Question

On which of the following does the moment of inertia of an object depend?

Check all that apply.

 

• linear speed
• linear acceleration
• angular speed
• angular acceleration
• total mass
• shape and density of the object
• location of the axis of rotation

 

Answer 1

The Moment of Inertia of a point mass m located at a distance r away from the axis of rotation is \(I=m r^{2}\)

For a system of particles, the moment of Inertia is \(\sum m_{i} r_{i}^{2}\) where \(m_{i}\) is the mass of the \(i^{t h}\) particle located at a distance \(r_{i}\) from the axis of rotation.

A solid object can be considered small mass elements dm; the Moment of Inertia of a solid object is the integration over these small mass elements.

\(I=\int r^{2} d m\)

where \(r\) is the distance of mass element dm from the axis of rotation. If the mass density of solid is given by

\(\rho\)

\(I=\int r^{2} \rho d V\)  ...... (1)

where \(\mathrm{d} \mathrm{V}\) is the volume of the small mass element.

The Moment of Inertia of an object does not depend on linear speed, linear acceleration, angular speed, angular acceleration.

It depends on total mass; among two uniform spheres of the same radius, the sphere with total mass will have greater Inertia momentum. So, a moment of Inertia depends on the total mass of an object.

As suggested by equation (1), the moment of Inertia depends on density. Also the moment of Inertia of disc of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) is \(1 / 2 \mathrm{MR}^{2}\), while the moment of inertia of a ring of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) is \(\mathrm{MR}^{2}\). Thus the moment of Inertia depends on the shape of the object.

The distance from the axis of rotation and moment of inertia depends on \(r\). Therefore, the moment of inertia depends on the axis of rotation.

The last three options are correct.

it depends upon

a)total mass

b)shape and density of the object

c)location of the axis of rotation