12.1aRadians and angular displacement
Definitions
- Radian
- the angle subtended at the centre of a circle by an arc equal in length to the radius.
- Angular displacement (θ)
- the angle through which an object has turned about the centre of the circle, measured in radians.
Key points
- A full circle is 2π rad = 360°; so 1 rad ≈ 57.3°.
- Degrees to radians: multiply by π/180.
- Arc length s = rθ only works with θ in radians.
- Small angles (θ in rad): sin θ ≈ tan θ ≈ θ.
Angle in radians = arc length / radius. When s = r, θ = 1 rad.
Watch out!
- Calculator in the wrong mode: set it to radians for any angle from θ = s/r or ωt.
12.1bAngular speed
Definitions
- Angular speed (ω)
- the rate of change of angular displacement (angle turned per unit time).
- Period (T)
- the time taken for one complete revolution.
- Frequency (f)
- the number of revolutions per unit time.
Key points
- Every point on a rigid rotating body has the same ω.
- Linear (tangential) speed increases with radius: v = rω.
- Uniform circular motion: constant speed, constant ω, but changing velocity (direction changes).
- Convert rev min-1 to rad s-1: multiply by 2π, divide by 60.
A rotating disc: points A and B turn through the same angle in the same time (same ω), but B at larger radius moves faster (v = rω).
Watch out!
- ω must be in rad s-1, not rev s-1 or degrees per second.
- Radius r is measured from the centre of the circle, not the diameter.
12.2aCentripetal acceleration and centripetal force
Definitions
- Centripetal acceleration
- the acceleration of an object moving in a circle, directed towards the centre of the circle.
- Centripetal force
- the resultant force on an object moving in a circle, directed towards the centre of the circle.
Key points
- At constant speed the velocity changes direction, so the object accelerates (Newton's first law needs a resultant force).
- Velocity is tangential; force and acceleration point along the radius to the centre.
- Force is perpendicular to velocity, so it does no work: kinetic energy and speed stay constant.
- Centripetal force is not an extra force: it is provided by tension, gravity, friction, normal contact or a magnetic force.
- If the force is removed, the object moves off along the tangent (not outwards along the radius).
Uniform circular motion: velocity is tangential; the centripetal force and acceleration point inwards along the radius.
Why the acceleration points inwards: the change in velocity Δv = v2 - v1 between two nearby points is directed towards the centre.
Watch out!
- There is no outward "centrifugal force" on the object. Draw only the real forces; their resultant is inwards.
- In F = mv2/r, F is the resultant force, not one of the forces added to it.
12.2bApplying circular motion
Key points
- Method: draw the real forces, find the resultant towards the centre, set it equal to mv2/r.
- Car on a flat bend: friction between tyres and road provides the centripetal force.
- Banked track: the horizontal component of the normal force provides it; N cos θ = mg.
- Mass on a string in a vertical circle: tension is greatest at the bottom, least at the top.
- At the top, the string stays taut only if v ≥ √(gr); at this minimum speed T = 0 and gravity alone provides the force.
- Satellites and planets: gravitational force provides the centripetal force (see topic 13).
- Conical pendulum / aircraft banking: horizontal component of tension or lift provides it.
Vertical circle: at the top, tension and weight both act towards the centre; at the bottom, tension must exceed weight.
Banked track: the horizontal component of the normal force, N sin θ, provides the centripetal force; N cos θ balances the weight.
Watch out!
- At the top of a vertical circle, weight helps (T + mg); at the bottom it opposes (T - mg). Do not use the same equation for both.
- Use the radius to the centre of mass of the object, e.g. string length plus the ball radius if needed.