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Rotational Motion

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

  • GCE A-Level H3 Physics 2027
Learning goals
  • 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
  • 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.
  • 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)

Rotational motion extends particle mechanics to rigid bodies turning about an axis of fixed orientation. This detailed route develops angular kinematics, mass distribution, torque, angular momentum, energy, and rolling without slipping in separate lessons.

Choose your route

Start with kinematics of angular motion, then follow the ordered lessons on this hub. Each lesson adds the model needed by the next.

Start here

Prerequisite check

You should be able to resolve forces, apply Newton’s laws, use radians, and conserve energy and linear momentum. Review Circular Motion (H2) and Work, Energy and Power (H2) if those foundations still need work.

Use this hub if translation-only mechanics is stable but the choice of axis, sign of torque, or translation–rotation split still becomes unclear in H3 structured questions. Every solution should name the axis before using I, L, or τ.

Deep-dive lessons

Study these in order. Moment of inertia appears before dynamics because τ = Iα cannot be applied until you understand the axis-dependent meaning of I.

  1. Kinematics of angular motion — define θ, ω, and α and apply the constant-α equations.
  2. Moment of inertia — calculate I from mass distribution, integration, and the parallel-axis theorem.
  3. Dynamics of angular motion — calculate torque and use a signed torque balance about one fixed axis.
  4. Torque and angular momentum — use vector τₑₓₜ = d vector L/dt and identify when angular momentum is conserved.
  5. Rotational kinetic energy — derive and apply E_(k,rot) = (1/2)Iω².
  6. Rolling motion — combine centre-of-mass translation with rotation and test the no-slip condition.

Revision

Quick reference
  • Radians only: s = rθ, with θ in radians.
  • Linear–angular links: v = rω, aₜ = rα, and a_c = rω².
  • Dynamics: τₑₓₜ = Iα for a rigid body about a fixed axis with constant I.
  • Angular momentum: L = Iω for fixed-axis rotation; more generally, vector τₑₓₜ = d vector L/dt.
  • Rotational energy: E_(k,rot) = (1/2)Iω².
  • Rolling without slip: v_cm = ω R in magnitude; use a consistent sign convention for angular quantities.
  • Parallel-axis theorem: I = I_cm + Md².
Problem templates

Moment of inertia by integration

  1. Name the rotation axis.
  2. Write I = ∫ r² dm.
  3. Express r and dm using one integration variable.
  4. Set physical limits, integrate, and check the units and limiting mass distribution.

Torque balance

  1. Name the axis and choose the positive rotational direction.
  2. Calculate every torque with τ = rF sin φ = Fd_⊥ and its sign.
  3. Apply ∑τₑₓₜ = Iα with I about that same axis.
  4. Check the limit ∑τₑₓₜ → 0.

Rolling down a slope

  1. Use Mgh = (1/2)Mv² + (1/2)I_cmω² when mechanical energy is conserved.
  2. Apply v = ω R for no slip.
  3. If friction data are given, calculate the required contact force and test it against F ≤ μ N using the syllabus convention.
Exam traps
  1. Using degrees in s = rθ or v = rω.
  2. Using an I calculated about a different axis from the torque or angular momentum.
  3. Omitting translational kinetic energy for a rolling body.
  4. Mixing up perpendicular components aₜ = rα and a_c = rω².
  5. Assuming a no-slip rolling problem must have non-zero friction instead of calculating the required contact force.

Practice

  1. Derive I for a uniform rod about its centre, then use the parallel-axis theorem for an end axis.
  2. Solve one signed torque-balance problem and check that α → 0 when the external torques cancel.
  3. Compare a ring and a disc rolling from the same height, explaining the result through I/(MR²).

Use the H3 Physics paper map for timed structured-question practice.

Continue learning

Continue to Electric and Magnetic Fields, the next maintained H3 topic. Return to Frames of Reference if axis or observer choices are still causing sign errors.

Course and syllabus information
Course
GCE A-Level H3 Physics
Edition
GCE A-Level H3 Physics 2027