Unit 09 - Grade 11-12 Physics

Rotational Motion and Torque

Learn how objects rotate, how torque changes rotational motion, how moment of inertia affects angular acceleration, and how angular momentum connects spinning systems, rolling objects, and equilibrium problems.

Lesson roadmap

What Students Should Master in This Unit

Rotational motion extends linear motion to objects that spin, roll, balance, or turn about an axis. Students learn how angle, angular velocity, angular acceleration, torque, moment of inertia, rotational energy, and angular momentum work together.

Translate linear motion into rotation

Use angular position, angular velocity, angular acceleration, and rotational kinematics.

Analyze torque and equilibrium

Calculate torque, choose a pivot, and solve balance problems using net force and net torque.

Connect rotation to energy and momentum

Use moment of inertia, rotational kinetic energy, rolling motion, and angular momentum conservation.

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Rotational Language Angular Variables Angular Kinematics Linear-Angular Links Torque Rotational Equilibrium Moment of Inertia Rotational Dynamics Rotational Energy Rolling Motion Angular Momentum Simulation Labs Lab Skills Worked Examples Practice Problems
Linear motion compared with rotation

1. The Big Idea of Rotational Motion

In linear motion, an object moves through distance. In rotational motion, an object turns through angle. Many formulas look similar, but the variables change.

Linear motion compared with rotational motion Linear motion has rotational partners linear object v, a, F rotating object θ, ω, α, τ
Rotation uses angular position, angular velocity, angular acceleration, and torque in the same way linear motion uses position, velocity, acceleration, and force.
Linear Quantity Rotational Quantity Meaning
Position xAngular position θWhere the object is along a path or through a turn.
Displacement ΔxAngular displacement ΔθChange in position or change in angle.
Velocity vAngular velocity ωHow fast position or angle changes.
Acceleration aAngular acceleration αHow fast velocity or angular velocity changes.
Mass mMoment of inertia IResistance to linear acceleration or angular acceleration.
Force FTorque τCause of linear acceleration or angular acceleration.
Momentum p = mvAngular momentum L = IωQuantity conserved when no external force or torque acts.
Core idea: A force changes linear motion. A torque changes rotational motion. The farther a force acts from the axis, the more turning effect it can produce.
Angles, rates, and units

2. Angular Variables

Rotational problems usually use radians because radians connect angle directly to arc length and radius.

Angular variables on a rotating disk Angles in radians connect circular motion to distance θ r s = rθ ω = change in angle / time α = change in ω / time
Radians make circular motion practical because one angle immediately gives the arc length traveled at any radius.
Radians in one circle 1 revolution = 2π rad Also equals 360 degrees.
Arc length s = rθ θ must be in radians.
Angular velocity ω = Δθ / Δt Unit: rad/s.
Angular acceleration α = Δω / Δt Unit: rad/s2.
Period connection ω = 2π / T For constant angular speed.
Frequency connection ω = 2πf f is revolutions per second.

Converting Revolutions and RPM

  • To convert revolutions to radians, multiply by 2π.
  • To convert rpm to rev/s, divide by 60.
  • To convert rev/s to rad/s, multiply by 2π.
Constant angular acceleration

3. Angular Kinematics

When angular acceleration is constant, rotational motion uses equations that match the linear kinematics equations from earlier units.

Angular kinematics during spin-up Constant angular acceleration changes angular velocity smoothly early middle later ωf = ωi + αt
Use angular kinematics when angular acceleration is constant, just like linear kinematics works when linear acceleration is constant.
Angular velocity after time ωf = ωi + αt Matches vf = vi + at.
Angular displacement θ = ωit + 1/2αt2 Use when time is known.
No-time equation ωf2 = ωi2 + 2αθ Use when time is not needed.
Average angular velocity θ = [(ωi + ωf)/2]t Works for constant angular acceleration.
Common mistake: Do not mix degrees with radian-based formulas. Convert degrees or revolutions to radians before calculating.
Connecting spinning to moving

4. Linear and Angular Relationships

Points farther from the axis travel a larger distance in the same angle. That is why the outside edge of a wheel has a greater tangential speed than a point closer to the center.

Linear and angular relationships The same angle gives more distance at a larger radius larger vt = rω smaller vt θ
A point near the rim has greater tangential speed than a point near the center because tangential speed depends on radius.
Arc length s = rθ Distance traveled along circular path.
Tangential speed vt = rω Linear speed at radius r.
Tangential acceleration at = rα Due to change in speed.
Centripetal acceleration ac = rω2 Due to change in direction.
Centripetal acceleration ac = v2/r Equivalent circular motion form.
Total acceleration a = √(at2 + ac2) When both tangential and centripetal parts exist.
The turning effect of force

5. Torque

Torque measures how strongly a force tends to rotate an object around an axis or pivot. Torque depends on force size, distance from the pivot, and the angle between the force and lever arm.

