Circular Motion
Circular Motion
Circular motion occurs when an object moves along a circular path. Even at constant speed, the object is accelerating because the direction of velocity is constantly changing.
Angular Displacement and Angular Velocity
Angular displacement, θ: The angle swept out by the radius, measured in radians (rad).
One complete revolution = 2π rad = 360°
Angular velocity, ω: The rate of change of angular displacement.
ω = θ/t = 2π/T = 2πf
where:
- ω = angular velocity (rad s⁻¹)
- T = period — time for one complete revolution (s)
- f = frequency — number of revolutions per second (Hz)
Relationship Between Linear and Angular Speed
For an object moving in a circle of radius r:
v = ωr
where v is the linear (tangential) speed.
Centripetal Acceleration
An object in uniform circular motion has a velocity that is constantly changing direction (always tangent to the circle). This requires an acceleration directed toward the centre of the circle.
Centripetal acceleration, a = v²/r = ω²r
Derivation of a = v²/r
Consider an object moving from point A to point B on a circle in time δt. The velocity changes from v at A to v at B (same magnitude, different direction).
The change in velocity δv points toward the centre. For a small angle δθ:
|δv| = vδθ (for small angles, arc ≈ chord)
a = δv/δt = vδθ/δt = v × ω = v × v/r = v²/r
Centripetal Force
By Newton's second law, the centripetal acceleration requires a centripetal force:
F = mv²/r = mω²r
This force is always directed toward the centre of the circle. It is not a new type of force — it is provided by whatever physical force keeps the object on its circular path:
| Situation | Force providing centripetal force |
|---|---|
| Planet orbiting the Sun | Gravitational attraction |
| Car turning a corner | Friction between tyres and road |
| Ball on a string (horizontal circle) | Tension in the string |
| Electron orbiting nucleus | Electrostatic (Coulomb) force |
| Clothes in a washing machine | Normal contact force from drum |
Worked Example — Car on a Bend
A car of mass 1200 kg travels around a circular bend of radius 40 m at 15 m s⁻¹. Calculate the centripetal force.
F = mv²/r = 1200 × 15² / 40 = 1200 × 225/40 = 6750 N
This force is provided by friction. If the road is wet and friction is reduced below 6750 N, the car will skid outward (not "fly outward" due to centrifugal force — there is simply insufficient centripetal force).
Conical Pendulum
A mass on a string swinging in a horizontal circle (the string traces a cone):
Resolving forces on the mass:
- Vertically: T cos θ = mg (no vertical acceleration)
- Horizontally: T sin θ = mv²/r = mω²r (centripetal force)
Dividing: tan θ = v²/(rg) = ω²r/g
Since r = L sin θ (where L is the string length): ω = √(g / L cos θ)
This shows that ω depends on the angle θ and the string length, but not on the mass.
Vertical Circular Motion
For an object on a string in a vertical circle, the speed varies (faster at the bottom, slower at the top) because of changes in gravitational potential energy.
At the top of the circle:
Weight and tension both act downward (toward centre):
mg + T = mv²_top/r
The minimum speed at the top occurs when T = 0:
mg = mv²_min/r → v_min = √(gr)
At the bottom of the circle:
Tension acts upward, weight acts downward:
T − mg = mv²_bottom/r → T = mv²_bottom/r + mg
The tension at the bottom is always greater than at the top (by at least 6mg for the critical case), which is why strings and chains are most likely to break at the bottom of vertical circular motion.
Banked Curves
On a banked road (tilted at angle θ to the horizontal), a component of the normal reaction force provides centripetal force, reducing the reliance on friction.
For the design speed (no friction needed):
N sin θ = mv²/r (centripetal)
N cos θ = mg (vertical equilibrium)
Dividing: tan θ = v²/(rg)
Worked Example — Banked Curve
An aircraft banks at 30° in a turn of radius 500 m. Find the speed.
tan 30° = v²/(rg) → v² = rg tan 30° = 500 × 9.81 × tan 30° = 500 × 9.81 × 0.577
v² = 2831 → v = 53.2 m s⁻¹
Common Misconception: Centrifugal Force
There is no outward force on an object in circular motion. What people call "centrifugal force" is the effect of inertia — the body's tendency to continue in a straight line. In the rotating reference frame it appears as a fictitious outward force, but in an inertial frame, the only real force is the centripetal force directed inward.