Q.A uniform disc of mass $m$ and radius $R$ stands vertically on its rim on a horizontal table; the coefficient of friction between the disc and the table is $\mu$. A horizontal force $\vec{F}$ is applied at the centre (the axle) of the disc, in the plane of the disc. Find the maximum value of $F$ for which the disc rolls without slipping.

Rotational Dynamics: The Physics of Spinning Things

Imagine you're trying to open a heavy door. You push near the hinge — it barely moves. Push near the handle — it swings open easily. Same force, different result. That's the first clue: rotation isn't just about how much you push, but where and in what direction.

Now think about a spinning bicycle wheel. Why is it so hard to tilt it sideways when it's spinning fast? And why does a figure skater spin faster when she pulls her arms in? These are the questions rotational dynamics answers.

The Core Idea

Rotational dynamics is the study of why things rotate and how their rotation changes. It's the spinning-world equivalent of Newton's laws for straight-line motion.

In linear motion, you have:

• Force ($F$) causes acceleration ($a$)
• Mass ($m$) resists acceleration

In rotational motion, you have:

• Torque ($\tau$) causes angular acceleration ($\alpha$)
• Moment of inertia ($I$) resists angular acceleration

The master equation is:

This is the rotational version of $F = ma$. Every term has a direct parallel.

Breaking It Down

Torque — The Rotational "Push"

Torque isn't just force — it's force multiplied by the distance from the pivot point (the lever arm). That's why the door handle works better than the hinge.

Where $r$ is the distance from the axis, $F$ is the force, and $\theta$ is the angle between them. Maximum torque happens when you push perpendicular to the lever arm ($\theta = 90^\circ$).

Moment of Inertia — The Rotational "Mass"

Mass resists linear acceleration. Moment of inertia resists angular acceleration. But unlike mass, moment of inertia depends on how the mass is distributed relative to the axis of rotation.

For a point mass $m$ at distance $r$ from the axis:

For extended objects, you sum (or integrate) over all mass elements:

Notice: a hoop has more moment of inertia than a disk of the same mass and radius because its mass is farther from the axis. That's why a hoop is harder to start spinning.

Angular Acceleration — How Fast Rotation Changes

Just as acceleration is the rate of change of velocity, angular acceleration $\alpha$ is the rate of change of angular velocity $\omega$:

And angular velocity is the rate of change of angular displacement $\theta$:

The Complete Picture: Rotational Analogues

The Key Insight: Conservation of Angular Momentum

This is where rotational dynamics gets beautiful. Just as linear momentum is conserved when no external force acts, angular momentum is conserved when no external torque acts:

$L = I\omega = \text{constant}$ …