Torque from a force on a lever arm Torque depends on force, distance, and angle lever arm r F θ turning effect τ = rF sin(θ)
A force makes the greatest torque when it acts perpendicular to the lever arm and farther from the pivot.
Torque τ = rF sin(θ) r is distance from pivot to where force acts.
Perpendicular force form τ = rF⊥ Use only the force component perpendicular to the lever arm.
Lever arm form τ = Fr⊥ r⊥ is the perpendicular distance to the line of action.

Torque Direction

  • Counterclockwise torque is often chosen as positive.
  • Clockwise torque is often chosen as negative.
  • The sign convention is flexible, but it must stay consistent in one problem.
  • A force through the pivot creates zero torque because the lever arm is zero.
Quick check: A small force far from the pivot can create the same torque as a large force close to the pivot.
Balanced forces and balanced torques

6. Rotational Equilibrium

An object is in static equilibrium when it is not accelerating linearly and not accelerating rotationally. That means both net force and net torque are zero.

Rotational equilibrium on a balance beam Static equilibrium means no net force and no net torque larger force smaller force ΣF = 0 and Στ = 0
A beam balances when clockwise torque equals counterclockwise torque, even if the forces are different sizes.
Translational equilibrium ΣFx = 0 and ΣFy = 0 No linear acceleration.
Rotational equilibrium Στ = 0 No angular acceleration.
Torque balance τclockwise = τcounterclockwise Useful for seesaws and balances.

How to Solve Equilibrium Problems

  1. Draw a free-body diagram.
  2. Choose a pivot. A smart pivot removes unknown forces from the torque equation.
  3. Measure each lever arm from the pivot.
  4. Assign clockwise and counterclockwise torque signs.
  5. Set Στ = 0, then solve.
Rotational mass

7. Moment of Inertia

Moment of inertia measures how difficult it is to change an object's rotation. Mass farther from the axis creates a larger moment of inertia.

Moment of inertia depends on mass distribution Mass farther from the axis is harder to spin up small r, smaller I large r, larger I I = Σmr2
Moment of inertia is not just mass. It also depends strongly on how far that mass sits from the rotation axis.
Point masses I = Σmr2 Add each mass times its distance squared.
Single point mass I = mr2 Useful for small objects orbiting a pivot.
Parallel-axis theorem I = Icm + Md2 Advanced tool for axes shifted from center of mass.

Common Moments of Inertia

Object Axis Moment of Inertia
Point massDistance r from axisI = mr2
Thin hoop or ringCenter axisI = MR2
Solid disk or cylinderCenter axisI = 1/2MR2
Solid sphereCenter axisI = 2/5MR2
Thin spherical shellCenter axisI = 2/3MR2
Thin rodThrough center, perpendicular to rodI = 1/12ML2
Thin rodThrough one end, perpendicular to rodI = 1/3ML2
Key idea: Two objects with the same mass can rotate very differently if their mass is spread differently around the axis.
Newton's second law for rotation

8. Rotational Dynamics

Net torque causes angular acceleration. The rotational version of Newton's second law is one of the main equations of the unit.

Net torque causes angular acceleration The rotational version of Newton's second law applied F α Στ = Iα same idea as ΣF = ma more net torque gives more angular acceleration
For the same moment of inertia, a larger net torque creates a larger angular acceleration.
Rotational Newton's second law Στ = Iα The rotational version of ΣF = ma.
Angular acceleration α = Στ / I Larger I means smaller angular acceleration for the same torque.
Net torque from multiple torques Στ = τ1 + τ2 + ... Use signs for clockwise and counterclockwise directions.

Problem-Solving Pattern

  • Choose an axis of rotation.
  • Calculate each torque about that axis.
  • Add torques with signs.
  • Use Στ = Iα to find angular acceleration.
Energy in spinning systems

9. Rotational Energy, Work, and Power

A spinning object has rotational kinetic energy. Rolling objects can have both translational kinetic energy and rotational kinetic energy at the same time.

Energy in rotation and rolling motion Rolling objects carry two kinds of kinetic energy spinning energy translation + rotation Krot = 1/2 Iω2 K = 1/2mv2 + 1/2Iω2
A wheel that rolls has energy from the center of mass moving forward and energy from the wheel spinning.
Rotational kinetic energy Krot = 1/2Iω2 Energy due to spinning.
Rotational work W = τθ For constant torque.
Rotational power P = τω Power delivered by torque.
Total kinetic energy for rolling K = 1/2mv2 + 1/2Iω2 Translational plus rotational energy.
Objects that rotate while moving

10. Rolling Motion

Pure rolling means the object rolls without slipping. The point touching the ground is momentarily at rest relative to the ground, and the center of mass moves forward.

