Question

Derive the relation between linear and angular velocity, or derive $v = r\omega$.

Answer 1

To solve this question, we have to need to use the formula of the length of an arc in terms of the radius and the angle subtended. Then using the definition of the angular velocity we can derive the given relation.

Formula used: The formula used to solve this question is given by
$\Rightarrow {{\theta }} = \dfrac{l}{r}$, here ${{\theta }}$ is the angle subtended by an arc of length $l$ and radius $r$.

Complete step by step answer
Let us consider a particle rotating in a circle of radius $r$ with the angular velocity $\omega$.
And let us consider a time interval $\Delta t$ in which the particle covers an angular displacement of $\Delta {{\theta }}$ and traverse a length of $\Delta s$ along the circle.
We know that the length of the curve is related to it radius as
$\Rightarrow {{\theta }} = \dfrac{l}{r}$
So the angular displacement and the length of circle traversed are related as
$\Rightarrow \Delta {{\theta }} = \dfrac{{\Delta s}}{r}$
$\Rightarrow \Delta s = r\Delta {{\theta }}$
Dividing both sides by the time interval $\Delta t$, we get
$\Rightarrow \dfrac{{\Delta s}}{{\Delta t}} = r\dfrac{{\Delta {{\theta }}}}{{\Delta t}}$
Now, taking the limit as $\Delta t$ tends to zero, we have
$\Rightarrow \mathop {\lim }\limits_{\Delta t \to 0} \dfrac{{\Delta s}}{{\Delta t}} = r\mathop {\lim }\limits_{\Delta t \to 0} \dfrac{{\Delta {{\theta }}}}{{\Delta t}}$
So we get
$\Rightarrow \dfrac{{ds}}{{dt}} = r\dfrac{{{{d\theta }}}}{{dt}}$
Now, we know that $\dfrac{{ds}}{{dt}} = v$, and $\dfrac{{{{d\theta }}}}{{dt}} = \omega$. Substituting these above we get
$\Rightarrow v = r\omega$
This is the required relation between the linear velocity and the angular velocity.

Note
The linear velocity so obtained comes out to be proportional to the radius of the circle in which the particle is rotating. The angular velocity is constant for each and every particle of a rotating body. From this relation, we can obtain the relation between all of the other linear and rotational variables. For example, the linear acceleration is also equal to the radius times the angular acceleration.