AP Physics 1 rotation, made clear
Units 5 and 6 of AP Physics 1 cover rotation, which most students find the hardest part of the course. The good news: it mirrors linear motion.
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Linear and rotational, side by side
- Position x ↔ angle θ
- Velocity v ↔ angular velocity ω
- Acceleration a ↔ angular acceleration α
- Force F ↔ torque τ
- Mass m ↔ rotational inertia I
- Momentum p = mv ↔ angular momentum L = Iω
Torque
Torque is the turning effect of a force: τ = rF sinθ. A force applied further from the axis, or more perpendicular to the lever arm, produces more torque.
Rotational inertia
Rotational inertia depends on mass and how far it is from the axis. The same object is harder to spin about an axis where its mass is spread further out.
Rotational dynamics and equilibrium
- Στ = Iα: net torque causes angular acceleration.
- For an object in equilibrium, both net force and net torque are zero.
Energy and angular momentum
- Rotational kinetic energy is ½Iω². A rolling object has both translational and rotational kinetic energy.
- Angular momentum is conserved when no net external torque acts, which is why a spinning skater speeds up when pulling their arms in.
How to study rotation
- Translate each problem into its linear twin first.
- Practise torque equilibrium problems with beams and ladders.
- Compare rolling objects with different rotational inertia racing down a slope.
Worked examples
Example 1: balancing a beam
A 40 N weight hangs 1.1 m left of a pivot. What force at 2.0 m on the right balances it?
- Net torque = 0: 40 × 1.1 = F × 2.0, so F = 22 N.
Example 2: angular acceleration
A 10 N·m net torque acts on a wheel with I = 2 kg·m². Find the angular acceleration and the angular speed after 3 s from rest.
- α = τ/I = 5 rad/s², so ω = αt = 15 rad/s.
Example 3: conservation of angular momentum
A skater spins at 2 rev/s with I = 4 kg·m², then pulls in her arms so I = 1.6 kg·m². Find the new spin rate.
- I1ω1 = I2ω2: 4 × 2 = 1.6ω2, so ω2 = 5 rev/s. Her kinetic energy increases, from the work her muscles do.
Example 4: rolling down a slope
A solid sphere and a hoop of the same mass and radius are released from the same height. Which reaches the bottom first?
- The solid sphere has less rotational inertia, so a larger share of its energy is translational. The sphere wins.
Practice questions
Try each one before opening the answer.
Q1. A 20 N force is applied 0.3 m from a hinge at 30° to the door. Find the torque.
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Q2. A disc with I = 0.5 kg·m² spins at 8 rad/s. Find its rotational kinetic energy.
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Q3. Why is it easier to open a door by pushing far from the hinges?
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Common mistakes
- Using the distance along the beam instead of the perpendicular lever arm.
- Forgetting that angular momentum is conserved only with no net external torque.
- Assuming rolling objects have only translational kinetic energy.
The key ideas in more depth
Rotational inertia
Rotational inertia measures how hard it is to change an object's rotation. It depends on how far the mass is from the axis, not just on how much mass there is. A long rod spun about its end has a larger rotational inertia than the same rod spun about its centre. In AP Physics 1 you will usually be given I or asked to compare objects, rather than calculate it from scratch.
Rolling without slipping
When an object rolls without slipping, the speed of its centre and its rotation are linked: v = rω. Its total kinetic energy is the sum of translational energy ½mv² and rotational energy ½Iω². Objects with more of their mass far from the centre put more energy into spinning and roll down slopes more slowly.
Angular momentum in everyday life
A diver tucks to spin faster and opens out to slow down before entering the water. A spinning bicycle wheel resists being tilted. In both cases, angular momentum is conserved because no significant external torque acts about the axis.
Equilibrium problems
For a ladder against a wall or a beam on supports, use two conditions: the net force is zero, and the net torque about any point is zero. Choose the pivot point where an unknown force acts, so that force drops out of the torque equation.
Exam tips
- Draw an extended free-body diagram, showing where each force acts on the object.
- State the axis you take torques about.
- For rolling questions, write the energy equation with both kinetic energy terms.
More practice
Try each one before opening the answer.
Q1. A uniform 4 m plank weighing 200 N rests on supports at each end. A 600 N person stands 1 m from the left end. Find the force from the right support.
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Q2. A wheel's angular speed increases from 4 rad/s to 10 rad/s in 2 s. Find its angular acceleration.
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Frequently asked questions
Is torque the same as work?
No, even though both can be measured in N·m. Torque is a turning effect; work is energy transferred. Work done by a torque is τθ, in joules.
Do I need to calculate rotational inertia in AP Physics 1?
Usually you are given it, or asked to compare objects qualitatively. You should know that mass further from the axis gives a larger rotational inertia.
Why do tightrope walkers carry long poles?
The pole has a large rotational inertia, which slows down any tipping rotation and gives the walker time to correct it.
Extra practice
Try each one before opening the answer.
Q1. A merry-go-round (I = 500 kg·m²) spins at 1 rad/s. A 50 kg child jumps onto its edge, 2 m from the centre. Find the new angular speed.
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Key takeaways
- Torque is the turning effect of a force: τ = rF sinθ.
- Rotational inertia depends on how mass is spread from the axis.
- Στ = Iα is Newton's second law for rotation.
- Angular momentum is conserved with no net external torque.
- Rolling objects share energy between translation and rotation.