Motion in a Circle

This guide covers Topic 12, Motion in a circle, in full — subtopics 12.1 Kinematics of uniform circular motion and 12.2 Centripetal acceleration — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is A Level content and is the first of the A Level topics, building directly on AS Level kinematics and dynamics.

Before studying this

This resource assumes the equations of motion from Kinematics: Equations of Motion and Newton’s laws from Dynamics: Newton’s Laws and Momentum.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — A Level, Topic 12

12.1 Kinematics of uniform circular motion — defining the radian, and converting between radians and degrees; understanding and using angular speed ω; recalling and using v = ωr, relating linear speed to angular speed and radius.

12.2 Centripetal acceleration — understanding that a resultant force (centripetal force) is needed to maintain circular motion, always directed toward the centre of the circle; deriving, using a = v²/r, the centripetal acceleration for an object moving in a circle at constant speed; recalling and using a = v²/r and a = ω²r; recalling and using F = mv²/r and F = mω²r.

The radian and angular speed

An angle measured in radians is the ratio of arc length to radius. One full circle (360°) is 2π radians, so:

θ (radians) = θ (degrees) × π/180

Angular speed ω is the rate of change of angle with time:

ω = θ/t = 2π/T = 2πf

where T is the period (time for one revolution) and f is the frequency of rotation.

Linear speed and angular speed

For an object moving in a circle of radius r, its linear (tangential) speed v is related to angular speed by:

v = ωr

This connects the “how fast around the circle” description (ω) to the “how fast along the path” description (v).

Centripetal acceleration

An object moving in a circle at constant speed is still accelerating, because its velocity — a vector — is continuously changing direction. This acceleration is directed toward the centre of the circle (hence centripetal, meaning “centre-seeking”) and has magnitude:

a = v²/r = ω²r

Worked example. A stone of mass 0.20 kg is whirled in a horizontal circle of radius 0.80 m at a constant speed of 4.0 m s⁻¹. Its centripetal acceleration:

a = v²/r = 4.0² / 0.80 = 20 m s⁻²

Centripetal force

By Newton’s second law, a centripetal acceleration requires a centripetal force, directed toward the centre of the circle:

F = mv²/r = mω²r

This force is not a new, separate type of force — it is provided by whatever real force (tension, gravity, friction, normal contact force) happens to act toward the centre in a given situation, e.g. tension in a string for a whirled object, or gravity for a satellite in orbit.

Banked tracks (beyond the syllabus — background only, not examinable)

Banked tracks are not part of the Cambridge 9702 Motion in a Circle content — they are included here as a worked extension of the centripetal-force ideas above, not as recall or exam-required material.

On a frictionless banked track at angle θ to the horizontal, only two forces act on a vehicle: its weight mg (vertically down) and the normal contact force N (perpendicular to the track surface). Resolving vertically, N cos θ = mg; resolving horizontally, the horizontal component of N provides the centripetal force, N sin θ = mv²/r. Dividing the two equations eliminates N:

tan θ = v² / (rg)

Worked example. A track is banked at 20° and designed for vehicles travelling at 25 m s⁻¹. The radius it should be built to:

r = v² / (g tan θ) = 25² / (9.81 × tan 20°) ≈ 175 m

At exactly this design speed and radius, no friction is needed at all; friction only becomes necessary if a vehicle’s actual speed differs from the design speed.

Vertical circles

For an object moving in a vertical circle — a bucket of water on a string, or a car passing over a hilltop — the centripetal force required is still mv²/r, but gravity’s contribution to it changes with position around the circle. At the top and bottom:

Top:     T + mg = mv²/r   =>   T = mv²/r − mg
Bottom:  T − mg = mv²/r   =>   T = mv²/r + mg

The tension (or normal force) is greatest at the bottom and least at the top. Subtracting the two equations gives a difference of 2mg only if the speed is the same at both points; for an object moving freely with energy conserved, v² at the bottom exceeds v² at the top by 4gr (from ½mv_bottom² = ½mv_top² + mg × 2r), and the difference is 6mg. At the top, the minimum speed occurs when T falls to zero, so that gravity alone supplies the entire centripetal force:

mg = mv²/r   =>   v_min = √(gr)

Below this minimum speed, the object cannot maintain the circular path — a string goes slack, or a car loses contact with the road — because gravity alone would now exceed the centripetal force actually required, and the path curves away from the intended circle.

Common mistakes

• Treating “centripetal force” as an additional force acting alongside the real forces — it is the name given to the resultant of the real forces acting toward the centre, not an extra force to add to a diagram.
• Confusing angular speed ω (rad s⁻¹) with linear speed v (m s⁻¹) — always check which one a question gives or asks for, and convert using v = ωr.
• Forgetting that constant speed does not mean constant velocity in circular motion, since direction is always changing — this is why there is still an acceleration.
• Mixing degrees and radians — angular speed formulas require radians, not degrees.
• Assuming a banked track needs friction to work — at the design speed for a given radius and angle, the normal force alone supplies the required centripetal force with no friction needed.
• Forgetting that tension cannot go negative — the minimum speed at the top of a vertical circle is the speed at which tension reaches zero, not some arbitrarily small value.

Quick revision checklist

• Radian measure and converting between radians and degrees
• ω = 2π/T = 2πf, and v = ωr
• a = v²/r = ω²r, directed toward the centre
• F = mv²/r = mω²r, and identifying which real force provides it
• T = mv²/r ∓ mg at the bottom/top of a vertical circle, and v_min = √(gr) at the top

Extension, beyond the syllabus: tan θ = v²/(rg) for a frictionless banked track — background only, not examinable.

Related resources

• Dynamics: Newton’s Laws and Momentum — the AS Level foundation for force and acceleration
• Gravitational Fields — the next A Level topic, applying circular motion to orbits
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.