Angular Speed & Linear Speed: Quick, Friendly Notes

1. What is Angular Speed (ω)?

Imagine a point on a spinning wheel. As the wheel turns through a tiny angle dθ in a tiny time dt, its angular speed is \( \displaystyle \omega = \frac{d\theta}{dt} \). :contentReference[oaicite:0]{index=0} Angular speed is a vector. Point your right-hand thumb along the axle in the direction a normal right-hand screw would move; that thumb shows the direction of ω. :contentReference[oaicite:1]{index=1}

2. Linking Angular Speed to Linear Speed

• For any particle on the spinning body, linear speed (tangential speed) equals \( v = r\,\omega \), where r is its perpendicular distance from the axis. :contentReference[oaicite:2]{index=2}
• In vector form you simply take a cross product: \( \mathbf v = \boldsymbol{\omega} \times \mathbf r \). :contentReference[oaicite:3]{index=3}
• A point right on the axis has r = 0, so its speed is zero; points on the axle stay still while everything else whirls around them. :contentReference[oaicite:4]{index=4}
• More generally, for the i-th particle: \( v_i = r_i\,\omega \) (Eq. 6.19). :contentReference[oaicite:5]{index=5}

3. Pure Rotation

If every part of the body shares the same ω at a given instant, we call the motion pure rotation. :contentReference[oaicite:6]{index=6}

4. Angular Acceleration (α)

Angular acceleration measures how fast ω itself changes: \( \displaystyle \boldsymbol{\alpha} = \frac{d\boldsymbol{\omega}}{dt} \). :contentReference[oaicite:7]{index=7} For rotation about a fixed axis, α points along that axis just like ω, and the scalar form is \( \displaystyle \alpha = \frac{d\omega}{dt} \). :contentReference[oaicite:8]{index=8}

5. Direction & Change of ω

With a fixed axis, the direction of ω stays steady; only its magnitude may rise or fall. In more complicated spins, both direction and size can vary from moment to moment. :contentReference[oaicite:9]{index=9}

High-Yield Points for NEET

1. The must-know link \( v = r\,\omega \) lets you jump between linear and angular motion problems. :contentReference[oaicite:10]{index=10}
2. The vector formula \( \mathbf v = \boldsymbol{\omega} \times \mathbf r \) shows direction instantly—handy for “find the velocity vector” questions. :contentReference[oaicite:11]{index=11}
3. Right-hand rule for the direction of ω (thumb along the axle) appears repeatedly in conceptual MCQs. :contentReference[oaicite:12]{index=12}
4. Points on the axis stay at rest (speed = 0), a classic trap-avoider in questions about rotating discs. :contentReference[oaicite:13]{index=13}
5. Angular acceleration definition \( \alpha = d\omega/dt \) mirrors linear acceleration, simplifying rotational kinematics problems. :contentReference[oaicite:14]{index=14}