Linear–angular relations

s = rθ, v = rω, a_t = rα — the bridge between rotation and translation.

Linear–angular relations0 min · Free lecture

In this lesson

  • s = rθ, v = rω, a_t = rα — the bridge between rotation and translation.

One spoke's tip moves faster than its hub — the radius is the gear ratio.

For a point at radius r: arc s = rθ, tangential speed v = rω, tangential acceleration a_t = rα. All radial quantities (centripetal) come separately.

These relations are exact for rigid rotation — every point has the same ω but v grows with r.

v=rω,at=rαv = r\omega,\quad a_t = r\alpha

Tangential speed and acceleration

Worked example

In uniform circular motion, the acceleration of the particle is directed:

  1. Speed is constant but velocity direction changes continuously.
  2. The change in velocity points toward the centre → centripetal acceleration a_c = v²/r toward the centre.

Answer: Toward the centre of the circle

Transcript (0 min)
WEBVTT

1
00:00:00.000 --> 00:00:05.000
Linear–angular relations — FemtoLearn.
2
00:00:05.000 --> 00:00:15.000
s = rθ, v = rω, a_t = rα — the bridge between rotation and translation.
Printable notes (free)

Practice

Free · 4 questions with full solutions
  1. Q1 · Numerical · difficulty 1/5

    A wheel rotates at 270 rpm. What is its angular velocity in rad/s?

  2. Q2 · Numerical · difficulty 1/5

    A point is at distance 1.7 m from the axis of a wheel rotating with angular velocity 11 rad/s. Find its linear speed.

  3. Q3 · Numerical · difficulty 2/5

    A particle moves on a circle of radius 1.5 m with constant speed 6 m/s. Find the magnitude of its centripetal acceleration.

  4. Q4 · MCQ · difficulty 1/5

    In uniform circular motion, the acceleration of the particle is directed:

    • Along the tangent to the circle
    • Toward the centre of the circle
    • Away from the centre of the circle
    • It is zero