gen-define-torque-and-angular-momentum-and-give-one-short-worked-af87

Define torque and angular momentum and give one short worked example.

# Torque and Angular Momentum ## Definitions **Torque (τ)**: A measure of the rotational force applied to an object. It's the cross product of the position vector and force: $$\vec{\tau} = \vec{r} \times \vec{F}$$ Magnitude: $\tau = rF\sin\theta$, where θ is the angle between **r** and **F**. **Angular Momentum (L)**: A measure of rotational motion, analogous to linear momentum. For a point particle: $$\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times m\vec{v}$$ Magnitude: $L = rmv\sin\theta$ **Key relationship**: $\vec{\tau} = \frac{d\vec{L}}{dt}$ (torque is the rate of change of angular momentum) ## Worked Example **Problem**: A 2 kg mass is attached to a string and swings in a horizontal circle of radius 0.5 m at 3 m/s. Find (a) the angular momentum about the center, and (b) if friction slows it to 2 m/s in 4 seconds, find the average torque. **Solution**: (a) Since **v** is perpendicular to **r** (θ = 90°): $$L = rmv = (0.5)(2)(3) = 3 \text{ kg·m}^2\text{/s}$$ (b) Final angular momentum: $$L_f = rmv_f = (0.5)(2)(2) = 2 \text{ kg·m}^2\text{/s}$$ $$\tau_{avg} = \frac{\Delta L}{\Delta t} = \frac{2-3}{4} = -0.25 \text{ N·m}$$ (The negative sign indicates the torque opposes the motion)