Centripetal force: (b) constant speed in a circle; v = ωr

The idea

An object moving in a circle at constant speed has a velocity that changes direction continuously, so its velocity is not constant even though its speed is. The linear speed v is related to the angular velocity ω and the radius r by v = ωr, where v is in metres per second (m s⁻¹), ω is in radians per second (rad s⁻¹) and r is in metres (m). For example, a point 0.30 m from the axis of a wheel rotating at ω = 4.0 rad s⁻¹ moves at v = 4.0 × 0.30 = 1.2 m s⁻¹. Because ω is the same for all points on a rigid body, v is proportional to r, so points further from the axis move faster in a straight-line sense. The direction of v is always tangent to the circle.

What earns the marks

• In uniform circular motion the speed is constant but the velocity changes direction continuously.
• The relationship between linear speed and angular velocity is v = ωr.
• v is measured in m s⁻¹, ω in rad s⁻¹ and r in m.
• For a rigid rotating body, ω is the same at all points, so v is proportional to r.
• The velocity vector is always tangent to the circular path.

Common mistakes and how to put them right

• Saying the velocity is constant.

  Put it right: The speed is constant but the velocity changes direction, so it is not constant.

• Using degrees in v = ωr.

  Put it right: ω must be in rad s⁻¹.

• Mixing up r and diameter.

  Put it right: r is the radius, half the diameter.

• Assuming all points on a rotating body have the same linear speed.

  Put it right: v = ωr, so points further from the axis move faster.

Exam tips

• Convert any diameter to a radius before substituting into v = ωr.
• Check that ω is in rad s⁻¹; if you are given a frequency, first use ω = 2πf.
• State that the velocity direction is tangential when explaining why the velocity is not constant.