Rotational Motion

1. Chapter in One View

Central idea: identify what is rotating, choose the relevant axis, determine what changes, identify the physical cause, and then select the governing relation.

• Describe rotation using angular displacement θ, angular velocity ω and angular acceleration α.
• Connect angular and linear quantities using the distance from the axis.
• Use net torque to explain angular acceleration; use force and torque balance for equilibrium.
• Choose moment of inertia only after identifying both the body and the axis.
• Check net external torque before applying angular-momentum conservation.
• Choose dynamics or energy according to the information given and the quantity asked.
• For rolling, combine translation and rotation using the no-slip constraint.
• For stability, locate the centre of mass relative to the support base.

2. Must-Know Concepts

• Fixed-axis rigid-body rotation: all points share the same angular displacement, angular velocity and angular acceleration, but generally have different linear speeds and accelerations.
• Angular displacement θ is measured in radians in the standard angular–linear relations.
• Angular velocity ω describes the rate of change of angular position; angular acceleration α describes the rate of change of angular velocity.
• Tangential acceleration changes speed; radial (centripetal) acceleration changes the direction of velocity.
• Torque depends on force and the perpendicular distance from the axis to the force's line of action.
• Static equilibrium of a rigid body requires both zero net external force and zero net external torque.
• A couple has zero net force but can produce a non-zero torque.
• Moment of inertia measures resistance to angular acceleration about a specified axis; it depends on mass distribution and the chosen axis.
• Angular momentum depends on motion and the chosen reference point/axis. Conservation requires zero net external torque about that point/axis.
• Rotational work is energy transferred by torque through angular displacement; power is the rate of energy transfer.
• Pure rolling combines centre-of-mass translation and rotation. On a stationary surface, vCM = Rω when rolling without slipping.
• Equilibrium can be stable, unstable or neutral depending on how the system responds to a small displacement.

3. Formula + Meaning

Angular and linear motion

Meaning: Arc length corresponding to angular displacement θ.

Use when: A point moves through angle θ at a fixed radius r.

Condition: θ must be in radians.

Do not use blindly when: The radius or axis changes without accounting for the changing geometry.

Exam trap: Do not substitute degrees directly.

Meaning: Tangential speed of a point at distance r from the axis.

Use when: Angular speed and the point's perpendicular distance from a fixed axis are known.

Condition: For fixed-axis rotation; r is the distance from the axis.

Do not use blindly when: You need the velocity direction as well as its magnitude.

Exam trap: All points share ω, not the same v.

Meaning: Tangential acceleration due to changing angular speed.

Use when: Angular acceleration is known for fixed-axis rotation.

Condition: Use the point's distance r from the axis.

Do not use blindly when: Only velocity direction changes while speed remains constant.

Exam trap: Tangential acceleration is not radial acceleration.

Meaning: Radial acceleration toward the centre of the circular path.

Use when: A point follows a circular path and its speed or angular speed is known.

Condition: Use the radius of that point's circular path.

Do not use blindly when: The path is not circular or the required quantity is tangential acceleration.

Exam trap: Uniform circular motion has radial acceleration even when aₜ = 0.

Meaning: Average and instantaneous angular acceleration.

Use when: Angular velocity changes over time.

Condition: Use the derivative for instantaneous acceleration.

Do not use blindly when: A constant-α equation is used when α varies.

Exam trap: Check signs against the chosen positive direction.

Constant angular acceleration

Meaning: Rotational counterparts of constant-acceleration kinematics.

Use when: Angular acceleration is constant and the initial state is known.

Condition: These equations require constant α.

Do not use blindly when: Torque or α varies so that angular acceleration is not constant.

Exam trap: Do not use them merely because the motion is

rotational.

Torque and rotational dynamics

Meaning: Turning effect of a force about a chosen point or axis.

Use when: A force acts at a known position and its line of action can be located.

Condition: φ is the angle between r and F; d⊥ is the perpendicular distance from the axis to the line of action.

Do not use blindly when: The line of action passes through the axis; torque is then zero.

Exam trap: The distance to the point of application is not necessarily the lever arm.

Recognition tip: use rF sinφ when r and the angle are known; use Fd⊥ when the perpendicular lever arm is easiest to identify.

Meaning: Net torque produces angular acceleration about a fixed axis.

Use when: The body rotates about a fixed axis and I is calculated about that same axis.

