# A.4 Rotational mechanics

> IB Physics HL · Theme A: Space, time and motion
> Source: https://www.owlsprep.com/study/ib-physics-hl-u1-a-4-rotational-mechanics/

This sub-topic introduces core concepts of rigid body rotational mechanics, connecting familiar linear motion quantities to their rotational equivalents. You will learn to solve problems involving torque, moment of inertia, and angular momentum conservation for rotating systems.

**Prerequisites:** [Linear kinematics and Newton's laws of motion](https://www.owlsprep.com/study/ib-physics-hl-u1-a-2-forces-and-momentum/); [Work, energy and power](https://www.owlsprep.com/study/ib-physics-hl-u1-a-3-work-energy-power/)

## Learning objectives

- Relate rotational kinematic quantities to their linear analogues
- Calculate moment of inertia for common rigid bodies using the parallel axis theorem
- Apply Newton's second law for rotation to solve rigid body problems
- Use conservation of angular momentum to analyze isolated rotational systems

## Rotational Kinematics: Linear-Rotational Analogues

All linear kinematic quantities have direct rotational equivalents, with the same form of kinematic equations for uniform acceleration.

**Rotational kinematic quantities** — Quantities that describe the rotation of a rigid body about a fixed axis, connected to linear quantities for a point at radius $r$ by $x = r\theta$, $v = r\omega$, $a = r\alpha$.

*Notation:* Angular displacement $\theta$ (rad), angular velocity $\omega$ (rad s⁻¹), angular acceleration $\alpha$ (rad s⁻²)

| Linear quantity | Rotational equivalent | Uniform acceleration equation |
| --- | --- | --- |
| Displacement $x$ | $\theta$ | $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$ |
| Velocity $v$ | $\omega$ | $\omega^2 = \omega_0^2 + 2\alpha\theta$ |
| Acceleration $a$ | $\alpha$ | $\omega = \omega_0 + \alpha t$ |

**Worked example:** A wheel of radius 0.5 m accelerates uniformly from rest to 12 rad s⁻¹ in 4 s. Find the final linear speed at the edge of the wheel and the total angular displacement.

1. Use the relationship between linear and angular speed for a point on the edge:
2. $$v = r\omega = 0.5 \times 12 = 6 \text{ m s}^{-1}$$
3. Calculate angular acceleration from the change in angular velocity:
4. $$\alpha = \frac{\omega - \omega_0}{t} = \frac{12 - 0}{4} = 3 \text{ rad s}^{-2}$$
5. Use the kinematic equation for angular displacement to get the final result:
6. $$\theta = \omega_0 t + \frac{1}{2}\alpha t^2 = 0 + \frac{1}{2} \times 3 \times 4^2 = 24 \text{ rad}$$

> **Exam tip:** Always convert angular velocity from revolutions per minute (rpm) to radians per second before substituting into formulas.

## Torque and Moment of Inertia

Torque is the rotational equivalent of force, and moment of inertia is the rotational equivalent of mass. The moment of inertia of a body depends on its mass distribution relative to the axis of rotation.

**Torque** — The turning effect of a force about a pivot, where $\theta$ is the angle between the position vector $r$ (from pivot to point of application) and the force $F$.

*Notation:* $\tau = rF\sin\theta$

**Moment of inertia** — A measure of a body's resistance to angular acceleration, equal to the sum of the product of each mass element and the square of its distance from the axis of rotation.

*Notation:* $I = \sum m_i r_i^2$

The parallel axis theorem lets you calculate $I$ for any axis parallel to one through the center of mass: $I = I_{cm} + Md^2$, where $M$ is total mass and $d$ is the distance between axes.

**Worked example:** Calculate the moment of inertia of a uniform rod of mass $M$ and length $L$ about an axis at one end, given that $I_{cm} = \frac{1}{12}ML^2$ for an axis through the center.

1. The distance between the center of mass axis and the end axis is $d = \frac{L}{2}$:
2. $$I = I_{cm} + Md^2$$
3. $$I = \frac{1}{12}ML^2 + M\left(\frac{L}{2}\right)^2$$
4. $$I = \frac{1}{12}ML^2 + \frac{3}{12}ML^2 = \frac{1}{3}ML^2$$

## Newton's Second Law for Rotation and Rotational KE

Just like linear motion, we have a simple form of Newton's second law for rotation, and can calculate the kinetic energy of a rotating body.

**Newton's second law for rotation** — The net torque on a rigid body equals the product of its moment of inertia and angular acceleration: $\tau_{net} = I\alpha$

**Rotational kinetic energy** — Kinetic energy due to rotation: $E_k = \frac{1}{2}I\omega^2$. For a body rolling without slipping, total kinetic energy is the sum of linear and rotational kinetic energy.

**Worked example:** A solid cylinder of mass $M$ and radius $R$ rolls without slipping down a slope inclined at angle $\theta$. Find its linear acceleration down the slope.