Rolling without slipping Pure rolling links center speed to angular speed vcm ω contact point momentarily at rest vcm = Rω
Rolling without slipping means the distance rolled equals the arc length turned by the wheel.
Rolling speed condition vcm = Rω Center-of-mass speed and angular speed.
Rolling acceleration condition acm = Rα For pure rolling without slipping.
Rolling down an incline a = g sin(θ) / [1 + I/(mR2)] More rotational inertia means slower acceleration.

Important Rolling Ideas

  • A solid sphere rolls down an incline faster than a hoop of the same mass and radius because its moment of inertia is smaller.
  • Static friction can provide the torque needed for rolling without slipping.
  • For pure rolling on a stationary surface, static friction does no work at the contact point.
Conservation in spinning motion

11. Angular Momentum

Angular momentum describes how much rotational motion a system has. If there is no net external torque, angular momentum is conserved.

Angular momentum conservation When external torque is zero, angular momentum stays constant arms out: large I, small ω arms in: small I, large ω Iiωi = Ifωf
Pulling mass closer to the axis lowers moment of inertia, so angular speed increases if angular momentum is conserved.
Rigid rotating object L = Iω For an object spinning about a fixed axis.
Point mass angular momentum L = mvr sin(φ) φ is the angle between r and v.
Angular impulse ΔL = τnetΔt Torque changes angular momentum.
Conservation of angular momentum Iiωi = Ifωf Use when external torque is zero.
Classic example: When a skater pulls their arms inward, moment of inertia decreases, so angular velocity increases to keep angular momentum constant.
Simulation labs

12. Simulation Labs for This Unit

These official PhET simulations help students visualize torque, balance, moment of inertia, angular acceleration, rotation, and angular momentum.

Simulation lab workflow Use simulations like real labs: change one variable at a time Change force, radius, mass Observe α, balance, ω Explain use τ and I Good lab evidence connects a graph, a formula, and a physical explanation.
Simulation work is strongest when students collect evidence and explain the trend using torque, moment of inertia, and angular acceleration.
Balance Act

Explore how mass and distance from the pivot determine whether a seesaw balances. This is excellent for torque and rotational equilibrium.

Lab idea: keep one object fixed, move the second object until clockwise torque equals counterclockwise torque.
Open Simulation
Torque

Investigate how applied force, radius, braking force, moment of inertia, and angular momentum affect the rotation of a platform.

Lab idea: apply the same force at different radii and compare angular acceleration.
Open Simulation
Investigation skills

13. Rotational Motion and Torque Lab Skills

Labs in this unit often involve measuring radius, force, angle, mass distribution, angular displacement, time, angular velocity, and angular acceleration.

Meter stick torque balance lab Measure force and lever arm before writing the torque equation r1 r2 hanging mass unknown mass Στ = 0 helps solve for unknown force or mass.
A professional lab setup labels the pivot, forces, lever arms, units, and torque sign convention before calculations begin.

Common Labs

  • Meter stick torque balance lab with hanging masses.
  • Seesaw or lever-arm equilibrium investigation.
  • Rotating platform torque and angular acceleration lab.
  • Rolling objects down an incline to compare moment of inertia.
  • Angular momentum conservation lab using a rotating stool or turntable.
  • Simulation-based balance and torque lab.

Useful Measurements

  • Lever arm distance from pivot in meters.
  • Force in newtons.
  • Angle between force and lever arm.
  • Angular displacement in radians.
  • Time interval in seconds.
  • Mass and radius for moment of inertia calculations.
Worked examples

14. Worked Examples

Example 1: Convert rpm to rad/s

A wheel spins at 120 rpm. Find angular speed in rad/s.

120 rpm = 120/60 = 2.0 rev/s.

ω = 2πf = 2π(2.0) = 12.6 rad/s.

Example 2: Tangential speed

A point on a wheel is 0.30 m from the center. The wheel spins at 10 rad/s. Find tangential speed.

v = rω = (0.30)(10) = 3.0 m/s.

Example 3: Torque from a perpendicular force

A 40 N force acts perpendicular to a wrench 0.25 m from the bolt. Find torque.

τ = rF = (0.25)(40) = 10 N·m.

Example 4: Torque at an angle

A 50 N force is applied 0.40 m from a pivot at 30 degrees to the lever arm. Find torque.

τ = rF sin(θ) = (0.40)(50)sin(30) = 10 N·m.

Example 5: Balance problem

A 20 N weight is 0.60 m left of a pivot. Where should a 30 N weight be placed on the right to balance it?

Set clockwise torque equal to counterclockwise torque.

(20)(0.60) = (30)x, so x = 0.40 m.

Example 6: Angular acceleration

A net torque of 8.0 N·m acts on a wheel with I = 2.0 kg·m2. Find angular acceleration.

α = Στ/I = 8.0/2.0 = 4.0 rad/s2.

