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Circular Motion and Centripetal Force

Classical Mechanics

Anything moving in a circle is constantly changing direction, which means it is constantly accelerating — even if its speed stays the same. That acceleration needs a force pointing toward the centre of the circle, called the centripetal force. From the Moon orbiting Earth to a car taking a bend, the same rule governs every turn.

Why circular motion needs a force

An object moving in a circle at a constant speed has a constant speed but a changing velocity, because velocity includes direction. Changing velocity means acceleration, and acceleration needs a net force. That acceleration always points toward the centre of the circle and is called **centripetal acceleration**. The force that produces it is the **centripetal force**. It is not a new kind of force: it is simply the name for whichever real force — tension, gravity, friction, or a normal force — happens to be pointing toward the centre in that situation.

The formulas

For an object of speed v moving in a circle of radius r, the centripetal acceleration is **a = v² / r**. Multiply by the mass and you get the centripetal force: **F = m v² / r**. These formulas show two important patterns: a faster object needs much more force (force grows with v²), and a tighter circle needs more force (force grows as r shrinks). That is why taking a bend too fast or too sharply is dangerous — the friction available from the tyres may not be enough to supply the centripetal force, and the car slides outward.

Where you see it

A car rounding a bend relies on friction between its tyres and the road for centripetal force. A satellite stays in orbit because Earth's gravity supplies the centripetal force, pulling it toward the centre even as it moves forward. A stone whirled on a string is held in by the tension in the string — let go, and the stone flies off in a straight line along the tangent, not outward from the centre. A roller coaster loop works because the track pushes on the cars toward the centre of the loop, and clothes in a washing machine are spun dry as water escapes through holes while the drum supplies the centripetal force.

Common mistakes

• Saying there is a centrifugal force pushing outward. Correction: in the ground frame there is no outward force — the object wants to go straight (inertia), and the centripetal force pulls it inward.
• Thinking constant speed in a circle means no acceleration. Correction: velocity changes because direction changes, so there is acceleration toward the centre.
• Calling centripetal force a separate type of force. Correction: it is a role played by a real force such as tension, gravity or friction.

Practice questions

1. A 2 kg ball on a string moves in a circle of radius 1 m at 4 m/s. What is the tension in the string?
   Show answer

   F = m v² / r = 2 x 16 / 1 = 32 N.

2. If the speed of an object in a circle doubles, what happens to the centripetal force?
   Show answer

   It quadruples, because F depends on v²: 2² = 4 times as much.

3. A satellite orbits Earth. What supplies its centripetal force?
   Show answer

   Gravity — Earth's gravitational pull acts toward the centre of the orbit.

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