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Circular Motion
Eduqas A Level Physics — Revision Notes
Component 1: Newtonian Physics · Topic 5
The Language of Circular Motion
When something moves in a circle at constant speed, it is not moving at constant velocity — the direction keeps changing. This means there is always an acceleration, and therefore always a net force. Before getting into the forces, you need to be comfortable with the vocabulary.
Key Definitions
Period (T ): The time for one complete revolution, in seconds. Frequency (f ): The number of complete revolutions per second (Hz). Angular velocity (ω): The angle swept out per second, in rad s−^1. These are linked by: T =
f
, ω =
2 π T
= 2πf
Radians
A radian is defined as the angle subtended at the centre of a circle when the arc length equals the radius. There are 2π radians in a full circle (360◦), so:
θ (radians) =
arc length r
, 1 rad ≈ 57. 3 ◦
The arc length for an angle θ is simply s = rθ.
Linking Linear and Angular Quantities
The linear (tangential) speed v relates to angular velocity ω via the radius r:
v = rω
Think about it physically: the larger the circle, the further you have to travel in the same time to complete the same angle, so v increases with r even when ω stays fixed.
Centripetal Acceleration
Even at constant speed, changing direction means changing velocity, which means acceleration. This acceleration always points towards the centre of the circle, which is why it is called centripetal (from Latin: “centre-seeking”).
Centripetal Acceleration
a =
v^2 r
= ω^2 r
Direction: Always towards the centre. The velocity is always perpendicular to this acceleration, which is why speed stays constant even though the object is accelerating.
Deriving a = v^2 /r: Consider a particle moving from point A to B over a tiny time δt, sweeping angle δθ. The change in velocity vector δv points towards the centre with magnitude v δθ. So:
a =
δv δt
v δθ δt
= vω =
v^2 r
Centripetal Force
By Newton’s second law, any acceleration requires a net force in the same direction. Centripetal force is simply the name given to this net inward force.
Centripetal Force
F =
mv^2 r
= mω^2 r
This is not a new type of force. It is whatever force happens to provide the inward push or pull in a given situation.
Common Misconception: “Centrifugal Force”
There is no outward “centrifugal force” acting on the object. In an inertial (non-rotating) frame of reference, the object simply tends to continue in a straight line (Newton’s 1st law). The feeling of being “pushed outward” in a car or a fairground ride is just your inertia resisting the centripetal acceleration — it is not a real force on the object.
What Provides the Centripetal Force?
Situation Force providing centripetal acceleration Ball on a string Tension in the string Planet orbiting the Sun Gravitational attraction Car rounding a bend Friction between tyres and road Electron in orbit Electrostatic attraction to nucleus Banked road/track Normal reaction component
Applications and Worked Examples
Worked Example 1: Car on a Circular Track
A car of mass 1200 kg travels around a circular track of radius 80 m at 20 m s−^1. Find (a) the centripetal acceleration, (b) the centripetal force, (c) the angular velocity.
(a) a =
v^2 r
= 5 m s−^2
(b) F = ma = 1200 × 5 = 6000 N
(c) ω =
v r
= 0. 25 rad s−^1 The friction from the road provides the 6000 N centripetal force.
vertical (balances weight) components.
