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
• Rotational Inertia: Why Shape Beats Mass16 min · 3 objectivesExplain rotational inertia as mass weighted by the square of its distance from the axis · Compare the standard rigid-body coefficients and predict which object is harder to spin · Show that the same object has different rotational inertia about different axes
• Static Equilibrium & Choosing a Pivot16 min · 3 objectivesApply both equilibrium conditions — zero net force and zero net torque — to a rigid body · Choose a pivot point that eliminates an unknown force from the torque equation · Solve beam, ladder and balance problems with distributed weight
• The Rolling Constraint & Pulleys With Mass15 min · 3 objectivesApply the rolling-without-slipping constraint v = ωR and a = αR · Explain why static rather than kinetic friction acts on a rolling object · Solve a system with a massive pulley, where the two string tensions differ

Formulas in Unit 5

Every term in Unit 5

All 44 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, the rotational effect of a force. Only the component perpendicular to the lever arm contributes; a force along the arm produces none.

Rolling without slipping

The contact point is instantaneously at rest, so v = ωr and a = αr. The condition that links translation and rotation in a rolling problem.

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.

Lever arm

The perpendicular distance from the axis to the force's line of action. Torque equals force times lever arm, which is often quicker than resolving the force.

Sign of torque

Choose a positive rotational direction — usually counterclockwise — and keep it. Opposing torques carry opposite signs.

Rotational equilibrium

Net torque zero, so angular acceleration is zero. An object can be in rotational equilibrium while accelerating linearly, and vice versa.

Choosing a pivot

Any point works. Choosing one where an unknown force acts removes it from the torque equation, which usually turns two unknowns into one.

Moment of inertia

The rotational analogue of mass, I = Σmr². Depends on how mass is distributed relative to the axis, not just on how much there is.

Why mass distribution matters

Mass far from the axis contributes as r², so a hoop has more rotational inertia than a disk of the same mass. This is why a hoop loses a race down a ramp.

Common moments of inertia

Hoop about its center MR²; solid disk or cylinder ½MR²; solid sphere ⅖MR²; rod about its center 1/12 ML² and about its end ⅓ML².

Newton's second law for rotation

Στ = Iα. Same structure as ΣF = ma with torque, moment of inertia and angular acceleration in place of force, mass and acceleration.

Angular displacement, velocity, acceleration

θ in radians, ω = Δθ/Δt in rad/s, α = Δω/Δt in rad/s². Every linear kinematic equation has an exact rotational twin.

Rotational kinematic equations

ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ. Identical in form to the linear equations, chosen by the same missing-quantity rule.

Linear and angular quantities

v = ωr, a_t = αr, and arc length s = θr — with θ in radians. Using degrees here is a silent factor-of-57 error.

Tangential vs centripetal acceleration

Tangential acceleration αr changes the speed; centripetal acceleration ω²r changes the direction. A point on a spinning-up wheel has both.

What provides the torque in rolling

Static friction at the contact point. It does no work, because the contact point is not sliding — which is why energy is conserved for rolling without slipping.

Torque from weight on an extended object

Acts as though the whole weight were at the center of mass. This is how a plank overhanging a table is analyzed.

Center of gravity and tipping

An object tips when its center of gravity passes beyond the base of support. Lower center of gravity and wider base both increase stability.

Angular acceleration of a pulley with mass

A massive pulley has different tensions on either side, because a net torque is needed to accelerate it. Assuming equal tensions is only valid for a massless pulley.

Why a wrench with a longer handle works

Torque is force times lever arm, so the same force at a greater distance produces more torque. The physics of every lever.

Why rotational inertia squares the distance

A particle twice as far out must travel twice as fast for the same angular speed, and kinetic energy goes as speed squared. Hence I = Σmr².

I depends on the axis, not just the object

A rod about its center has ML²/12; about its end, ML²/3 — four times larger. Rotational inertia is a property of an object AND an axis.

Ranking the shape coefficients

I = cMR² with c = 2/5 for a solid sphere, ½ for a disk, 2/3 for a hollow sphere, 1 for a hoop. More mass at the rim means a larger c.

The rotational translation table

x→θ, v→ω, a→α, m→I, F→τ, p→L. Every rotational equation is a translational one with the symbols swapped, so no new equations need memorizing.

Two equilibrium conditions, not one

ΣF = 0 AND Στ = 0. Two equal opposite forces at different points give zero net force but spin the object — that is a couple.

Choosing the pivot to erase an unknown

Torque balance holds about any point, so pivot at an unknown force and its lever arm is zero. Two unknowns become one equation in one unknown.

Two ways to compute a torque

τ = rF sin θ resolves the force; τ = F·r_⊥ uses the perpendicular distance to its line of action. Identical statements — the second is usually faster to read off a diagram.

A distributed weight acts at the center of mass

A uniform beam's whole weight can be placed at its midpoint for torque purposes. Placing it at the pivot or at the end is the standard beam-problem error.

A force through the pivot makes no torque

However large it is. This is why a door handle sits at the edge rather than beside the hinge, and why pivoting at an unknown force works.

Radians are dimensionless

A radian is a length divided by a length. That is why s = r theta and v = r omega work without a conversion factor, and why omega can be written as 1/s.

Angular velocity is shared across a rigid body

Every point on a rotating rigid body has the SAME omega. The linear speed v = r omega differs, which is why the rim moves faster than the hub.

Torque is in newton-meters, never joules

The units match energy but the quantity does not. Torque is a force applied about an axis; writing it in joules confuses a turning effect with an amount of energy.

Direction of angular velocity

Curl the right hand along the rotation and the thumb gives the direction of omega, along the axis. This is why angular quantities add as vectors along the axis rather than around the circle.

Converting revolutions to radians

One revolution is 2 pi radians. A wheel at 300 rpm turns 300 x 2 pi / 60 = 31.4 rad/s, and forgetting the 60 is the usual slip.

A couple turns without pushing

Two equal, opposite forces on different lines of action give zero net force and a non-zero net torque. The object angularly accelerates while its center of mass does not move.

Zero net torque does not mean not rotating

It means the angular velocity is constant. A wheel spinning steadily has no net torque on it, exactly as an object moving at constant velocity has no net force.

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.
• The AP equation sheet gives you the rotational inertia coefficients, so do not memorize them — but do learn their *order* (sphere < disk < hoop). Ranking questions ask which object wins a race or needs more torque, and the ranking alone answers them.
• State your pivot explicitly and label the sign convention (counterclockwise positive is standard). Rubrics award the correct torque equation, and an equation with an unstated pivot cannot be graded even when the numbers are right.
• Whenever a problem says "rolls without slipping", write v = ωR and a = αR immediately. That constraint is the equation that closes the system, and rubrics award it as a separate point from the dynamics equations it connects.

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