Q.Obtain Eq. (6.36), $\omega = \omega_0 + \alpha t$, from first principles.

The key idea is that angular acceleration $\alpha$ is the rate of change of angular velocity $\omega$, just as linear acceleration is the rate of change of linear velocity.

Step 1: By definition, for constant angular acceleration,

Step 2: Separate variables and integrate from initial time $t=0$ (where $\omega = \omega_0$) to a general time $t$:

Step 3: Since $\alpha$ is constant, it factors out of the integral:

$\omega - \omega_0 = \alpha (t - 0)$ …

Why This Derivation Matters

The equation $\omega = \omega_0 + \alpha t$ is the rotational analogue of $v = u + at$ in linear motion. It connects angular velocity, initial angular velocity, angular acceleration, and time — but only when angular acceleration $\alpha$ is constant. Understanding where it comes from, rather than just memorising it, builds the foundation for all rotational kinematics.

The key idea: angular acceleration is the rate of change of angular velocity. That's a definition, not a derived result. If you know how fast $\omega$ is changing ($\alpha$) and for how long ($t$), you can find the new $\omega$.

Step-by-Step Derivation

1. Start with the definition of angular acceleration

Angular acceleration $\alpha$ is defined as the instantaneous rate of change of angular velocity $\omega$ with respect to time:

This is the rotational equivalent of $a = dv/dt$. It tells us: "At any instant, how rapidly is the angular velocity changing?"

2. Rearrange to separate variables

We want to find $\omega$ as a function of time. Multiply both sides by $dt$:

This is a differential equation. It says: a small change in angular velocity equals the angular acceleration multiplied by the small time interval during which it acts.

3. Integrate both sides

We integrate from the initial state (time $t=0$, angular velocity $\omega_0$) to the final state (time $t$, angular velocity $\omega$):

The left side is straightforward: the integral of $d\omega$ is just $\omega$ evaluated between the limits.

4. Handle the right side — the crucial assumption

Here's where the assumption of constant angular acceleration enters. If $\alpha$ is constant, it can be taken outside the integral:

5. Equate the two sides

Putting the left and right sides together:

6. Rearrange to the standard form …

Concept: Deriving the First Equation of Rotational Kinematics

Step 1: Start from the definition of angular acceleration

Step 2: Separate variables

Step 3: Integrate from the initial state ($t=0,\ \omega=\omega_0$) to a general time $t$

Step 4: Evaluate, using the fact that $\alpha$ is constant …