Example 7: Rotational kinetic energy

A disk has I = 0.50 kg·m2 and spins at 6.0 rad/s. Find rotational kinetic energy.

Krot = 1/2Iω2 = 1/2(0.50)(6.0)2 = 9.0 J.

Example 8: Angular momentum conservation

A skater has I = 4.0 kg·m2 and spins at 2.0 rad/s. After pulling in their arms, I becomes 2.0 kg·m2. Find final angular speed.

Iiωi = Ifωf.

(4.0)(2.0) = (2.0)ωf, so ωf = 4.0 rad/s.

Independent practice

15. Practice Problems

Try each problem first. Then open the answer check and compare formulas, units, signs, and rotational reasoning.

1. Convert 3.0 revolutions to radians.

Answer

θ = 3.0(2π) = 6π rad = 18.8 rad.

2. A wheel turns 20 rad in 4.0 s. Find average angular velocity.

Answer

ω = Δθ/Δt = 20/4.0 = 5.0 rad/s.

3. A disk speeds up from 2.0 rad/s to 10 rad/s in 4.0 s. Find angular acceleration.

Answer

α = Δω/Δt = (10 - 2.0)/4.0 = 2.0 rad/s2.

4. A wheel has radius 0.50 m and angular speed 8.0 rad/s. Find tangential speed at the rim.

Answer

v = rω = (0.50)(8.0) = 4.0 m/s.

5. A wheel has radius 0.40 m and angular acceleration 6.0 rad/s2. Find tangential acceleration.

Answer

at = rα = (0.40)(6.0) = 2.4 m/s2.

6. A 25 N force acts perpendicular to a lever 0.80 m from a pivot. Find torque.

Answer

τ = rF = (0.80)(25) = 20 N·m.

7. A 60 N force acts 0.30 m from a pivot at 90 degrees. Find torque.

Answer

τ = rF sin(90) = (0.30)(60)(1) = 18 N·m.

8. A 60 N force acts 0.30 m from a pivot at 0 degrees along the lever arm. Find torque.

Answer

τ = rF sin(0) = 0. The force acts through the lever arm and creates no rotation.

9. A 10 N force acts 0.50 m left of a pivot. A 5.0 N force acts 1.0 m right of the pivot. Is the system balanced?

Answer

Left torque = (10)(0.50) = 5.0 N·m. Right torque = (5.0)(1.0) = 5.0 N·m. Yes, it is balanced.

10. A 12 N weight is 0.40 m from a pivot. How far from the pivot should an 8.0 N weight be placed to balance it?

Answer

(12)(0.40) = (8.0)x, so x = 0.60 m.

11. A point mass of 3.0 kg is 2.0 m from an axis. Find moment of inertia.

Answer

I = mr2 = (3.0)(2.0)2 = 12 kg·m2.

12. A solid disk has mass 4.0 kg and radius 0.50 m. Find I about its center.

Answer

I = 1/2MR2 = 1/2(4.0)(0.50)2 = 0.50 kg·m2.

13. A net torque of 15 N·m acts on an object with I = 3.0 kg·m2. Find angular acceleration.

Answer

α = Στ/I = 15/3.0 = 5.0 rad/s2.

14. A wheel starts from rest and has α = 4.0 rad/s2 for 3.0 s. Find final angular speed.

Answer

ωf = ωi + αt = 0 + (4.0)(3.0) = 12 rad/s.

15. A wheel starts from rest and has α = 2.0 rad/s2 for 5.0 s. Find angular displacement.

Answer

θ = ωit + 1/2αt2 = 0 + 1/2(2.0)(5.0)2 = 25 rad.

16. A rotating object has I = 0.80 kg·m2 and ω = 5.0 rad/s. Find rotational kinetic energy.

Answer

Krot = 1/2Iω2 = 1/2(0.80)(5.0)2 = 10 J.

17. A rolling wheel has radius 0.25 m and angular speed 12 rad/s. Find center-of-mass speed.

Answer

vcm = Rω = (0.25)(12) = 3.0 m/s.

18. A student pulls their arms inward while spinning and no external torque acts. What happens to angular speed?

Answer

Moment of inertia decreases, so angular speed increases to conserve angular momentum.

Final review

16. What to Know Before Moving On

  • Rotational motion uses angular position, angular velocity, and angular acceleration.
  • Radians are the best angle unit for rotational calculations.
  • Linear and angular motion connect through s = rθ, v = rω, and at = rα.
  • Torque is the turning effect of force: τ = rF sin(θ).
  • Rotational equilibrium requires ΣF = 0 and Στ = 0.
  • Moment of inertia depends on both mass and how far that mass is from the axis.
  • Newton's second law for rotation is Στ = Iα.
  • Rotational kinetic energy is Krot = 1/2Iω2.
  • For rolling without slipping, vcm = Rω.
  • If no external torque acts, angular momentum is conserved.
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