1. Torque from friction about the center of the cylinder equals $I\alpha$. For solid cylinder, $I = \frac{1}{2}MR^2$, and $\alpha = \frac{a}{R}$ for rolling without slipping:
2. $$\tau = fR = I\alpha = \frac{1}{2}MR^2 \times \frac{a}{R} \rightarrow f = \frac{1}{2}Ma$$
3. Apply linear Newton's second law along the slope:
4. $$Mg\sin\theta - f = Ma$$
5. Substitute $f$ and solve for $a$:
6. $$Mg\sin\theta - \frac{1}{2}Ma = Ma \rightarrow a = \frac{2}{3}g\sin\theta$$

> **Exam tip:** Always add rotational kinetic energy to linear kinetic energy when calculating total energy for rolling objects.

## Angular Momentum and Conservation

Angular momentum is the rotational equivalent of linear momentum, and it is conserved when the net external torque on a system is zero. This is one of the most fundamental conservation laws in physics.

**Angular momentum** — The total angular momentum of a system is constant if the net external torque acting on the system is zero.

*Notation:* $L = I\omega$ (for rigid body rotation)

**Worked example:** An ice skater spins at 1.5 rev s⁻¹ with an initial moment of inertia of 4.8 kg m². They pull their arms inwards, reducing their moment of inertia to 2.4 kg m². Find their new angular speed in rev s⁻¹.

1. There is negligible external torque on the skater, so angular momentum is conserved: $L_1 = L_2$
2. $$I_1\omega_1 = I_2\omega_2$$
3. Rearrange to solve for $\omega_2$:
4. $$\omega_2 = \frac{I_1}{I_2}\omega_1 = \frac{4.8}{2.4} \times 1.5 = 3 \text{ rev s}^{-1}$$

## Common pitfalls

- **Wrong:** Forgetting to convert angular velocity from rpm to radians per second
  - Why it fails: All rotational formulas require angular quantities to be in radians, not revolutions
  - Correct: Multiply rpm by $\frac{2\pi}{60}$ to convert to rad s⁻¹
- **Wrong:** Using the wrong moment of inertia for a given shape and axis
  - Why it fails: Common confusion between standard moment of inertia values for different axes
  - Correct: Memorize IB standard values, or derive using the parallel axis theorem from center of mass values
- **Wrong:** Ignoring rotational kinetic energy for rolling objects in energy conservation problems
  - Why it fails: Only accounting for linear kinetic energy leads to incorrect final speeds
  - Correct: Total kinetic energy for rolling without slipping is $\frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2$
- **Wrong:** Calculating torque as $rF$ without the $\sin\theta$ term
  - Why it fails: Torque depends on the perpendicular component of force, not the full force magnitude
  - Correct: Always include $\sin\theta$, where $\theta$ is the angle between $r$ and $F$
- **Wrong:** Applying conservation of angular momentum when net external torque is non-zero
  - Why it fails: The conservation law only holds for isolated systems with zero net external torque
  - Correct: Check for external torques (e.g. gravity on an off-center pivot) before using conservation

## Cheatsheet

| Quantity | Linear | Rotational | Relationship |
| --- | --- | --- | --- |
| Displacement | $x$ | $\theta$ | $x = r\theta$ |
| Velocity | $v$ | $\omega$ | $v = r\omega$ |
| Acceleration | $a$ | $\alpha$ | $a = r\alpha$ |
| Inertia | $m$ | $I$ | $I = \sum m_i r_i^2$ |
| Force/Torque | $F$ | $\tau$ | $\tau = rF\sin\theta$ |
| Newton's 2nd | $F_{net}=ma$ | $\tau_{net}=I\alpha$ | - |
| Momentum | $p=mv$ | $L=I\omega$ | Conserved if $\tau_{net,ext}=0$ |
| Kinetic Energy | $\frac{1}{2}mv^2$ | $\frac{1}{2}I\omega^2$ | $KE_{total} = KE_{lin} + KE_{rot}$ |

## What's next

Rotational mechanics is a foundational core concept that extends to nearly all areas of IB Physics HL. You will use its principles when analyzing orbital motion in astrophysics, torsional oscillations in simple harmonic motion, and collision problems involving rotating bodies. The pattern of linking linear and rotational quantities you learn here also generalizes to more advanced mechanics concepts you may encounter in university study. Mastering problem-solving for rotation now will help you avoid common errors in multi-topic exam questions that combine energy, momentum, and rotational motion.

- [A.5 Special relativity: energy and momentum (AHL)](https://www.owlsprep.com/study/ib-physics-hl-u1-a-5-special-relativity-energy/)
- [A.6 Circular motion and gravitation (AHL)](https://www.owlsprep.com/study/ib-physics-hl-u1-a-6-circular-motion-and/)
- [Theme B: The particulate nature of matter](https://www.owlsprep.com/study/ib-physics-hl-u2-overview/)

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