Condition: Use the net torque component about the rotation axis.

Do not use blindly when: The axis is not fixed or a general three-dimensional angular-momentum analysis is required.

Exam trap: Use signed torques consistently; one force's torque is not automatically the net torque.

Meaning: Conditions for static equilibrium of a rigid body.

Use when: The body remains at rest in an inertial frame.

Condition: Include every external force and its torque about a consistent point.

Do not use blindly when: Only one of the two conditions has been checked.

Exam trap: Zero net force alone does not guarantee rotational equilibrium.

Efficient method: choose the torque point so that unknown forces acting through that point contribute zero torque.

Meaning: Torque of a couple, where d is the perpendicular separation between the two force lines.

Use when: Two equal, opposite, parallel forces act along distinct lines.

Condition: d is the perpendicular distance between their lines of action.

Do not use blindly when: The forces do not form a couple.

Exam trap: A couple can turn a body even though its net force is zero.

Moment of inertia

Meaning: Measures how mass is distributed relative to an axis.

Use when: Masses and their perpendicular distances are known, or a continuous distribution is given.

Condition: r is the perpendicular distance from the axis.

Do not use blindly when: The body or axis has not been identified.

Exam trap: Moment of inertia belongs to the body–axis combination.

Meaning: Defines radius of gyration k.

Use when: An equivalent effective distance for the mass distribution is required.

Condition: k refers to the same axis as I.

Do not use blindly when: You assume k must equal the physical radius of the object.

Exam trap: Radius of gyration is an effective distance, not necessarily an actual edge.

Meaning: Parallel-axis theorem.

Use when: The moment about a parallel axis through the centre of mass is known.

Condition: Axes are parallel; d is their perpendicular separation.

Do not use blindly when: The known axis is not through the centre of mass or axes are not parallel.

Exam trap: Use the correct M and axis separation.

Meaning: Perpendicular-axis theorem for a planar lamina.

Use when: x and y axes lie in the plane and z is perpendicular to it; all pass through the same point.

Condition: The body is a planar lamina.

Do not use blindly when: The object is an arbitrary three-dimensional solid.

Exam trap: Do not apply it to every solid body.

Standard moments of inertia

Axis check: before selecting a table value, match both the object's geometry and the specified axis.

Angular momentum, work, energy and power

Meaning: Angular momentum of a particle about a chosen point.

Use when: Position relative to the reference point and momentum are known.

Condition: φ is the angle between r and p.

Do not use blindly when: The reference point is unspecified.

Exam trap: Angular momentum depends on the chosen reference point.

Meaning: Angular momentum of a rigid body in the standard fixed-axis model.

Use when: Fixed-axis rotation and I about that axis are known.

Condition: Use the actual rotation axis and the applicable rigid-body model.

Do not use blindly when: A general three-dimensional rotation requires more general treatment.

Exam trap: Do not substitute I about a different axis.

Meaning: Net external torque changes angular momentum; its time integral gives angular-momentum change.

Use when: Angular-momentum change over a time interval is required.

Condition: Use the same reference point/axis for torque and L.

Do not use blindly when: You assume L is conserved despite a significant net external torque.

Exam trap: Zero net force is not the angular-momentum conservation condition.

Meaning: Angular momentum is conserved when net external torque about the chosen point/axis is zero or negligible.

Use when: The system and interval have negligible net external torque about that point/axis.

Condition: Check system boundary, reference point and torque during the whole interval.

Do not use blindly when: You have checked only that net external force is zero.

Exam trap: Conservation depends on torque, not force.

Meaning: Kinetic energy associated with rotation about an axis.

Use when: A rigid body rotates with angular speed ω.

Condition: I is about the rotation axis.

Do not use blindly when: Rolling translation is present and total kinetic energy is requested.

Exam trap: For rolling, add translational kinetic energy.

Meaning: Work done by torque through angular displacement.

Use when: Torque acts through a known angular displacement.

Condition: The product τΔθ applies when torque is constant.

Do not use blindly when: Torque varies and a single value is used without justification.

Exam trap: Work and torque are different physical quantities.

Meaning: Instantaneous rotational power for torque and angular velocity about the same axis.

Use when: The rate of rotational energy transfer is required.

Condition: Use the torque component along the rotation axis.

Do not use blindly when: Total work rather than rate is requested.

