A.4 Rotational mechanics

IB Physics HL· 45 min read

1. Rotational Kinematics: Linear-Rotational Analogues★★☆☆☆⏱ 10 min

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

📘 Definition

Rotational kinematic quantities

Angular displacement $θ$ (rad), angular velocity $ω$ (rad s⁻¹), angular acceleration $α$ (rad s⁻²)

Quantities that describe the rotation of a rigid body about a fixed axis, connected to linear quantities for a point at radius $r$ by $x = r θ$ , $v = r ω$ , $a = r α$ .

Linear quantity	Rotational equivalent	Uniform acceleration equation
Displacement $x$	$θ$	$θ = ω_{0} t + \frac{1}{2} α t^{2}$
Velocity $v$	$ω$	$ω^{2} = ω_{0}^{2} + 2 α θ$
Acceleration $a$	$α$	$ω = ω_{0} + α t$

📐 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.

2. Torque and Moment of Inertia★★★☆☆⏱ 15 min

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.

📘 Definition

Torque

$τ = r F sin θ$

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

📘 Definition

Moment of inertia

$I = \sum m_{i} r_{i}^{2}$

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.

The parallel axis theorem lets you calculate $I$ for any axis parallel to one through the center of mass: $I = I_{c m} + M d^{2}$ , 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 $I_{c m} = \frac{1}{12} M L^{2}$ 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
$I = \frac{1}{12}ML^2 + \frac{3}{12}ML^2 = \frac{1}{3}ML^2$

3. Newton's Second Law for Rotation and Rotational KE★★★★☆⏱ 20 min

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.

📘 Definition

Newton's second law for rotation

The net torque on a rigid body equals the product of its moment of inertia and angular acceleration: $τ_{n e t} = I α$

📘 Definition

Rotational kinetic energy

Kinetic energy due to rotation: $E_{k} = \frac{1}{2} I ω^{2}$ . 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 $θ$ . Find its linear acceleration down the slope.

1
Torque from friction about the center of the cylinder equals $I α$ . For solid cylinder, $I = \frac{1}{2} M R^{2}$ , and $α = \frac{a}{R}$ 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.

4. Angular Momentum and Conservation★★★☆☆⏱ 15 min

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.

📘 Definition

Angular momentum

$L = I ω$ (for rigid body rotation)

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

📐 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 $ω_{2}$ :
4
$\omega_2 = \frac{I_1}{I_2}\omega_1 = \frac{4.8}{2.4} \times 1.5 = 3 \text{ rev s}^{-1}$

5. Common Pitfalls

Wrong move:

Forgetting to convert angular velocity from rpm to radians per second

Why:

All rotational formulas require angular quantities to be in radians, not revolutions

Correct move:

Multiply rpm by $\frac{2 π}{60}$ to convert to rad s⁻¹

Wrong move:

Using the wrong moment of inertia for a given shape and axis

Why:

Common confusion between standard moment of inertia values for different axes

Correct move:

Memorize IB standard values, or derive using the parallel axis theorem from center of mass values

Wrong move:

Ignoring rotational kinetic energy for rolling objects in energy conservation problems

Why:

Only accounting for linear kinetic energy leads to incorrect final speeds

Correct move:

Total kinetic energy for rolling without slipping is $\frac{1}{2} M v^{2} + \frac{1}{2} I ω^{2}$

Wrong move:

Calculating torque as $r F$ without the $sin θ$ term

Why:

Torque depends on the perpendicular component of force, not the full force magnitude

Correct move:

Always include $sin θ$ , where $θ$ is the angle between $r$ and $F$

Wrong move:

Applying conservation of angular momentum when net external torque is non-zero

Why:

The conservation law only holds for isolated systems with zero net external torque

Correct move:

Check for external torques (e.g. gravity on an off-center pivot) before using conservation

6. Quick Reference Cheatsheet

Quantity	Linear	Rotational	Relationship
Displacement	$x$	$θ$	$x = r θ$
Velocity	$v$	$ω$	$v = r ω$
Acceleration	$a$	$α$	$a = r α$
Inertia	$m$	$I$	$I = \sum m_{i} r_{i}^{2}$
Force/Torque	$F$	$τ$	$τ = r F sin θ$
Newton's 2nd	$F_{n e t} = ma$	$τ_{n e t} = I α$
Momentum	$p = m v$	$L = I ω$	Conserved if $τ_{n e t, e x t} = 0$
Kinetic Energy	$\frac{1}{2} m v^{2}$	$\frac{1}{2} I ω^{2}$	$K E_{t o t a l} = K E_{l in} + K E_{r o t}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · Paper 2
10-mark rolling cylinder problem
2024 · Paper 1
Conservation of angular momentum MCQ
2023 · Paper 2
Torque and rotation problem

Going deeper

IB SyllabusIB Physics HL Guide 2025Topic A.4 pages 45-47

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.