Rotational motion

Angular kinematics, rotational dynamics, moment of inertia, angular momentum, rotational energy, and rolling motion for H3 Physics.

GCE A-Level H3 Physics 2027
8 lessons

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Before you begin

A spinning wheel, a rolling cylinder and a skater pulling in their arms are all rigid bodies turning about an axis. The lessons describe rotation with angular displacement, velocity and acceleration, then show how the moment of inertia depends on mass distribution and how to calculate it. They go on to torque and angular acceleration, angular momentum, rotational kinetic energy, and finally rolling without slipping.

Be comfortable with: circular motion, including radians and v = rω, and work, energy and power.

Learning goals

Use angular displacement, angular velocity and angular acceleration for a rigid body rotating about a fixed axis
Solve problems with the equations of motion for uniform angular acceleration
Use angular momentum and moment of inertia for a rotating rigid body
Calculate the moment of inertia of simple bodies about an axis, using calculus or the parallel-axis theorem
Use torque as the rate of change of angular momentum to solve problems, including bodies with a changing moment of inertia such as an ice-skater
Apply Eₖ,rot = ½Iω² to the rotational kinetic energy of a rigid body.
Derive Eₖ,rot = ½Iω² for a rigid body from the equations of motion.
Treat rigid-body motion as translation of the centre of mass plus rotation about it, including rolling without slipping with F ≤ µN

Syllabus statements covered

show an understanding of and use the terms angular displacement, angular velocity, and angular acceleration of a rigid body with respect to a fixed axis
solve problems using the equations of motion for uniform angular acceleration that are analogous to the equations of motion for uniform linear acceleration
show an understanding of and use the terms angular momentum and moment of inertia of a rotating rigid body
calculate the moment of inertia about an axis for simple bodies by using calculus, the parallel-axis theorem or otherwise (knowledge of the perpendicular-axis theorem and mathematical derivation of the moment of inertia for spheres are not required)
show an understanding of torque produced by a force relative to a reference point, and apply the principle that torque is related to the rate of change of angular momentum to solve problems, such as those involving point masses, rigid bodies, or bodies with a variable moment of inertia e.g. an ice-skater
derive, from the equations of motion, and apply the formula Eₖ,rot = ½Iω² for the rotational kinetic energy of a rigid body
recall and apply the result that the motion of a rigid body can be regarded as translational motion of its centre of mass with rotational motion about an axis through the centre of mass to solve problems, including the use of F ⩽ µN for solid surfaces in no-slip contact (no distinction is made between the coefficient of static and kinetic friction)

Lessons

Work through them in order.

Practise and check

Or choose

Topic reference

Quick reference

Idea	Relation	Notes
linear–angular links	s = rθ, v = rω, aₜ = rα	angles in radians
centripetal acceleration	aᵣ = rω²	perpendicular to aₜ
constant angular acceleration	ω = ω₀ + α t, ω² = ω₀² + 2αΔθ	like the linear equations
moment of inertia	I = ∑ mᵢrᵢ² = ∫ r² dm	depends on the axis
parallel-axis theorem	I = I_cm + Md²	I_cm about a parallel axis through the COM
torque	τ = rF sin φ = Fd_⊥	N m
Newton’s second law for rotation	∑τₑₓₜ = Iα	fixed axis, constant I
angular momentum	L = Iω, ∑τₑₓₜ = dL/dt	conserved when the external torque is zero
rotational kinetic energy	Eₖ = (1/2)Iω²
rolling without slipping	v_cm = ω R, Eₖ = (1/2)Mv_cm² + (1/2)I_cmω²

Core knowledge to remember

Every point of a rigid body shares the same ω and α, but points farther from the axis move faster.
The moment of inertia depends on how the mass is distributed about the chosen axis; mass farther from the axis counts for more.
Of a set of parallel axes, the one through the centre of mass gives the smallest moment of inertia.
A net external torque changes angular momentum. With no external torque, angular momentum is conserved even if the moment of inertia changes.
A rolling body has both translational and rotational kinetic energy. A body with more of its mass near the rim rolls down a slope more slowly.
For rolling without slipping, the required friction must not exceed μ N.

Common mistakes

Using degrees or revolutions instead of radians.
Using a moment of inertia about a different axis from the torque or angular momentum.
Adding Md² to a moment of inertia that is not about the centre-of-mass axis.
Leaving out the translational kinetic energy of a rolling body.
Confusing tangential acceleration aₜ = rα with centripetal acceleration aᵣ = rω².
Assuming friction equals μ N in rolling, instead of calculating the friction needed.