Theme A — Space, Time and Motion

A.4 Rigid Body Mechanics — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 42 of 63 1 quiz

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Lesson summary

A.4 Rigid Body Mechanics — IB PHYSICS HL notes: Apply torque, rotational equilibrium, moment of inertia, angular momentum and rotational kinetic energy to rigid bodies.Moment of a force = force × perpendicular distance from the pivot, in N m.If the force is not perpendicular, res…

Principle of moments
Choose the pivot point to remove unknown forces
Angular displacement θ
Angular velocity ω
What the symbols mean
Angular acceleration α

Part 1 of 8Moments, couples and torque

Moment of a force

Moment of a force force × perpendicular distance from the pivot, in N m.

If the force is not perpendicular, resolve it with trigonometry.

Couple

Couple a pair of equal, opposite, coplanar forces, perpendicular to the distance between them, that produce rotation only.

Moment of a couple = force × perpendicular distance between the lines of action — it does not depend on the pivot.
Resultant force is zero, so the body is in translational equilibrium but has a net torque.

Torque τ

Fr sin θ, with θ the angle between the force and the line from the axis.

τ = Fr for a perpendicular force; the maximum torque occurs at 90°.
A couple of two forces F at radius r gives τ = 2Fr.

Part 2 of 8Rotational equilibrium

Rotational equilibrium

Rotational equilibrium zero resultant torque — the body stays at rest or rotates at constant angular velocity.

Principle of moments

Principle of moments total clockwise torque = total anticlockwise torque.

Choose the pivot point to remove unknown forces

Choose the pivot point to remove unknown forces through which their lines of action pass.

A resultant torque causes angular acceleration in its direction.

At this level only the clockwise or anticlockwise sense of torque is needed.

Part 3 of 8Angular displacement, velocity and acceleration

Angular displacement θ

Angular displacement θ change in angle in radians — s = rθ.

Angular velocity ω

Δθ/Δt in rad s^-1 — v = ωr,

and ω = 2πf = 2π/T.
What the symbols mean: v = speed (m s⁻¹) · ω = angular speed (rad s⁻¹) · r = radius (m) · T = period (s)

Angular acceleration α

Δω/Δt in rad s^-2 — a = αr.

Variable	Linear	Angular
Displacement	s = rθ	θ = s/r
Velocity	v = ωr	ω = v/r
Acceleration	a = αr	α = a/r

Angular displacement, velocity and acceleration

Graphs.

Angular displacement is the area under the ω–t graph.
Angular velocity is the gradient of the θ–t graph and the area under the α–t graph.
Angular acceleration is the gradient of the ω–t graph.

Angular acceleration is not the same as centripetal acceleration.

Part 4 of 8Rotational kinematic equations

Linear equations map onto angular ones by swapping s, u, v, a for θ, ω_i, ω_f, α.

ωf

ω_i + αt

ω_it + ½αt²

ωf2

ω_i² + 2αθ

½(ω_i + ω_f)t

One rotation is 2π rad

One rotation is 2π rad so the number of rotations is θ/2π.

Part 5 of 8Moment of inertia and Newton's second law

1Moment of inertia I

Moment of inertia I the resistance to a change in rotational motion — depends on the distribution of mass about the chosen axis.

Unit kg m²; it depends on shape, density and orientation of the axis.

2Point mass

I = mr²; system — I_tot = Σmr².

3Moments of inertia of shapes are given in exam questions

Moments of inertia of shapes are given in exam questions you need not memorise them.

4Newton's second law for rotation

τ = Iα.

Linear	Rotational
Force F	Torque τ
Mass m	Moment of inertia I
Acceleration a	Angular acceleration α
F = ma	τ = Iα

Part 6 of 8Angular momentum and angular impulse

Angular momentum L

Iω, in kg m² s^-1.

For a point mass L = mvr — a body moving in a straight line can have angular momentum about an axis.

Conservation

Conservation total angular momentum is constant unless a resultant torque acts: I_iω_i = I_fω_f.
A skater spins faster when arms are pulled in, because I decreases.
Planets move faster nearer the body they orbit; tornados speed up as the radius decreases.

