Summary — What This Chapter Covers
- The radian: angle subtended at the centre by an arc equal in length to the radius. 2π rad = 360°.
- Angular speed ω is the angle swept out per unit time: ω = 2π/T and v = rω.
- A force of constant size that is always perpendicular to the velocity causes centripetal acceleration and circular motion at constant angular speed.
- a = rω2 = v2/r and F = mrω2 = mv2/r, both directed towards the centre.
1. Kinematics of Uniform Circular Motion
12.1
Radians and angular speed
One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. So an angle θ in radians is arc length ÷ radius, θ = s/r, and a full turn is 2π rad. To convert degrees, multiply by π/180.
Angular displacement is the angle (in radians) through which the radius has turned. Angular speed ω is the rate of change of angular displacement, measured in rad s−1. In one period T the object turns through 2π, and the linear (tangential) speed is the arc length covered per second.
Defining equations
θ = s/r ω = Δθ/Δt = 2π/T = 2πf v = rω
Worked Example
A wheel of radius 0.30 m makes 4.0 revolutions per second. Find its period, its angular speed and the speed of a point on the rim.
Practice Question
Express 60° in radians, and find the arc length it cuts off on a circle of radius 2.0 m.
2. Centripetal Acceleration
12.2
Force towards the centre
Velocity is a vector. In uniform circular motion its magnitude is constant but its direction changes continuously, so the object accelerates. The acceleration is directed towards the centre (centripetal). By Newton’s second law there must be a resultant force towards the centre. Because this force is always perpendicular to the velocity it does no work, so the speed stays constant and only the direction changes.
The centripetal force is not a new kind of force. It is supplied by tension, friction, gravity, a normal contact force, or a combination; in a free-body diagram never draw an extra “centripetal force” arrow.
Centripetal acceleration and force
a = rω2 = v2/r F = mrω2 = mv2/r
Memorise