This physics grades 11-12 lesson on circular motion notes gives the key ideas first, then worked examples and diagrams. After that, practise with the worksheet on circular motion and check yourself with the quiz.
Quick answer: Circular motion at constant speed involves a constant angular velocity ω = 2π/T and a centripetal acceleration a = v²/r = ω²r directed towards the centre. The resultant force that causes it, F = mv²/r, is supplied by tension, friction, gravity or a normal reaction, and does no work.
Level: Physics grades 11-12 (A Level · IB · Class 11-12) · Curriculum: A Level 3.6
Circular motion notes: the key ideas
How do you describe motion in a circle?
An object going round a circle keeps changing direction, even when its speed is constant. You describe how fast it turns with the angular velocity ω, measured in radians per second. One full turn is 2π radians, so ω = 2π ÷ T = 2π f, where T is the period and f the frequency.
Angular velocity ω is the angle turned per unit time (rad/s). The linear speed along the circle is v = r ω, where r is the radius. One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius, so 180° = π rad.
A merry-go-round in Kumasi makes one turn in 8.0 s. A child sits 2.5 m from the axis.
ω = 2π ÷ T = 6.283 ÷ 8.0 = 0.785 rad/s.
v = r ω = 2.5 × 0.785 = 2.0 m/s.
Why is there an acceleration in uniform circular motion?
Velocity is a vector. When its direction changes, the object accelerates, even if the speed does not change. The acceleration points towards the centre. It is called centripetal (“centre-seeking”) acceleration, and the resultant force that causes it is the centripetal force.
a = v² ÷ r = ω² r
F = m v² ÷ r = m ω² r
a = centripetal acceleration (m/s²), F = resultant force towards the centre (N), m = mass (kg), v = speed (m/s), ω = angular velocity (rad/s), r = radius (m).
Centripetal force is not a new kind of force. It is the name for whichever real force, or combination, points to the centre: tension in a string, friction on a tyre, gravity on a planet, or a normal reaction. Because the force is always at right angles to the velocity, it does no work and the speed stays constant.
A stone of mass 0.30 kg is whirled on a string in a horizontal circle of radius 0.80 m at 4.0 m/s.
a = v² ÷ r = 16 ÷ 0.80 = 20 m/s².
F = m a = 0.30 × 20 = 6.0 N, which is the tension in the string.
| Situation | Force that provides the centripetal force |
|---|---|
| Car on a flat bend | friction between the tyres and the road |
| Moon orbiting the Earth | gravitational attraction |
| Stone on a string | tension in the string |
| Rider pressed on the wall of a spinning drum | normal reaction of the wall |
How do flat and banked bends work?
On a flat bend the friction at the tyres provides the centripetal force. The greatest value is μ m g, so the car just keeps its circle when μ m g = m v² ÷ r. The mass cancels, giving vmax = √(μ g r).
On a banked track the normal reaction N has a horizontal component N sin θ that points to the centre. With no friction, N cos θ = m g and N sin θ = m v² ÷ r, so tan θ = v² ÷ (r g).
A frictionless banked bend has radius 80 m and a design speed of 20 m/s.
tan θ = v² ÷ (r g) = 400 ÷ (80 × 9.81) = 0.510, so θ = 27°.
What changes in a vertical circle?
In a vertical circle the weight sometimes helps and sometimes opposes the centripetal force, so the speed and the tension vary. At the bottom, T − m g = m v² ÷ r. At the top, T + m g = m v² ÷ r.
Wanjiru swings a bucket of water in a vertical circle of radius 0.80 m. The water stays in the bucket if the normal reaction at the top is not negative. At the limit T = 0, so m g = m v² ÷ r.
vmin = √(g r) = √(9.81 × 0.80) = 2.8 m/s.
Many students draw an outward “centrifugal force” on the object. No such real force acts on it. The object tends to move in a straight line, and the inward resultant force bends its path. Draw only real forces, then say which direction is the net force.
Start every circular motion problem by drawing the real forces, then pick the direction to the centre as positive. Write “net force towards the centre = m v² ÷ r” and fill in the forces with their signs.
- Angular velocity ω = 2π/T = 2πf in rad/s; linear speed v = r ω.
- Centripetal acceleration a = v²/r = ω²r, directed towards the centre.
- Centripetal force F = m v²/r = m ω² r; it is a real force or a resultant, not an extra force.
- Flat bend: vmax = √(μ g r); frictionless banked bend: tan θ = v²/(r g).
- Vertical circle: T − m g = m v²/r at the bottom, T + m g = m v²/r at the top.
Angular velocity, centripetal force and circular paths: frequently asked questions
What is the difference between angular velocity and linear speed?
Angular velocity ω is the angle turned per second in rad/s, the same for every point on a rotating disc. Linear speed v = r ω depends on the distance from the axis, so points farther out move faster.
Why does a satellite in a circular orbit not speed up?
Gravity is always at right angles to the velocity, so it changes the direction but does no work. The kinetic energy and so the speed stay constant.
Is centrifugal force real?
No real force pushes an object outward. What you feel in a turning car is your body tending to continue in a straight line, while the seat and door push you inward as the real centripetal force.
How do you convert revolutions per minute to rad/s?
Multiply by 2π to get radians per minute, then divide by 60. For example, 120 rpm is 120 × 2π ÷ 60 = 12.6 rad/s.
Why do roads have banked bends?
Banking lets the normal reaction supply part of the centripetal force. At the design speed no friction is needed at all, so the bend is safe even on ice or when it is wet.