Angular impulse

τΔt = ΔL = Δ(Iω), in N m s.

The area under a torque–time graph is the angular impulse.
A small torque for a long time has the same effect as a large torque for a short time.

Part 7 of 8Rotational kinetic energy

½Iω² = L²/2I.

Rolling without slipping

Rolling without slipping rotation plus translation.
The contact point is at rest, the centre of mass has v = ωr and the top point has 2v.
If a disc slips, every point has the same linear velocity and ω is zero.

Rolling down a slope

Rolling down a slope mgh = ½mv² + ½Iω².
For a solid sphere I = (2/5)mr², giving mgh = (7/10)mv².
A solid cylinder has total kinetic energy ¾mv².

Part 8 of 8More from the specification

1Torque and Applied Force

The torque acting on a rigid body is defined as the turning effect of an applied force, calculated from tau = r * F * sin(theta),
where r is the distance from the pivot and theta is the angle of application.

2Action of a Couple

A couple consists of two parallel forces of equal magnitude acting in opposite directions along different lines of action on an object,
producing a pure rotational torque with zero resultant linear force.

3Newton's Second Law for Rotation

The net resultant torque acting on a rigid object equals its moment of inertia multiplied by its angular acceleration,
- (Sigma tau = I * alpha)
which allows engineers to determine rotational motions.

4Rotational Kinematics with Constant Angular Acceleration

Using rotational equations of motion, angular displacement Delta theta is determined from initial angular velocity omega_i, angular acceleration alpha,
and time t via Delta theta = omega_i * t + 1/2 * alpha * t².

5Torque and Applied Force

The torque acting on a rigid body is defined as the turning effect of an applied force, calculated from tau = r * F * sin(theta),
where r is the distance from the pivot and theta is the angle of application.

6Action of a Couple

A couple consists of two parallel forces of equal magnitude acting in opposite directions along different lines of action on an object,
producing a pure rotational torque with zero resultant linear force.

7Newton's Second Law for Rotation

The net resultant torque acting on a rigid object equals its moment of inertia multiplied by its angular acceleration,
- (Sigma tau = I * alpha)
which allows engineers to determine rotational motions.

8Rotational Kinematics with Constant Angular Acceleration

Using rotational equations of motion, angular displacement Delta theta is determined from initial angular velocity omega_i, angular acceleration alpha,
and time t via Delta theta = omega_i * t + 1/2 * alpha * t².

9Torque and Applied Force

The torque acting on a rigid body is defined as the turning effect of an applied force, calculated from tau = r * F * sin(theta),
where r is the distance from the pivot and theta is the angle of application.

10Action of a Couple

A couple consists of two parallel forces of equal magnitude acting in opposite directions along different lines of action on an object,
producing a pure rotational torque with zero resultant linear force.

11Newton's Second Law for Rotation

The net resultant torque acting on a rigid object equals its moment of inertia multiplied by its angular acceleration,
- (Sigma tau = I * alpha)
which allows engineers to determine rotational motions.

12Rotational Kinematics with Constant Angular Acceleration

Using rotational equations of motion, angular displacement Delta theta is determined from initial angular velocity omega_i, angular acceleration alpha,
and time t via Delta theta = omega_i * t + 1/2 * alpha * t².

Key terms

Term	Meaning
Torque	Turning effect of a force, τ = Fr sin θ.
Couple	Pair of equal and opposite forces producing rotation only.
Moment of inertia	Resistance to change in rotational motion, Σmr².
Angular momentum	L = Iω.
Angular impulse	Torque multiplied by time, equal to change in angular momentum.
Rotational equilibrium	State of zero resultant torque.
Rotational kinetic energy	Energy of rotation, ½Iω^2.

Avoid these mistakes

Using the full distance from the pivot instead of the perpendicular distance.

Thinking a couple gives a resultant force — it gives torque only.

Forgetting to use radians in angular equations.

Mixing angular acceleration with centripetal acceleration.

Ignoring the change in moment of inertia when applying conservation of angular momentum.

Last updated: 18 October 2026