A.4 Rigid body mechanics

UNIT SYLLABUS

HL 7 hours · HL only

Everything you learned about linear motion has a rotational twin, and this HL unit introduces the full dictionary: force becomes torque, mass becomes moment of inertia, momentum becomes angular momentum, and Newton's second law becomes $\tau = I\alpha$. The deep new idea is that how mass is distributed matters as much as how much there is — a hollow cylinder is harder to spin than a solid one of equal mass. Conservation of angular momentum explains why an ice skater spins faster when pulling in their arms, why neutron stars rotate hundreds of times a second, and why helicopters need tail rotors.

Guiding Questions

? How is the rotational motion of a rigid body analogous to the translational motion of a point mass?
? What quantities are conserved in rotational dynamics, and under what conditions?

✓ What the IB expects you to master

Calculate the torque of a force about an axis: $\tau = Fr\sin\theta$ .
Apply the conditions for rotational equilibrium (resultant torque zero) and recognise that an unbalanced torque produces angular acceleration.
Use the rotational suvat equations for uniform angular acceleration ( $\omega_f = \omega_i + \alpha t$ , $\Delta\theta = \omega_i t + \frac{1}{2}\alpha t^2$ , $\omega_f^2 = \omega_i^2 + 2\alpha\Delta\theta$ ).
Explain how moment of inertia depends on the distribution of mass about the axis, and compute $I = \Sigma mr^2$ for point-mass systems.
Apply Newton's second law for rotation, $\tau = I\alpha$ .
Use angular momentum $L = I\omega$ , its conservation in the absence of external torque, and angular impulse $\Delta L = \tau\Delta t$ .
Calculate rotational kinetic energy $E_k = \frac{1}{2}I\omega^2$ , including bodies that roll (translation + rotation).

1 Key Formulas

Torque

\tau = Fr\sin\theta

Rotational second law

\tau = I\alpha

Moment of inertia (point masses)

I = \Sigma mr^{2}

Rotational kinematics

\omega_{f} = \omega_{i} + \alpha t

Angular displacement

\Delta\theta = \omega_{i}t + \tfrac{1}{2}\alpha t^{2}

Angular velocity–displacement

\omega_{f}^{2} = \omega_{i}^{2} + 2\alpha\Delta\theta

Angular momentum

L = I\omega

Angular impulse

\Delta L = \tau\Delta t

Rotational kinetic energy

E_{k} = \tfrac{1}{2}I\omega^{2} = \frac{L^{2}}{2I}

2 Exam Preparation & Topic Explanations

The linear–rotational dictionary

Almost every rotational problem is a linear problem you already know, translated word for word. Build the dictionary until it is reflexive: $s \to \theta$ , $v \to \omega$ , $a \to \alpha$ , $m \to I$ , $F \to \tau$ , $p \to L$ .

For rolling bodies, total kinetic energy is $\frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$ with $v = \omega r$ — energy conservation with both terms is the standard hard question.

Pro Exam Strategy

Torque depends on where the force acts: $\tau = Fr\sin\theta$ uses the perpendicular distance from the axis.
Conservation of angular momentum questions signal themselves with "no external torque" — skaters, collapsing stars, merging discs.
Moments of inertia for standard shapes are given in exams; only $I = \Sigma mr^2$ for point masses must be computed from scratch.
When something rolls without slipping, friction does no work — mechanical energy is still conserved.

3 MCQ Practice

Q1. An ice skater spinning with arms outstretched pulls her arms in. Which row is correct?

Angular momentum increases; kinetic energy constant
Angular momentum constant; kinetic energy increases
Both angular momentum and kinetic energy constant
Angular momentum constant; kinetic energy decreases

Q2. Two children of equal mass sit on a seesaw, one at 2.0 m and one at 1.5 m from the pivot. For balance, an extra downward force must act on the shorter side. If each child weighs 400 N, what extra torque is needed?

$100\ \text{N m}$
$200\ \text{N m}$
$400\ \text{N m}$
$800\ \text{N m}$

Q3. A solid sphere and a hollow sphere of equal mass and radius roll from rest down the same incline. Which reaches the bottom first?

The solid sphere
The hollow sphere
They arrive together
It depends on the angle of the incline

4 Short Answer Questions

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