AP Physics 1 · Lesson 1

Rotational kinematics and linear motion

Know Know it

Rotation has its own versions of position, velocity and acceleration. The angular displacement Δθ\Delta\theta is measured in radians (one revolution is 2π2\pi rad), the angular velocity is ω=Δθ/Δt\omega = \Delta\theta/\Delta t in rad/s, and the angular acceleration is α=Δω/Δt\alpha = \Delta\omega/\Delta t in rad/s².

With constant α\alpha, the constant-acceleration equations carry over letter for letter: ω=ω0+αt\omega = \omega_0 + \alpha t, Δθ=ω0t+12αt2\Delta\theta = \omega_0 t + \frac{1}{2}\alpha t^2 and ω2=ω02+2αΔθ\omega^2 = \omega_0^2 + 2\alpha\Delta\theta. On a graph of ω\omega against time, the slope is α\alpha and the area is Δθ\Delta\theta.

Every point on a rigid spinning object shares the same ω\omega and α\alpha, but a point at distance r from the axis moves along an arc s=rθs = r\theta at speed v=rωv = r\omega, with tangential acceleration at=rαa_t = r\alpha.

Linear motion and its rotational partner
LinearRotationalLink for a point at radius r
position x (m)angle θ\theta (rad)s=rθs = r\theta
velocity v (m/s)angular velocity ω\omega (rad/s)v=rωv = r\omega
acceleration a (m/s²)angular acceleration α\alpha (rad/s²)at=rαa_t = r\alpha

Calculator On the exam

On the exam you may use a four-function, scientific or graphing calculator on both sections, and the testing app also has built-in scientific and graphing calculators. You get an equation sheet with reference tables too, so your job is choosing the right relationship and setting it up, not memorizing it. Convert revolutions with 2π2\pi rad per revolution, and check the angle mode before you take a sine or cosine: degrees when a problem gives angles in degrees.

Worked example Easy

A potter’s wheel turns at a steady 2.5 revolutions per second. How fast is a lump of clay moving if it sits 0.15 m from the axis?

  1. 0.38 m/s
  2. 2.4 m/sAnswer
  3. 15.7 m/s
  4. 105 m/s

How to solve it

  1. Convert: ω=(2.5 rev/s)(2π rad/rev)≈15.7\omega = (2.5 \text{ rev/s})(2\pi \text{ rad/rev}) \approx 15.7 rad/s.
  2. Link to the point: v=rω=(0.15)(15.7)≈2.4v = r\omega = (0.15)(15.7) \approx 2.4 m/s.

Why each choice is right or wrong

  • A. Incorrect. 0.38 m/s multiplies 2.5 by 0.15 without changing revolutions into radians, leaving out the factor 2π2\pi.
  • B. Correct. ω=2.5(2π)=15.7\omega = 2.5(2\pi) = 15.7 rad/s, so v=rω=(0.15)(15.7)≈2.4v = r\omega = (0.15)(15.7) \approx 2.4 m/s.
  • C. Incorrect. 15.7 m/s is the angular speed in rad/s, not the clay’s speed; it still has to be multiplied by 0.15 m.
  • D. Incorrect. 105 m/s divides the angular speed by the radius instead of multiplying by it.

Trap Same angular speed, different linear speeds

All points of a rigid wheel turn through the same angle in the same time, so they share ω\omega. Their linear speeds differ, growing with distance from the axis. Do not give a rim point and a hub point different ω\omega values, and do not give them the same v.

Explain Explain it

Task verbs: calculate, explain

To solve a constant-α\alpha problem, list the three quantities you know among ω0\omega_0, ω\omega, α\alpha, Δθ\Delta\theta and t, and pick the equation that leaves out the one you neither know nor want. Work in radians, and change to revolutions only at the end.

A point on a speeding-up wheel has two parts of acceleration: the tangential part rαr\alpha, which changes its speed, and the centripetal part ω2r\omega^2 r, which points toward the axis and changes its direction.

Worked example Medium

A ceiling fan starts from rest and speeds up with constant angular acceleration, reaching 12 rad/s after 4.0 s. How many revolutions does it make in those 4.0 s?

  1. 3.8 revAnswer
  2. 7.6 rev
  3. 24 rev
  4. 151 rev

How to solve it

  1. With constant α\alpha, the average angular velocity is (0+12)/2=6.0(0 + 12)/2 = 6.0 rad/s.
  2. Δθ=(6.0)(4.0)=24\Delta\theta = (6.0)(4.0) = 24 rad.
  3. Revolutions: 24/(2π)≈3.824/(2\pi) \approx 3.8 rev.

Why each choice is right or wrong

  • A. Correct. Δθ=12(0+12)(4.0)=24\Delta\theta = \frac{1}{2}(0 + 12)(4.0) = 24 rad, and 24/(2π)≈3.824/(2\pi) \approx 3.8 rev.
  • B. Incorrect. 7.6 rev multiplies the final speed by the time, as if the fan had turned at 12 rad/s all along.
  • C. Incorrect. 24 rev is the angle in radians, 24 rad, reported without dividing by 2π2\pi.
  • D. Incorrect. 151 rev multiplies the 24 rad by 2π2\pi instead of dividing by it.

Calculate Your turn

A wind-turbine blade is 35 m long and turns at a steady 0.80 rad/s about the hub. Calculate the speed of the blade tip, and explain why a point halfway along the blade has the same angular speed but half the tip’s linear speed.

Write it, then compare. Nothing you type here is saved or graded.

Compare with a model answer

A full answer does this

  • Gives the tip speed as 28 m/s
  • Says the halfway point has the same angular speed because the whole blade turns through the same angle in the same time
  • Says the halfway point moves at half the tip speed because v=rωv = r\omega and its radius is half as large

Model answer

The tip moves at v=rω=(35 m)(0.80 rad/s)=28v = r\omega = (35 \text{ m})(0.80 \text{ rad/s}) = 28 m/s. The blade is rigid, so every point on it sweeps through the same angle in the same time and has the same angular speed, 0.80 rad/s. Linear speed is v=rωv = r\omega, and the halfway point is at 17.5 m, half the tip’s radius, so it moves at half the tip’s speed, 14 m/s.

Apply Apply it

Practice this topic. AP Practice questions on exactly what this lesson covers, with an explanation for every answer.

Practice this topic