A one-page summary of Torque and Rotational Dynamics — every angular formula, the linear ↔ rotational analogy table, and the recipe for solving equilibrium problems.
What's covered: Rotational kinematics, connecting linear and rotational motion, torque, rotational inertia, rotational equilibrium, and Newton's second law in rotational form.
Exam weight: 10–15%.
The big trick: Every linear physics equation has a rotational version — just swap the variables.
The master idea: τ_net = Iα is the rotational F = ma. Solve it the same way.
🔄 The linear ↔ rotational translation table
Memorize this. Every linear concept has a rotational analog with the same form of equation.
Concept
Linear
Rotational
Position / Angle
x
θ (radians)
Velocity
v
ω = Δθ/Δt
Acceleration
a
α = Δω/Δt
Mass / Inertia
m
I = Σmr²
Force / Torque
F
τ = rF·sin(θ)
Newton's 2nd Law
F = ma
τ_net = Iα
Kinematic 1
v = v₀ + at
ω = ω₀ + αt
Kinematic 2
x = x₀ + v₀t + ½at²
θ = θ₀ + ω₀t + ½αt²
Kinematic 3
v² = v₀² + 2aΔx
ω² = ω₀² + 2αΔθ
📐 Key equations
ω = Δθ/Δt
Angular velocity. Rate of change of angle. Units: rad/s. Rotational analog of v = Δx/Δt.
α = Δω/Δt
Angular acceleration. Rate of change of ω. Units: rad/s². Rotational analog of a = Δv/Δt.
v = rω, a_t = rα
Linear ↔ rotational link. Tangential velocity and acceleration of a point at radius r from the axis.
a_c = ω²r
Centripetal acceleration. Always toward the center for circular motion. Also equal to v²/r.
τ = rF·sin(θ)
Torque. Lever arm × perpendicular component of force. Units: N·m. Zero when force is along the lever arm.
I = Σmr² (point masses)
Rotational inertia. Sum mass × distance from axis², for each piece. Larger r matters more (r is squared).
Στ = 0
Rotational equilibrium. Sum of clockwise torques equals sum of counterclockwise torques. Object's ω is constant.
τ_net = Iα
Newton's 2nd law (rotational). Net torque equals rotational inertia times angular acceleration. Use it like F = ma.
The 6 topics at a glance
5.1 Rotational Kinematics
Angular position (θ), angular velocity (ω), angular acceleration (α). Same kinematic equations as Unit 1 with rotational variables. Constant α required.
5.2 Linear ↔ Rotational
v = rω, a_t = rα, s = rθ. The radius r is the bridge. A point farther from the axis moves faster (tangentially) for the same rotation rate.
5.3 Torque
τ = rF·sin(θ). Only the perpendicular component of force creates torque. Bigger lever arm = bigger torque. Doors have handles far from hinges for this reason.
5.4 Rotational Inertia
I = Σmr². Depends on mass AND mass distribution. Mass far from axis matters much more (r is squared). Skaters pull in arms to reduce I and spin faster.
5.5 Equilibrium
Στ = 0. Object has constant ω (usually zero). Static equilibrium requires BOTH ΣF = 0 AND Στ = 0. Choose your pivot wisely.
5.6 Newton's 2nd (Rot.)
τ_net = Iα. The direct rotational analog of F = ma. Same logic, same solution process — just with angular variables throughout.
🧠 How to solve a torque problem (5 steps)
1. Draw a force diagram. Mark the pivot, all forces, and where each acts.
2. Choose a pivot. Pick wisely — putting it through an unknown force eliminates that force from your torque equation.
4. Write the equation. For equilibrium: Στ = 0. For acceleration: Στ = Iα.
5. Solve for the unknown. Usually a force, distance, angle, or angular acceleration.
⚠️ Common exam traps
Use the perpendicular component of force, not the full force. The sin(θ) in τ = rF·sin(θ) is essential when force isn't perpendicular to the lever arm.
Don't forget that I depends on the AXIS chosen. The same object has different rotational inertias about different axes.
Mass is squared in I. Wait — no. The DISTANCE is squared in I = Σmr². Mass matters linearly; distance matters quadratically.
For a uniform beam, weight acts at the center. Don't try to split it among multiple points — just put the entire weight at the center of mass.
Equilibrium means constant ω, NOT necessarily ω = 0. A wheel spinning at constant speed (no net torque) is in rotational equilibrium.
Torque and energy share units (N·m) but ARE NOT the same. Torque is a vector; energy is a scalar. Don't confuse the two.
θ must be in radians for v = rω and friends. If you use degrees, the relationship breaks.