1bRotational Motion and Torque
Explore angular kinematics, torque, moment of inertia, and conservation of angular momentum.
Rotational motion governs everything from spinning tops to collapsing stars. Torque and angular momentum are essential tools for analyzing engines, gyroscopes, and orbital mechanics.
Why does a longer wrench make it easier to loosen a bolt — and how does a spinning figure skater speed up by pulling in her arms?
Lesson Overview
Rotational motion mirrors linear motion: angular displacement (θ) corresponds to position, angular velocity (ω) to linear velocity, and angular acceleration (α) to linear acceleration. Torque (τ = rF sinθ) is the rotational equivalent of force — it measures how effectively a force causes rotation about a pivot. The moment of inertia (I) is the rotational equivalent of mass, measuring resistance to angular acceleration. Newton's second law for rotation becomes τ = Iα, and angular momentum L = Iω is conserved when no external torques act.
Key Equations
Worked Examples
A wrench applies a force of F = 40 N at r = 0.3 m from the pivot. The force is perpendicular to the lever arm (θ = 90°). Find the torque.
A bolt is tightened by applying F = 25 N at r = 0.2 m from the center. The force makes an angle of θ = 60° with the wrench handle. Find the torque.
A figure skater spins with I₁ = 4 kg·m² at ω₁ = 2 rad/s. She pulls her arms in, reducing her moment of inertia to I₂ = 1 kg·m². Find her new angular velocity ω₂.
A wheel with moment of inertia I = 2 kg·m² is accelerated from rest by a net torque of τ = 6 N·m. Find the angular acceleration α and the angular velocity after t = 4 s.
A uniform disk (I = ½mr²) of mass m = 3 kg and radius r = 0.2 m starts from rest and reaches ω = 10 rad/s in t = 5 s. Find the net torque applied.
Guided Problems
A force of F = 30 N is applied perpendicular to a wrench at r = 0.5 m from the pivot. Find the torque.
Hint: Use τ = rF sinθ. Since the force is perpendicular, θ = 90° and sinθ = 1.
A spinning top has moment of inertia I = 0.005 kg·m² and angular velocity ω = 50 rad/s. Find its angular momentum L.
Hint: Angular momentum is simply L = Iω. Multiply moment of inertia by angular velocity.
A diver tucks into a ball, reducing her moment of inertia from I₁ = 12 kg·m² to I₂ = 3 kg·m². If her initial angular velocity is ω₁ = 1.5 rad/s, find her final angular velocity.
Hint: Use conservation of angular momentum: I₁ω₁ = I₂ω₂. Solve for ω₂.
A wheel starts from rest and reaches ω = 20 rad/s in t = 4 s. Find the angular acceleration α and the total angle θ rotated.
Hint: Use α = Δω/Δt for angular acceleration, then θ = ω₀t + ½αt² for angular displacement.
A net torque of τ = 10 N·m acts on a flywheel with I = 5 kg·m². Find the angular acceleration and the angular velocity after 3 s, starting from rest.
Hint: Use τ = Iα to find α, then ω = ω₀ + αt.
Key Vocabulary
Angular Displacement (θ)
The angle through which an object rotates, measured in radians. One full revolution = 2π radians = 360°.
Example: A wheel rotating through half a turn has θ = π radians.
Angular Velocity (ω)
The rate of change of angular displacement: ω = Δθ/Δt, measured in rad/s. Related to tangential speed by v = ωr.
Example: A wheel completing one revolution per second has ω = 2π ≈ 6.28 rad/s.
Angular Acceleration (α)
The rate of change of angular velocity: α = Δω/Δt, measured in rad/s². The rotational analog of linear acceleration.
Example: A wheel speeding from 0 to 10 rad/s in 5 s has α = 2 rad/s².
Torque (τ)
The rotational equivalent of force — a measure of how effectively a force causes rotation about a pivot. τ = rF sinθ, where θ is the angle between the force and the lever arm.
Example: Applying 40 N perpendicularly at 0.3 m from a pivot produces τ = 12 N·m.
Moment of Inertia (I)
The rotational equivalent of mass — a measure of an object's resistance to angular acceleration. Depends on both mass and how it is distributed: I = Σmr².
Example: A solid disk has I = ½mr²; a hollow ring has I = mr². The ring is harder to spin up.
Angular Momentum (L)
The rotational equivalent of linear momentum: L = Iω. In the absence of external torques, angular momentum is conserved.
Example: A figure skater pulling in her arms decreases I, so ω increases to keep L = Iω constant.
Workbook Check — Interactive Quiz
Interactive Practice — 5 Questions
A force of 30 N is applied perpendicular to a wrench at 0.5 m from the pivot. What is the torque?
A skater with I₁ = 6 kg·m² spinning at ω₁ = 3 rad/s pulls in her arms to I₂ = 2 kg·m². What is her new angular velocity?
A net torque of τ = 8 N·m acts on a wheel with I = 4 kg·m². What is the angular acceleration?
A force is applied to a wrench at angle θ = 0° (parallel to the lever arm). What is the torque?
Which quantity is conserved when a spinning skater pulls in her arms (no external torques)?
Independent Practice
A force of 50 N is applied perpendicular to a door at r = 0.8 m from the hinge. Find the torque.
A wheel with I = 3 kg·m² is accelerated from rest by τ = 9 N·m. Find α and ω after t = 5 s.
A solid disk (I = ½mr²) of mass 4 kg and radius 0.3 m rotates at ω = 8 rad/s. Find its angular momentum L.
A gymnast has I₁ = 10 kg·m² and ω₁ = 2 rad/s. She tucks to I₂ = 2.5 kg·m². Find ω₂.
★ A star of radius R₁ = 5 × 10⁸ m and ω₁ = 2 × 10⁻⁷ rad/s collapses to R₂ = 1 × 10⁴ m. Using I = ⅖MR² and conservation of angular momentum, find ω₂. (Hint: M cancels.)
ChallengeCommon Mistakes
Applying τ = rF without accounting for the angle between r and F
Torque is τ = rF sinθ where θ is the angle between the position vector and the force. Maximum torque occurs at θ = 90°
Confusing angular velocity ω (rad/s) with linear speed v (m/s)
They are related by v = ωr. Angular velocity is the same for all points on a rigid body; linear speed increases with radius
Forgetting to convert RPM to rad/s before using rotational equations
Convert: ω (rad/s) = RPM × 2π/60. Always use radians in rotational formulas
Thinking rotational KE is conserved when a skater pulls in her arms
Angular momentum is conserved (no external torque), but KE increases because the skater does work pulling her arms inward
Math Tips
Rotational–linear analogies: θ↔x, ω↔v, α↔a, I↔m, τ↔F, L↔p. The rotational kinematic equations mirror the linear ones exactly.
Torque is maximized when force is perpendicular to the lever arm (θ = 90°, sinθ = 1). Zero torque when force is parallel (θ = 0°).
Moment of inertia depends on mass AND distribution: I = mr² (point mass), ½mr² (solid disk), ⅔mr² (hollow sphere). More spread-out mass → larger I.
Angular momentum L = Iω is conserved when net torque = 0. If I decreases (arms pulled in), ω must increase proportionally.