Quick answer: Uniform circular motion is travel at constant speed along a circle. The velocity direction keeps changing, so the object accelerates toward the centre with a = v²/r. The net force causing this is m v²/r. In an orbit, gravity F = G m₁ m₂ / r² supplies that net force.
Level: AP Physics 1 (algebra-based) · Curriculum: AP P1 Unit 2
What is uniform circular motion?
An object in uniform circular motion travels around a circle at constant speed. Its velocity is not constant, because the direction changes at every instant. A changing velocity means acceleration, so the object accelerates even though its speed never changes.
Centripetal acceleration is the acceleration of an object moving on a circular path. It points toward the centre of the circle and is perpendicular to the velocity, so it changes the direction of the motion but not the speed.
How do you calculate centripetal acceleration and force?
ac = v² / r = 4π² r / T² and ΣF = m ac = m v² / r. Here ac is centripetal acceleration (m/s²), v is speed (m/s), r is the radius (m), T is the period (s), m is mass (kg) and ΣF is the net force toward the centre (N).
The phrase centripetal force names a role, not a new kind of force. Add up the real forces that point toward the centre, subtract those that point away, and the result must equal m v² / r. Never draw an arrow labelled centripetal force on a free-body diagram.
Kiran in Chennai whirls a 0.40 kg stone on a 0.75 m string in a horizontal circle on a smooth table, two revolutions every second. Find the tension.
Period T = 1 ÷ 2 = 0.50 s. Speed v = 2π r / T = 2π(0.75) ÷ 0.50 = 9.42 m/s.
ac = v² / r = (9.42)² ÷ 0.75 = 118 m/s².
Tension = m ac = 0.40 × 118 = 47 N, directed toward the centre.
What supplies the centripetal force?
Each circular motion has its own source of inward force. Identify it before you write any equation.
| Situation | Real force that supplies the centripetal force |
|---|---|
| Stone whirled on a string | Tension in the string |
| Car turning on a flat road | Static friction between tyres and road |
| Moon orbiting the Earth | Gravitational attraction of the Earth |
| Rider pinned to the wall of a spinning drum | Normal force from the wall |
| Bucket at the top of a vertical circle | Weight plus the normal force or tension, both pointing down |
In a vertical circle the weight always points down, but the centre can be above or below the object. Choose the direction toward the centre as positive and the equations follow directly. At the very top, the least speed that keeps a string taut or water in a bucket is found by setting the tension or normal force to zero, which gives v = √(g r).
Students say that a centrifugal force pushes the object outward. Nothing pushes outward. The object tends to continue in a straight line, so if the inward force disappears it moves along the tangent.
How does universal gravitation work?
Every pair of masses attracts with a force F = G m₁ m₂ / r², where G = 6.67 × 10⁻¹¹ N m²/kg² and r is the distance between the centres. The two forces are equal in size and opposite in direction, as Newton’s third law requires.
Dividing the force on a small mass by that mass gives the gravitational field strength g = G M / r². At the surface of the Earth this is about 9.81 N/kg, and it falls to one quarter at a distance of two Earth radii from the centre.
How fast does a satellite move in a circular orbit?
For a satellite, gravity is the only force, so it supplies the whole centripetal force: G M m / r² = m v² / r. The satellite mass cancels, so the orbit speed does not depend on it.
v = √(G M / r) and T² = 4π² r³ / (G M). Here M is the mass of the central body (kg), r is the orbit radius measured from the centre (m) and T is the period (s).
A survey satellite orbits 600 km above the Earth’s surface. Use G M = 3.98 × 10¹⁴ m³/s² and Earth radius 6.37 × 10⁶ m. Find its speed and period.
r = 6.37 × 10⁶ + 6.0 × 10⁵ = 6.97 × 10⁶ m.
v = √(G M / r) = √(3.98 × 10¹⁴ ÷ 6.97 × 10⁶) = 7.56 × 10³ m/s.
T = 2π r / v = 2π(6.97 × 10⁶) ÷ 7.56 × 10³ = 5.79 × 10³ s = 96.6 min.
A larger orbit therefore means a lower speed and a longer period. Astronauts float because they and the station fall together with the same acceleration, not because gravity has vanished.
Start every circular-motion problem by sketching the object and the centre, then write Newton’s second law along the line to the centre. Only after that decide which numbers to insert.
- Centripetal acceleration: a = v²/r = 4π²r/T², directed toward the centre (m/s²).
- Net force toward the centre: ΣF = m v²/r, not a separate force.
- Gravity: F = G m₁ m₂ / r², with G = 6.67 × 10⁻¹¹ N m²/kg².
- Circular orbit: v = √(G M / r) and T² = 4π² r³ / (G M).
Centripetal acceleration and gravity in orbit: frequently asked questions
Why does an object moving at constant speed on a circle accelerate?
Acceleration means a change in velocity, and velocity includes direction. On a circle the direction changes continuously, so the velocity changes and the acceleration points toward the centre.
Is centripetal force a separate force?
No. It is the name for the net force toward the centre, made up of real forces such as tension, friction, gravity or the normal force. It never appears as an extra arrow on a free-body diagram.
What happens if the string breaks?
With no inward force the object moves in a straight line along the tangent to the circle at the point of release, at the speed it had at that moment.
How does the gravitational force change with distance?
It follows an inverse-square law. Double the distance between the centres and the force becomes one quarter; triple it and the force becomes one ninth.
Why are astronauts weightless in orbit?
Gravity at 400 km is still about 89 % of its surface value. The astronauts and the station are in free fall together, so there is no contact force between them, which feels like weightlessness.
