Q.Arrive at an expression for the time period of simple pendulum. [a QR code is printed alongside this question in the source paper]

Angular Kinematics Derivation

Imagine a ceiling fan. When you switch it on, the blades don't just move — they rotate. A car's wheel, a spinning top, the Earth itself — all these objects turn around a fixed axis. The physics of how they turn is angular kinematics.

The Core Intuition

Linear kinematics describes motion in a straight line: position $x$, velocity $v$, acceleration $a$. Angular kinematics describes rotation: angular position $\theta$, angular velocity $\omega$, angular acceleration $\alpha$.

The beautiful thing? The equations are identical in structure. Every linear quantity has an angular twin. If you know how to handle $v = u + at$, you already know how to handle $\omega_f = \omega_i + \alpha t$ — you just swap letters.

Why Radians?

Angular kinematics only works if angles are measured in radians, not degrees. Why? Because the arc length $s$ along a circle of radius $r$ is:

This simple relation holds only when $\theta$ is in radians. If you used degrees, you'd need an ugly conversion factor. Radians make the math clean.

Deriving the Angular Kinematics Equations

We start from the definitions, exactly as in linear kinematics.

Step 1: Angular velocity

Average angular velocity:

Instantaneous angular velocity (the limit as $\Delta t \to 0$):

Step 2: Angular acceleration

Average angular acceleration:

Instantaneous:

Step 3: Constant angular acceleration

If $\alpha$ is constant, we can integrate just like in linear motion.

From $\alpha = \frac{d\omega}{dt}$, integrate:

This is the first equation.

Step 4: Angular displacement

Since $\omega = \frac{d\theta}{dt}$, and $\omega$ changes linearly with time (because $\alpha$ is constant), the average angular velocity is:

Then:

(Here $\theta$ means $\Delta \theta$, the angular displacement.)

Step 5: Displacement from initial velocity and acceleration

Substitute $\omega_f$ from equation (1) into equation (2):

Step 6: Velocity-displacement relation

Eliminate $t$ from equations (1) and (3). From (1): $t = \frac{\omega_f - \omega_i}{\alpha}$. Substitute into (3):

Simplify:

Multiply through by $2\alpha$:

The $2\omega_i\omega_f$ terms cancel:

Connecting to Linear Quantities

Every point on a rotating rigid body also has linear motion. The relations are: