Dynamics of Rotational Motion About a Fixed Axis

Opening the Dynamics of Rotational Motion

The previous sections built the language of rotational motion — angular displacement, velocity, acceleration, moment of inertia, and torque. Now we put these tools to work. The central question: What happens when a torque acts on a rigid body rotating about a fixed axis? The answer mirrors Newton's second law for linear motion, but with rotational analogues: torque replaces force, moment of inertia replaces mass, and angular acceleration replaces linear acceleration.

But there is a subtlety. In linear motion, force and acceleration are always in the same direction. In rotation, torque and angular acceleration are related by the moment of inertia, which depends on how mass is distributed. The dynamics are richer, and we must account for the work done by torques, the power delivered, and the energy stored in the rotating body.

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The Rotational Analogue of Newton's Second Law

Consider a rigid body rotating about a fixed axis. Take a small mass element $m_i$ at a perpendicular distance $r_i$ from the axis. When a net external torque $\tau_{\text{ext}}$ acts on the body, each mass element experiences a tangential force. For the $i$-th element, Newton's second law in the tangential direction gives:

where $\alpha$ is the angular acceleration (same for all particles in a rigid body). The torque on this element about the axis is:

Summing over all particles:

The left side is the net external torque $\tau_{\text{ext}}$ (internal torques cancel in pairs by Newton's third law). The sum in parentheses is the moment of inertia $I$ about the fixed axis. Thus:

This is the rotational analogue of $F = ma$. It holds only when the axis is fixed and the moment of inertia is constant.

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Work Done by a Torque

When a torque rotates a body through an angular displacement, it does work. Consider a force $\mathbf{F}$ acting at a point on a rotating body. The force has a tangential component $F_{\text{tan}}$ that does work, and a radial component $F_{\text{rad}}$ that does no work (it points toward the axis).

For a small angular displacement $\mathrm{d}\theta$, the point of application moves a distance $\mathrm{d}s = r \, \mathrm{d}\theta$ along the arc. The work done by the tangential force is:

But $F_{\text{tan}} \, r$ is the torque $\tau$ about the axis. Therefore:

For a finite angular displacement from $\theta_1$ to $\theta_2$:

If the torque is constant, this simplifies to $W = \tau (\theta_2 - \theta_1) = \tau \Delta\theta$.

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Kinetic Energy of Rotation

A rotating rigid body stores kinetic energy. Each mass element $m_i$ moves with speed $v_i = r_i \omega$, so its kinetic energy is $\frac{1}{2} m_i v_i^2 = \frac{1}{2} m_i r_i^2 \omega^2$. Summing over all particles:

This is the rotational kinetic energy. It is the exact analogue of $\frac{1}{2} m v^2$ for linear motion.

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The Work-Energy Theorem for Rotation

The work-energy theorem states that the net work done on a body equals the change in its kinetic energy. For rotation, we can derive this directly. Starting from $\tau = I \alpha = I \frac{\mathrm{d}\omega}{\mathrm{d}t}$:

Integrating from initial angular speed $\omega_1$ to final $\omega_2$:

Thus:

The work done by the net external torque equals the change in rotational kinetic energy. This is the rotational work-energy theorem.

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Power Delivered by a Torque

Power is the rate at which work is done. From $\mathrm{d}W = \tau \, \mathrm{d}\theta$:

This is the rotational analogue of $P = F v$ for linear motion.

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Conservation of Mechanical Energy in Rotational Motion

When only conservative forces (like gravity) do work, mechanical energy is conserved. For a rotating body, the total mechanical energy includes both rotational kinetic energy and potential energy:

where $h$ is the height of the centre of mass above a reference level. …