Question

A uniform solid sphere has mass M and radius R. If these are increased to 2M and 3R, what happens to the sphere's moment of inertia about a central axis?

Answer 1

Answer:

The moment of inertia about central axis becomes 18 times to its original moment of inertia when its mass doubled and radius tripled, I₁ = 18I.

Explanation:

Moment of inertia of a uniform solid sphere about its central axis is given by the relation :

$I=\frac{2}{5}MR^{2}$    ....(1)

Here I is moment of inertia, M is mass of the solid sphere and R is the radius of the solid sphere.

Now, the mass of the sphere becomes twice and radius becomes thrice i.e.

New mass of sphere = 2M

New radius of new sphere = 3R

The new moment of inertia is:

$I_{1} =\frac{2}{5}(2M)(3R)^{2}$

$I_{1} =\frac{2}{5}\times18MR^{2}$

Substitute equation (1) in the above equation.

$I_{1} =18I$

Answer 2

A uniform solid sphere has mass M and radius R. If these are increased to 2M and 3R, the sphere's moment of inertia about a central axis - will be 18 times its original moment of inertia.

Given:

mass = M

radius = R.

later change in mass = 2M

in radius = 3R

solution:

• The moment of inertia of a uniform solid sphere about its central axis is given by the relation :  

$I=\frac{2}{5}MR^{2}$  . .. 1

Here,

I = moment of inertia,

M = mass of the solid sphere

R is the radius of the solid sphere.  

Now, the mass of the sphere becomes twice and the radius becomes thrice

The new mass of sphere = 2M  

The new radius of the new sphere = 3R  

The new moment of inertia is:  

$I_{1} =\frac{2}{5}(2M)(3R)^{2}\\I_{1} =\frac{2}{5}\times18MR^{2}$  

Substitute equation (1) in the above equation.  

$I_{1} =18I$  

Thus, A uniform solid sphere has mass M and radius R. If these are increased to 2M and 3R, the sphere's moment of inertia about a central axis - will be 18 times its original moment of inertia.  

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