Exam trap: Power can be negative when torque opposes angular motion.

Rolling motion

Meaning: Translation–rotation constraint for pure rolling.

Use when: A circular body rolls without slipping on a stationary surface.

Condition: Use the radius R and compatible signed directions.

Do not use blindly when: The body slips or surface motion has not been accounted for.

Exam trap: This is a kinematic constraint, not a force law.

Meaning: Total kinetic energy of a rolling rigid body.

Use when: The body both translates and rotates about its centre of mass.

Condition: Use ICM for the rotational term in this decomposition.

Do not use blindly when: One component of motion is omitted.

Exam trap: Do not count the same energy twice.

Meaning: Acceleration down an incline for ideal rolling without slipping.

Use when: A body rolls down a fixed incline under gravity.

Condition: Pure rolling holds and no additional force/torque changes the model.

Do not use blindly when: The body slides or an applied force changes the dynamics.

Exam trap: For the same M and R, larger ICM generally gives smaller acceleration.

Meaning: Gravitational potential energy converted into translational and rotational kinetic energy.

Use when: A rolling body descends through height h and mechanical energy is conserved.

Condition: Pure rolling; no dissipative energy loss; v = Rω.

Do not use blindly when: Slipping or other significant energy transfer occurs.

Exam trap: Use the correct ICM for the body.

Equilibrium describes a state in which the net force and net torque are zero. Stability describes what happens after a small displacement.

• Stable equilibrium: a small displacement produces a tendency to return to the original position.
• Unstable equilibrium: a small displacement produces a tendency to move farther away.
• Neutral equilibrium: a small displacement leaves the body in a new equilibrium position.
• For support on a base, stability is maintained while the vertical line through the centre of mass falls within the support base.

5. Formula Selection Guide

6. When to Use / When Not to Use

Torque and equilibrium

• Use torque balance when a rigid body is in equilibrium and forces act at different positions.
• Do not infer rotational equilibrium from ΣF = 0 alone; a couple can have zero net force and non-zero torque.
• Choose a torque point that eliminates unknown forces whose lines of action pass through that point.

Moment of inertia

• Use a standard value only after matching the body, axis and geometry.
• Use the parallel-axis theorem only for parallel axes when the known axis passes through the centre of mass.
• Use the perpendicular-axis theorem only for a planar lamina, with all three axes through the same point.

Important reminders

• Moment of inertia depends on the chosen axis, not only on the object's shape.
• Parallel-axis theorem: I = ICM + Md², where d is the perpendicular distance between parallel axes and the known axis passes through the centre of mass.
• For a planar lamina, perpendicular-axis theorem: Iz = Ix + Iy, with x and y in the plane and z perpendicular to it; all axes pass through the same point.
• Use the listed formulas for the idealized uniform bodies and axes stated in the table.

Angular momentum

• Conserve angular momentum only when net external torque about the selected point/axis is zero or negligible over the interval.
• For a collision, choose the system and reference point first, then check external impulsive torque.

Rolling and friction

• Use vCM = Rω only for pure rolling without slipping, accounting for any motion of the surface.
• Static friction adjusts to the value required by the motion, up to its limiting magnitude fₛ,max = μₛN; it is not always equal to μₛN.
• The instantaneous contact point is useful for instantaneous velocity relative to a stationary surface. Do not automatically use it as the pivot for every torque or energy calculation.

7. Common Misconceptions

• Wrong idea: “Every point on a rotating body has the same speed.” → Correct: Angular velocity is common in fixed-axis rigid rotation, but linear speed depends on distance from the axis.
• Wrong idea: “Zero net force means no rotation.” → Correct: A couple can have zero net force and non-zero torque.
• Wrong idea: “Moment of inertia belongs only to the object.” → Correct: It depends on the axis as well as mass distribution.
• Wrong idea: “Zero external force guarantees angular-momentum conservation.” → Correct: Check net external torque about the chosen point or axis.
• Wrong idea: “Static friction always equals μₛN.” → Correct: μₛN is the limiting magnitude; actual static friction adjusts as required within the limit.
• Wrong idea: “The contact point of a rolling wheel is permanently at rest.” → Correct: The material point in contact changes continuously; the instantaneous contact point has zero ground-frame velocity in ideal pure rolling.

