# Rotational Kinematics and Torque

Physics I · Principles of Mechanics · https://tryals.app/learn/physics-i/rotational-kinematics-and-torque

## Rotational Kinematics

Door handles and wrenches exploit one core idea: rotational effect equals force times lever arm. Rotation has a full parallel dictionary to linear motion where $\theta, \omega, \alpha$ mirror $x, v, a$, torque mirrors force, and **moment of inertia** mirrors mass.

| Translational | Rotational | Relation |
| --- | --- | --- |
| $x$ (position) | $\theta$ (angle) | $s = r\theta$ |
| $v$ (velocity) | $\omega$ (angular velocity) | $v = r\omega$ |
| $a$ (acceleration) | $\alpha$ (angular acceleration) | $a_t = r\alpha$ |

For **constant angular acceleration**, use these direct kinematic equations:

$$\omega = \omega_0 + \alpha t, \qquad \theta = \theta_0 + \omega_0 t + \tfrac{1}{2}\alpha t^2$$

## Torque & Moment of Inertia

**Torque** is the rotational analogue of force, defined as $\vec{\tau} = \vec{r}\times\vec{F}$ with magnitude $|\tau| = rF\sin\theta$. **Newton's second law for rotation** is $\tau_{\text{net}} = I\alpha$, where $I = \sum m_i r_i^2$ is the **moment of inertia**.

**Common shapes and their inertia**

| Shape | Formula |
| --- | --- |
| Solid cylinder / disk | $I = \tfrac{1}{2}MR^2$ |
| Solid sphere | $I = \tfrac{2}{5}MR^2$ |
| Thin rod (centre) | $I = \tfrac{1}{12}ML^2$ |
| Thin hoop / ring | $I = MR^2$ |

The **parallel axis theorem** ($I = I_{\text{cm}} + Md^2$) shifts any axis from the centre of mass.

**Common pitfall:** $I$ is not just mass, but **where** mass sits. Rim mass counts far more than hub mass because of $r^2$ weighting.

## Practice questions

1 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. A disk and a hoop of equal mass and radius roll down the same ramp, and the disk wins. Why does mass cancel while the shape does not?

A. Because the disk is denser and therefore accelerates faster
B. Because mass divides out and only its DISTRIBUTION remains
C. Because the hoop’s mass is concentrated at the centre
D. Because the hoop has more surface area and so more friction

**Answer:** B. Because mass divides out and only its DISTRIBUTION remains

**Why:** Only $\beta$ survives: 1/2 for the disk, 1 for the hoop. Distribution, not amount, decides the race, which is why the result holds for any mass and any radius.

Page: https://tryals.app/practice/physics-i/rotational-kinematics-and-torque/a-disk-and-a-hoop-of-equal-mass-and-radius-roll-down-the-same-ramp