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Connecting Linear and Rotational Motion

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A rigid body's points all share one angular velocity ω, but their linear motion depends on how far they sit from the axis. The bridge equations make the link: a point's tangential speed is v = rω, its tangential acceleration is at = rα, and the arc it sweeps is s = rθ. Turning toward the axis there is also a centripetal acceleration ac = ω²r. Each of these grows with the radius r, and each only works when the angle is in radians.

One shared spin, many linear speeds. all points share ω, but the tangential speed v = rω grows straight out from the axis v inner: small r, small v v same ω, but the outer point is faster: v = rω v = rω tangential speed at = rα changes the speed ac = ω²r turns the direction s = rθ arc length swept at and ac are perpendicular pieces. Even at α = 0, ac still pulls toward the axis. every bridge equation needs the angle in radians: one full turn is 2π rad
Every point on a rigid body shares one angular velocity, but its tangential speed v = rω, tangential acceleration at = rα, and arc length s = rθ all grow with the radius. The same point also accelerates toward the axis with ac = ω²r, a separate perpendicular piece. Every one of these relations needs the angle measured in radians.
Linear-Rotational Bridge · Open the sandbox →

The bridge equations fail in three ways. Treating the tangential speed as the same at every radius, instead of letting v = rω grow with r; confusing the tangential acceleration at = rα with the centripetal acceleration ac = ω²r, or thinking a steady spin means no acceleration at all; and slipping on units, plugging degrees or revolutions into equations that hold only in radians. Fix those three and the bridge between linear and rotational motion becomes routine.

The work

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Lesson
Connecting Linear and Rotational Motion ›

Builds the bridge between linear and angular motion: a point's tangential speed v = rω, its tangential acceleration at = rα, the arc length s = rθ, and the centripetal acceleration ac = ω²r toward the axis. Shows why every linear quantity grows with the radius, how at and ac are different perpendicular pieces, and why the angle must be in radians. Closes with a ten-scenario skill check.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items on the three misconceptions for Topic 5.2: treating tangential speed as equal at every radius instead of v = rω, confusing tangential acceleration at = rα with centripetal ac = ω²r, and using degrees or revolutions where the bridge equations need radians. Take it cold to see what is still tangled, or after the lesson to confirm it is not.

Not started · 10 items · ~15 min
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Targeted Practice
Drill a single misconception ›

Pick one misconception you keep missing and drill it on its own. The round adapts: two correct in a row clears it and you move on.

Take the diagnostic to identify your misconceptions
Diagnostic first