Torque and Rotational Dynamics

What this unit covers

The topics below follow the published Physics 1 course framework for Unit 5. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Rotational Kinematics13 min · 3 objectivesDescribe rotation with angular position, velocity, and acceleration · Relate linear and angular quantities through v = rω · Apply the rotational kinematic equations that mirror the linear ones
• Torque13 min · 3 objectivesDefine torque as τ = rF sinθ and identify the lever arm · Explain how distance and angle affect turning effectiveness · Determine the direction of a torque (clockwise or counterclockwise)
• Rotational Inertia & Newton’s Second Law for Rotation14 min · 3 objectivesDefine rotational inertia and its dependence on mass distribution · Apply the rotational form of Newton’s second law, τ_net = Iα · Compare the rotational inertia of different shapes
• Rotational Equilibrium13 min · 3 objectivesState the two conditions for static equilibrium · Solve balanced-beam and seesaw problems · Choose a pivot point to simplify a torque equation

Formulas in Unit 5

Every term in Unit 5

All 17 terms we publish for Torque and Rotational Dynamics, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Torque

τ = rF sin θ, or force times lever arm. Only the force component perpendicular to the radius produces torque.

Rolling without slipping

v_cm = rω and the contact point is instantaneously at rest, so static friction acts and does no work.

Lever arm

The perpendicular distance from the axis to the line of action of the force. Increasing it increases torque for the same force.

Rotational equilibrium

Στ = 0 about any axis. Combined with ΣF = 0, this is the condition for static equilibrium and solves balance problems.

Choosing a pivot

Any point works, so choose one where an unknown force acts — its torque becomes zero and disappears from the equation.

Moment of inertia

Rotational analogue of mass, depending on how mass is distributed relative to the axis. Mass further out gives larger I.

Common moments of inertia

Point mass mr²; solid cylinder ½mr²; hoop mr²; solid sphere ⅖mr². The hoop is hardest to spin for the same mass and radius.

Newton's second law for rotation

Στ = Iα, the direct analogue of ΣF = ma with torque, moment of inertia and angular acceleration.

Angular kinematics

ω = ω₀ + αt and θ = θ₀ + ω₀t + ½αt², valid only for constant angular acceleration — the same restriction as the linear versions.

Relating linear and angular quantities

v = rω and a = rα for a point at radius r. Points further from the axis move faster at the same angular speed.

Torque sign convention

Counterclockwise is conventionally positive. Consistency matters more than the choice.

Why a door handle is far from the hinge

Torque is force times lever arm, so a longer lever arm produces the same torque with less force.

Static equilibrium of a beam

Take torques about a support to eliminate its unknown reaction force, leaving one equation in one unknown.

Parallel axis theorem qualitatively

Moving the rotation axis away from the center of mass increases the moment of inertia, so the object is harder to spin.

Angular acceleration from torque

α = τ/I. The same torque produces less angular acceleration on a body whose mass sits further from the axis.

Rotational vs translational analogues

x↔θ, v↔ω, a↔α, m↔I, F↔τ, p↔L. Every translational relation has a rotational twin with these substitutions.

Friction in rolling

Static friction acts at the contact point of a rolling object and does no work, since that point is instantaneously at rest.

What examiners penalize here

• Watch the difference between angular speed ω (shared by the whole object) and linear speed v (larger the farther you are from the axis). A question about "a point on the rim" almost always wants v = rω, not ω.
• When a force is not perpendicular, do not forget the sinθ. A common error is using τ = rF and ignoring the angle; only the perpendicular component turns the object.
• On the AP formula sheet you are given I for standard shapes (hoop MR², solid disk ½MR², sphere ⅖MR²). You are not expected to derive them — just choose the right one and remember the pattern: mass farther out gives a bigger coefficient.
• Free-response equilibrium problems almost always need both conditions. Write ΣF = 0 and Στ = 0 as two separate equations — many setups are unsolvable from the force equation alone.

Practice Physics 1

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 8 units of AP Physics 1: Algebra-Based

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations
8. Unit 8 · Fluids