Rotation of Rigid

Linear inertia is the tendency of a body to resist a change in its linear velocity. In other words: objects do not change their state of linear motion unless acted upon by some external force.

Rotational inertia is the tendency of a body to resist a change in its angular velocity. In other words: objects do not change their rotational motion unless acted upon by some external torque. This property is also known as the moment of inertia.

The moment of inertia of a body about an axis is a measure of the difficulty in starting, stopping, or changing the rotation of the body about that axis. It is denoted by the symbol $I$.

The greater the difficulty in starting or stopping rotation, the greater the moment of inertia about that axis, and vice-versa.

A body rotates under the action of a net external torque. The greater the moment of inertia of a body about an axis of rotation, the greater the torque required to:

• Start rotation
• Stop rotation
• Change the rotational speed or direction

Moment of inertia of a point mass:

$I = mr^2$

Where:

• $I$ = moment of inertia
• $m$ = mass of the object
• $r$ = perpendicular distance from the axis of rotation

Consider a rigid body rotating about the axis $yy-1$ with an angular speed $\omega$ as shown in figure 1 below.

Although each particle of the body has the same angular speed $\omega$, the linear velocity ($v$) of each particle depends upon the particle's distance from the axis of rotation. Thus a particle of mass $m_i$ follows a circular path of radius $r$. The linear velocity of this particle is $v_i$.

The rotational kinetic energy (R.K.E) of a particle of mass $m_i$ moving in a circle of radius $r_i$ with angular velocity $\omega$ is:

$K_i = \frac{1}{2} m_i (r_i \omega)^2 = \frac{1}{2} m_i r_i^2 \omega^2$

The total rotational kinetic energy of the rigid body composed of $n$ particles is:

$K_r = \sum_{i=1}^n \frac{1}{2} m_i r_i^2 \omega^2 = \frac{1}{2} I \omega^2$

Where $I = \sum_{i=1}^n m_i r_i^2$ is the moment of inertia about the axis of rotation.

The radius of gyration $K$ is defined by the equation:

$I = MK^2$

It represents the distance from the axis at which the entire mass $M$ could be concentrated without changing the moment of inertia.

• $\omega = \omega_0 + \alpha t$
• $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$
• $\omega^2 = \omega_0^2 + 2\alpha \theta$

These are rotational analogues of linear motion equations.

Torque $\tau$ is related to angular acceleration $\alpha$ as:

$\tau = I \alpha$

This is analogous to Newton's second law $F = ma$.

Angular momentum $L$ of a rigid body is:

$L = I\omega$

And torque is the rate of change of angular momentum:

$\tau = \frac{dL}{dt}$

Power delivered by torque is:

$P = \tau \omega$

The work done by torque over angle $\theta$ is:

$W = \int \tau \, d\theta = \Delta K_r = \frac{1}{2}I\omega^2 - \frac{1}{2}I\omega_0^2$

If no external torque acts:

$L = I\omega = \text{constant} \quad \Rightarrow \quad I_1\omega_1 = I_2\omega_2$

1. Uniform rod (length $L$, mass $M$), about center: $I = \frac{1}{12}ML^2$
2. Thin ring (radius $R$, mass $M$), about center: $I = MR^2$
3. Solid cylinder (radius $R$, mass $M$), about center axis: $I = \frac{1}{2}MR^2$
4. Hollow cylinder (inner $R_1$, outer $R_2$): $I = \frac{1}{2}M(R_1^2 + R_2^2)$

If a body consists of $n$ particles each of mass $m$:

$K = \sqrt{\frac{\sum_{i=1}^n r_i^2}{n}}$

This represents the root mean square distance from the axis.