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# Moment of Inertia

Moment of inertia, also known as rotational inertia or angular mass, is a physical quantity that resists a rigid body’s rotational motion. It is analogous to mass in translational motion. It determines the torque required to rotate an object by a given angular acceleration. Moment of inertia does not restrict itself to a rigid body only. It also applies to a system of particles rotating about a common axis.

## How to Calculate Moment of Inertia

Point Mass

For a point mass (single body), the moment of inertia formula is given by the product of mass and the square of the object’s perpendicular distance from the axis of rotation.

$I = mr^2$

Where

I is the moment of inertia

m is the mass

r is the perpendicular distance from the axis of rotation

System of Particles

Suppose n particles of masses m1, m2, …, mn rotate about a common axis at perpendicular distances r1, r2, …, rn from the axis. Then the moment of inertia of the system is

$I = \sum m_i r_i\hspace{0.01 cm}^2$

The moment of inertia of a system of particles is the sum of the moments of inertia of the individual particles taken about a common axis.

Units and Dimensions

The SI unit of moment of inertia is kgˑm2, and the cgs unit is gˑcm2. The dimension is [M L2 T0].

### Integral Moment of Inertia

For a continuous mass distribution, the integral form of moment of inertia is given by

$I = \int r^2 dm$

Where dm is the instantaneous mass.

The above equation can also be written in terms of density ρ and instantaneous volume dV as follows.

$I = \int r^2 \rho dV$

## Moment of Inertia Table

Several everyday objects, such as rotating disks, cylinders, and spheres, have well-defined moment of inertia formulas. A chart consisting of the different formulas is listed in the table below. Here M represents mass, R represents radius, and L represents the length.

## Moment of Inertia and Rotational Kinematics

The moment of inertia of a rotating object about a fixed axis is useful in calculating a few key quantities in rotational motion. Newton’s second law for rotation gives a relationship between torque, moment of inertia, and angular acceleration. According to this law,

$\tau = I\alpha$

Where

τ is the applied torque

I is the moment of inertia

α is the angular acceleration

The relationship between moment of inertia and rotational kinetic energy is given by

$K = I \omega^2$

And with angular momentum, the equation is

$L = I\omega$

Where

K is the rotational kinetic energy

L is the angular momentum

ω is the angular velocity

## Factors Affecting Moment of Inertia

The moment of inertia depends upon

• The shape of the object
• Mass distribution or density
• Location of the axis of rotation

Article was last reviewed on Wednesday, August 2, 2023