow cylinder is given M and the radius is given R. The moment of inertia of a given hollow cylinder about an axis of rotation is \[M{R^2}\]. We need to find the radius of gyration of the cylinder about the axis. Let the radius of gyration of the cylinder about the axis of rotation is \[K\]. As we know that the moment of inertia is the product of mass and the square of the distance from the axis of rotation.

If whole of the mass of cylinder is assumed to be concentrated at this point, then the moment of inertia of the hollow cylinder will be as,
\[I = M{K^2}\]
The distance K will be said to be the radius of gyration if both the moments of inertias are equal.
\[M{K^2} = M{R^2}\]
\[ \Rightarrow {K^2} = {R^2}\]
\[ \Rightarrow K = \pm R\]
As K is the distance, so we take only positive values. Hence, the radius of gyration of the hollow cylinder about the axis of rotation is equal to R.

Therefore, the correct option is C.

Note: We should be careful about the distribution of the mass of the body while calculating moment of inertia. When the body is made of discrete mass distribution then the moment of inertia of the body will be calculated using arithmetic sum, otherwise we need to use integration of mass over the geometry of the body.