AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/5/5-2)
Unit 5 · Topic 5.2
5.2 Connecting Linear and Rotational Motion
Every point on a spinning rigid object turns through the same angle in the same time, but points farther from the axis travel farther and faster. Three equations link the two descriptions: s = rθ, v = rω and a_T = rα, all with angles in radians.
Key terms
- arc length
- tangential velocity
- tangential acceleration
- distance from the axis
Same angle, different distances
Picture a merry-go-round. A child at the edge and a child near the center both go around once in the same time, so they share the same angular displacement, angular velocity and angular acceleration. But the child at the edge travels around a bigger circle, so they cover more distance and move faster.
That's true of every rigid system: all points share θ, ω and α, but the linear quantities depend on the distance r from the axis.
The three linking equations
A point a distance r from a fixed axis moves along a circle. As the object turns through Δθ, the point travels an arc length s = rΔθ.
Divide by time and you get the point's speed: v = rω. This velocity is tangential, meaning it points along the circle, perpendicular to the radius.
If ω is changing, the point's speed is changing too, and its tangential acceleration is a_T = rα. It also points along the circle: forward if the spin is speeding up, backward if it's slowing.
These equations only work with θ in radians, ω in rad/s and α in rad/s². That's why radians matter: they're defined so that s = rθ comes out with no extra conversion factor.
| Rotational (same for every point) | Linear (grows with r) |
|---|---|
| angular displacement θ (rad) | arc length s = rθ (m) |
| angular velocity ω (rad/s) | tangential speed v = rω (m/s) |
| angular acceleration α (rad/s²) | tangential acceleration a_T = rα (m/s²) |
Tangential and centripetal acceleration
A point on a spinning object is moving in a circle, so it always has a centripetal acceleration pointing toward the axis: a_c = v²/r, which you met in 2.9. Substituting v = rω gives a_c = ω²r. The point has this acceleration even if ω is constant, because its direction of motion keeps changing.
Tangential acceleration is different. It's only there when the angular speed is changing. A point on a wheel that's speeding up has both: a_T along the circle and a_c toward the center.
Because a_c = ω²r, points farther out have bigger centripetal acceleration at the same ω. That's why mud flies off the outer edge of a spinning tire first.
Ropes, belts and pulleys
When a rope wraps around a pulley or drum without slipping, the rope moves with the rim. So the rope's speed equals the rim speed, Rω, and the rope's acceleration equals the rim's tangential acceleration, Rα, where R is the radius where the rope touches.
This link is what lets you connect a hanging block's motion to a pulley's spin in 5.6. The same idea applies to a bike chain: the chain moves at the same speed over both sprockets, so the smaller sprocket spins with the larger ω.
Rolling wheels use similar equations, but there the center of the wheel moves too. You'll see that case in 6.5.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Two riders on a merry-go-round
A merry-go-round turns at a steady 1.5 rad/s. Ana sits 2.0 m from the axis and Ben sits 1.0 m from it. Find each rider's speed and centripetal acceleration.
Show the solutionHide the solution
- Step 1: Both share ω = 1.5 rad/s because the merry-go-round is rigid.
- Step 2: Speeds: v = rω. Ana: (2.0)(1.5) = 3.0 m/s. Ben: (1.0)(1.5) = 1.5 m/s.
- Step 3: Centripetal acceleration: a_c = ω²r. Ana: (1.5)²(2.0) = 4.5 m/s². Ben: (1.5)²(1.0) = 2.25 ≈ 2.3 m/s².
- Step 4: Doubling r doubles both v and a_c at the same ω. Neither rider has tangential acceleration, since ω is constant.
Answer: Ana: 3.0 m/s and 4.5 m/s²; Ben: 1.5 m/s and about 2.3 m/s²
- Example 2Calculator allowed
A bucket on a windlass
A rope is wrapped around a drum of radius 0.10 m. A bucket on the rope starts from rest and moves down with a constant acceleration of 0.50 m/s². The rope doesn't slip. Find the drum's angular acceleration, its angular velocity after 3.0 s and how much rope unwinds in that time.
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- Step 1: The rope moves with the drum's rim, so the bucket's acceleration equals the rim's tangential acceleration: a = Rα.
- Step 2: α = a/R = 0.50 ÷ 0.10 = 5.0 rad/s².
- Step 3: ω = ω₀ + αt = 0 + (5.0)(3.0) = 15 rad/s. Check: the bucket's speed is at = 1.5 m/s, and Rω = (0.10)(15) = 1.5 m/s.
- Step 4: Rope unwound = distance the bucket falls = ½at² = ½(0.50)(3.0)² = 2.25 m. As an angle: Δθ = s/R = 22.5 rad.
Answer: α = 5.0 rad/s²; ω = 15 rad/s; about 2.3 m of rope (22.5 rad of drum rotation)
- Example 3Calculator allowed
Comparing two points (classic trap)
Point P is 0.10 m from the axis of a disk and point Q is 0.30 m from it. The disk is speeding up. Compare Q to P for angular velocity, angular acceleration, speed, tangential acceleration and centripetal acceleration.
Show the solutionHide the solution
- Step 1: ω and α are the same for every point on a rigid disk, so the ratios are 1. The trap is thinking the outer point spins faster.
- Step 2: v = rω and a_T = rα are proportional to r, so Q's are 0.30/0.10 = 3 times P's.
- Step 3: a_c = ω²r is also proportional to r at the same moment, so Q's is 3 times P's.
Answer: Same ω and α; Q has 3 times the speed, tangential acceleration and centripetal acceleration
Common mistakes
- Saying points farther from the axis have greater angular velocity. They have greater linear speed; ω is the same for the whole rigid object.
- Using v = rω with ω in rpm or rev/s. Convert to rad/s first.
- Forgetting centripetal acceleration when ω is constant. A point on a steadily spinning wheel still accelerates toward the axis.
- Mixing up tangential and centripetal acceleration. a_T = rα points along the circle and exists only when ω changes; a_c = ω²r points to the center.
On the exam
- Comparison questions (point near the rim versus near the axis) are common. Name which quantities are shared and which scale with r.
- On free-response questions with ropes on pulleys, write the no-slip link a = Rα explicitly; it's often a point in the derivation.
Connected topics
Videos
Check yourself
4 questions on 5.2 Connecting Linear and Rotational Motion. Pick an answer to see if you got it, and why.
Ana sits 1.0 m from the center of a merry-go-round and Ben sits 2.0 m from the center. The merry-go-round turns at a constant rate. Which statement correctly compares their motion?
A bicycle wheel of radius 0.35 m spins on a fixed stand at 20 rad/s. What is the speed of a point on the rim?
A wheel of radius 0.25 m turns through exactly 3.0 revolutions. How far does a point on its rim travel?
A grinding wheel's angular velocity increases uniformly from 10 rad/s to 30 rad/s in 5.0 s. What is the tangential acceleration of a point 0.10 m from the axis?
0 of 4 answered