8. Exam Traps

• Using point-to-force distance instead of the perpendicular lever arm.
• Mixing clockwise and anticlockwise torque signs.
• Choosing the wrong moment of inertia because the axis differs from the table's axis.
• Using constant-angular-acceleration equations when α is not constant.
• Applying angular-momentum conservation without checking net external torque.
• Using rotational kinetic energy alone when a rolling body's total kinetic energy is required.
• Assuming static friction always equals μₛN.
• Using v = Rω when slipping occurs.
• Confusing instantaneous contact-point velocity with centre-of-mass velocity.

9. See This → Think This

• Same ω at different radii → Compare linear speeds using v = rω.
• Force applied at an angle → Draw its line of action and find the perpendicular lever arm.
• Beam or ladder at rest → Draw a free-body diagram; use force balance and torque balance.
• Net force zero but body tends to turn → Check for a couple or non-zero net torque.
• Shape and axis given → Select I only after matching both.
• Axis shifted parallel to a centre-of-mass axis → Check the parallel-axis theorem.
• Planar lamina and perpendicular in-plane axes → Check the perpendicular-axis theorem.
• Person moves inward on a rotating platform → If external torque is negligible, I decreases and ω increases.
• Collision produces rotation → Choose the point/axis and check external impulsive torque.
• Torque and angular displacement given → Consider the work–energy route.
• Rolling body descends through a known height → Consider conservation of mechanical energy.
• Small displacement from equilibrium → Ask whether the system returns, moves farther away, or remains in its new position.

10. Quick Checks

• Q: Do all points of a fixed-axis rigid body have the same angular velocity? → Yes.
• Q: What determines the linear speed of a point? → v = rω.
• Q: When is torque of a force zero about a pivot? → When its line of action passes through the pivot.
• Q: What two conditions are needed for static rigid-body equilibrium? → ΣF = 0 and Στ = 0.
• Q: What must be identified before selecting a moment-of-inertia formula? → The body and the axis.
• Q: What condition permits angular-momentum conservation? → Zero or negligible net external torque about the chosen point/axis.
• Q: What is the pure-rolling constraint on a stationary surface? → vCM = Rω.
• Q: What terms appear in the kinetic energy of a rolling body? → Translational and rotational kinetic energy.
• Q: Is static friction always μₛN? → No; that is the limiting magnitude.

11. Mini Exam Reasoning

Clue: A force acts at an angle on a rod pivoted at one end. → Situation: turning effect about the pivot. → Model: torque geometry. → Relation: τ = rF sinφ or Fd⊥. → Check: φ is between r and F; d⊥ is perpendicular to the line of action.

Clue: A person moves closer to the axis of a rotating platform. → Situation: mass distribution changes. → Model: angular-momentum conservation if external torque is negligible. → Relation: I₁ω₁ = I₂ω₂. → Check: if I decreases, ω increases when L is conserved.

Clue: A solid sphere rolls down a height h without slipping. → Situation: gravitational energy becomes translational and rotational kinetic energy. → Model: mechanical energy conservation. → Relation: Mgh = ½Mv² + ½ICMv²/R². → Check: use the correct ICM and v = Rω.

Clue: A ladder rests against a wall. → Situation: static equilibrium. → Model: free-body diagram and balance laws. → Relations: ΣFx = 0, ΣFy = 0, Στ = 0. → Check: choose a torque point that removes as many unknown support forces as possible.

12. One-Minute Recall

• Rotation state → θ, ω, α.
• Angular–linear link → s = rθ; v = rω; aₜ = rα; aᵣ = rω².
• Constant α → rotational kinematics equations.
• Torque → force × perpendicular lever arm.
• Fixed-axis dynamics → Στ = Iα.
• Static equilibrium → ΣF = 0 and Στ = 0.
• Moment of inertia → body + axis; mass farther from axis contributes more.
• Parallel axis → I = ICM + Md²; planar lamina → Iz = Ix + Iy when conditions fit.
• Angular momentum → L = r × p; fixed-axis form L = Iω.
• Conservation → check net external torque, not merely net force.
• Rotational energy → Krot = ½Iω²; work W = ∫τ dθ; power P = τω.
• Pure rolling → vCM = Rω; total kinetic energy has translation and rotation.
• Static friction adjusts up to μₛN; it is not always at its limiting value.
• Stability → stable returns; unstable moves away; neutral remains in a new equilibrium.

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