Question

Define uniform circular motion. Show that linear speed of particle performing circular motion is the product of radius of circle and angular speed of particle.

Answer 1

When a body is moving in a circular path with a constant linear speed or constant angular speed is known as uniform circular motion.

Let's consider a body performing uniform circular motion, along a circumference of radius r with constant linear speed v and constant angular speed $\omega$ moving in anticlockwise direction as shown in figure.

here,in small time of interval $\delta{t}$ particle moves from P to Q through a distance $\delta{x}$ and subtends an angle $\delta{\theta}$at the centre O as shown in figure.
we know, $\theta=\frac{L}{r}$ from geometry
so, $\delta{x}=r\delta{\theta}$

now, $v=\lim_{x\to 0}\frac{\delta{x}}{\delta{t}}\\\\=\lim_{x\to 0}\frac{r\delta{\theta}}{\delta{t}}\\\\=r\lim_{x\to 0}\frac{\delta{\theta}}{\delta{t}}\\\\=r\omega$

hence, $v=\omega r$