Obtain an expression for torque acting on a rigid body rotating with constant angular acceleration.

Answer: \(\tau = I\alpha\)

Consider a rigid body rotating about a fixed axis with constant angular acceleration \(\alpha\). Let it consist of particles of masses \(m_1, m_2, \dots, m_n\) at perpendicular distances \(r_1, r_2, \dots, r_n\) from the axis. All particles have the same angular acceleration \(\alpha\).

The tangential (linear) acceleration of the \(i\)-th particle is \(a_i = r_i\alpha\), so the force on it is \(F_i = m_ia_i = m_ir_i\alpha\). This force is perpendicular to \(r_i\), so its torque about the axis is

The total torque on the body is

Since \(\sum m_ir_i^2 = I\), the moment of inertia of the body about the axis,

In vector form \(\vec\tau = I\vec\alpha\); torque plays the role of force and moment of inertia that of mass in rotational motion (compare \(F = ma\)).