recall and use ω = 2π / T and v = rω

Kinematics of Uniform Circular Motion 🚀

Key Concepts

Uniform circular motion occurs when an object travels around a circle at a constant speed. Even though the speed is constant, the direction changes continuously, so the motion is not “uniform” in the usual sense.

Important Formulas

Angular speed is related to the period by:

$$\omega = \frac{2\pi}{T}$$

The linear speed is related to the angular speed and radius by:

$$v = r\omega$$

Analogy: The Ferris Wheel 🎡

Imagine a Ferris wheel that takes 10 s to make one full rotation. The period $T$ is 10 s. The angular speed is:

$$\omega = \frac{2\pi}{10}\approx 0.628\ \text{rad/s}$$

If a point on the rim is 5 m from the centre, its linear speed is:

$$v = 5 \times 0.628 \approx 3.14\ \text{m/s}$$

Step‑by‑Step Example

1. Identify the period $T$ (time for one revolution).
2. Use $\omega = \dfrac{2\pi}{T}$ to find angular speed.
3. Measure or know the radius $r$.
4. Calculate linear speed $v = r\omega$.

Quick Quiz 🎯

• What is the angular speed of a wheel that completes a revolution every 5 s?
• Given $r = 3$ m and $\omega = 2$ rad/s, what is the linear speed?

Summary

Remember:

1. $\omega = \dfrac{2\pi}{T}$ – angular speed depends on how fast the object goes around.
2. $v = r\omega$ – linear speed increases with both radius and angular speed.