Angular and Linear Motion
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
Every point on a spinning object turns through the same angle, so they all share the same ω. But points farther from the axis travel a bigger circle, so they move faster: v = rω. Likewise, s = rθ and atangential = rα.
1. Same ω, different speeds
2. When to use each
These connect the rotation (θ, ω, α) to the motion of one point (s, v, a). They only work with angles in radians.
A wheel that rolls without slipping moves forward at v = rω, the same speed as a point on its rim.
A point on a spinning object can have two accelerations at once: tangential (rα), if the spin rate changes, and centripetal (ω²r), always, because its direction keeps changing.
3. Example, then try it
A merry-go-round turns at 1.5 rad/s. How fast do riders 1.0 m and 2.0 m from the center move?
v = rω: (1.0)(1.5) = 1.5 m/s and (2.0)(1.5) = 3.0 m/s.
Same ω for both; the outer rider moves twice as fast.
A bicycle wheel has a radius of 0.35 m. The bike rolls at 7.0 m/s without slipping. Find the wheel’s angular velocity, its revolutions per second, and how far the bike goes in 10 s.
Angular velocity: ω = v ÷ r = 7.0 ÷ 0.35 = 20 rad/s.
Revolutions per second: 20 ÷ 2π ≈ 3.2 rev/s.
Distance: in 10 s the wheel turns θ = 200 rad, so s = rθ = (0.35)(200) = 70 m. (Check: 7.0 m/s × 10 s = 70 m. ✓)
Answer: 20 rad/s, about 3.2 rev/s, and 70 m in 10 s.
4. Common mistakes
Two dots are on a spinning record: one at the edge, 0.15 m from the center, and one 0.05 m from the center. Compared with the inner dot, the outer dot has…
They turn together, so ω is the same. With v = rω, the dot 3 times as far out moves 3 times as fast.
Ready to practice?
5 questions on Angular and Linear Motion, each with a hint and an explanation for every wrong answer.