Equations of rotational motion and their derivation

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equations of rotational motion

We have the equations of motion for a linear(or translational) motion as:

$\begin{align*} v &= u + at \\ v^2 &= u^2 + 2as \\ s &= ut + \frac{1}{2}at^2 \end{align*}$

For the rotational motion, we replace $v$ by $\omega$ , $u$ by $\omega_0$ , $s$ by $\theta$ and $a$ by $\alpha$ in the above equations of motion. To know more about rotational motion, click here.

The equations of rotational motion become:

$\begin{align*} \omega &= \omega_0 + \alpha t \\ \omega^2 &= \omega_0^2 + 2\alpha \theta \\ \theta &= \omega_0 t + \frac{1}{2} \alpha t^2 \end{align*}$

Here is the derivation of the rotational motion equation.

equations of rotational motion

1] Proof of the first equation of rotational motion

Consider a rigid body moving with constant angular acceleration ‘ $\alpha$ ‘ such that its angular velocity reaches ‘ $\omega$ ‘ from ‘ $\omega_0$ ‘ in time ‘ $t$ ‘ covering an angular displacement ‘ $\theta$ ‘.

$\begin{align*}\alpha &= \frac{d\omega}{dt} \\ d\omega &= \alpha \,dt \\ \end{align*}$

Integrating the above equation with the corresponding limit.

$\begin{align*} \int_{\omega_0}^{\omega} d\omega &= \alpha \int_{0}^{t} dt \\ [\omega]_{\omega_0}^{\omega} &= \alpha [t]_{0}^{t} \\ \omega - \omega_{0} &= \alpha (t-0) \\ \omega - \omega_0 &= \alpha t \end{align*}$

$\boxed{\omega = \omega_0 + \alpha t}$

2] Proof of second equation of rotational motion

Let’s start with the angular velocity to prove another equation.

$\begin{align*} \omega &= \frac{d\theta}{dt} \\ d\theta &= \omega \, dt \\ \end{align*}$

Integrating the above equation with the corresponding limit.

$\begin{align*} \int_{0}^{\theta} d\theta &= \int_{0}^{t} \omega \, dt \\ \int_{0}^{\theta} d\theta &= \int_{0}^{t} (\omega_0 + \alpha t)dt \\ \int_{0}^{\theta} d\theta &= \int_{0}^{t} \omega_0 \, dt + \int_{0}^{t} \alpha t \, dt \\ \int_{0}^{\theta} d\theta &= \omega_0 \int_{0}^{t} dt + \alpha \int_{0}^{t} t \, dt \\ [\theta]_{0}^{\theta} &= \omega_0 [t]_{0}^{t} + \alpha \left[\frac{t^2}{2}\right]_{0}^{t} \\ \end{align*}$

$\boxed{\theta = \omega_{0} t + \frac{1}{2}\alpha t^2 }$

3] Proof of third equation of rotational motion

We know, that angular acceleration is the rate of change of angular velocity. So,

$\begin{align*} \alpha &= \frac{d\omega}{dt} \\ \alpha &= \frac{d\omega}{d\theta} \frac{d\theta}{dt} \\ \alpha &= \frac{d\omega}{d\theta} \omega \\ \alpha &= \omega \frac{d\omega}{d\theta} \\ \end{align*}$

Now,

$\begin{align*}\alpha \, d\theta&= \omega \, d\omega \\ \end{align*}$

Integrating within a corresponding limit,

$\begin{align*} \alpha \int_{0}^{\theta} d\theta &= \int_{\omega_0}^{\omega}\omega \, d\omega \\ \alpha \left[ \theta\right ]{0}^{\theta} &= \left[\frac{\omega^2}{2}\right]{\omega_0}^{\omega}\\ \alpha \theta &= \frac{1}{2}(\omega^2-\omega_0)\\ 2\alpha \theta &= \omega^2 - \omega_0 \\\end{align*}$

$\boxed{\omega^2 = \omega_0 + 2 \alpha \theta}$

These three equations can define the rotational motion of a rigid body.

Also, check out the table that shows the quantities used in both translational as well as rotational motion.

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equations of rotational motion

1] Proof of the first equation of rotational motion

2] Proof of second equation of rotational motion

3] Proof of third equation of rotational motion

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