Question

Define angular velocity (ù). Derive v = r ω.

Answer 1

angular velocity is the rate of change of angular position per unit time.

e.g., $\omega=\frac{\textbf{change in angular position}}{\textbf{change in time}}$

or, $\omega=\frac{\triangle\theta}{\triangle t}$

Let a particle is rotating in circle of radius r and after time t it makes $\theta$ angle with horizontal line as shown in figure.

so, $x=r(cos\theta\hat{i}+sin\theta\hat{j})$
differentiate with respect to t,
$\frac{dx}{dt}=r(-sin\theta\hat{i}+cos\theta\hat{j})\frac{d\theta}{dt}$
we know, $\frac{dx}{dt}=v$
and $\frac{d\theta}{dt}=\omega$
we also know, linear velocity is perpendicular upon plane of angular velocity and radius vector .

so, $v=r\omega(-sin\theta\hat{i}+cos\theta\hat{j})$

so, magnitude of $v=r\omega$

Answer 2

Hii dear,


◆ Angular velocity -
Angular velocity of a particle is defined as rate of change of angular displacement with respect to time.


◆ Proof-
Consider a circle of radius r.
Suppose in time t, a particle covers angular displacement θ, and path l.

Relation between angular displacement and arc length is given by-
θ = l/r
l = r×θ

Taking derivative wrt time on both sides,
dl/dt = r × dθ/dt

But we know, dl/dt = v and dθ/dt = w
Therefore,
v = rw

In vector form-
v⃗ = w⃗ × r⃗

Hope, this helps you...