Δs = rΔθ states that the linear distance (arc length) traveled by a point on a rotating rigid system equals its distance from the axis of rotation times the angular displacement in radians. It is the foundational link between linear and rotational motion in AP Physics 1 Topic 5.2 (LO 5.2.A).
Δs = rΔθ is the translation rule between the rotational world and the linear world. When a rigid system rotates through an angle Δθ, every point on it sweeps through that same angle, but the actual distance each point travels along its circular path depends on how far it sits from the axis of rotation. A point twice as far out travels twice the distance in the same rotation. That's all the equation says: linear distance equals radius times angle, as long as the angle is in radians.
Think of two people on a spinning merry-go-round, one near the center and one at the edge. Both complete the same angle in the same time, but the person at the edge covers way more ground. The CED builds the rest of Topic 5.2 directly from this idea. Take the rate of change of both sides and you get v = rω; do it again for the tangential direction and you get a_T = rα. All three equations are the same relationship at different levels: position, velocity, acceleration.
This equation lives in Topic 5.2 (Connecting Linear and Rotational Motion) inside Unit 5: Torque and Rotational Dynamics, and it directly supports learning objective 5.2.A, which asks you to describe the linear motion of a point on a rotating rigid system that corresponds to its rotational motion, and vice versa. That two-way translation is the whole job of Unit 5's opening topics. Everything you learned about kinematics in Unit 1 gets a rotational twin (θ for x, ω for v, α for a), and Δs = rΔθ is the dictionary that converts between the two languages. The CED also pins down a fact this equation makes intuitive: in a rigid system, all points share the same ω and α, but their linear quantities scale with r.
Keep studying AP® Physics 1 Unit 5
Rigid system (Unit 5)
Δs = rΔθ only works cleanly because a rigid system can't stretch or bend, so every point rotates through the same Δθ. That shared angle, combined with each point's own r, is exactly what the equation exploits.
Axis of rotation (Unit 5)
The r in the equation is measured from the axis of rotation, not from the edge or the center of mass. If a problem moves the axis (like a rod pivoting at its end instead of its middle), every point's r changes and so does every point's linear distance.
v = rω and a_T = rα (Unit 5)
These two equations are Δs = rΔθ taken per unit time, once for velocity and once for tangential acceleration. If you remember the pattern 'linear quantity equals r times angular quantity,' you've memorized all three at once.
Linear kinematics (Unit 1)
Rotational kinematics is Unit 1 kinematics with the variables swapped (θ, ω, α instead of x, v, a). Δs = rΔθ is what lets you move an answer back and forth, like finding how far a wheel rolls from how many radians it turned.
Expect this relationship in multiple-choice stems about points at different radii on the same rotating object, like 'point A is twice as far from the axis as point B; compare their speeds' or 'how far does a point on the rim travel when the wheel rotates 3 radians?' The trap answers usually come from forgetting that ω and α are shared across the rigid system while v, a_T, and Δs scale with r. No released FRQ has used the equation verbatim as a prompt, but it shows up constantly as a setup step in rotation FRQs, especially rolling problems where you need to connect a wheel's angular motion to the linear distance it covers. The two-sided skill in LO 5.2.A is what gets tested: given rotational info, find linear, and given linear info, find rotational. Always check that your angle is in radians before you multiply.
Δs is arc length, the distance traveled along the curved circular path. Linear displacement is the straight-line vector from start to finish. For a point that rotates half a circle, Δs = πr (the curve it traced), but its displacement is just 2r (straight across the diameter). On the exam, Δs = rΔθ gives you distance, never the straight-line displacement, and the difference matters whenever the rotation covers a large angle.
Δs = rΔθ says the arc length a point travels equals its distance from the axis of rotation times the angular displacement, and the angle must be in radians.
All points on a rigid system rotate through the same Δθ with the same ω and α, but points farther from the axis travel farther and move faster.
The same pattern repeats at every level of motion: s = rθ, v = rω, and a_T = rα all say 'linear equals r times angular.'
Δs is distance along the circular path, not straight-line displacement, so don't use it when a question asks for the displacement vector.
Measure r from the axis of rotation, which is not always the center of the object, especially when something pivots at its end.
This equation is the workhorse for rolling problems, converting how many radians a wheel turns into how far it travels along the ground.
It's the equation connecting linear and rotational motion: the linear distance Δs a point travels on a rotating object equals its radius r from the axis times the angular displacement Δθ in radians. It's the core of Topic 5.2 in Unit 5 and supports learning objective 5.2.A.
Yes. The equation only works in radians because a radian is literally defined as the angle where arc length equals radius. If a problem gives degrees or revolutions, convert first (1 revolution = 2π radians, 180° = π radians).
No. Δs is the arc length, the curved distance traveled along the circular path. Displacement is the straight line from start to end. After half a rotation, Δs = πr but the displacement magnitude is only 2r.
No, and that's the point of this equation. All points share the same angular velocity ω because the system is rigid, but linear speed is v = rω, so a point at the rim moves faster than a point near the axis.
They're the velocity and acceleration versions of the same relationship. Looking at how Δs = rΔθ changes over time gives v = rω, and doing the same for the tangential direction gives a_T = rα. The CED lists all three together under Topic 5.2.
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