Centripetal force

The force that keeps things going round

• A circular path requires a centripetal acceleration toward the centre, so by Newton's second law it requires a net force toward the centre.
• That inward net force is called the centripetal force:

$$F_c = ma_c = \frac{mv^2}{r}$$

• $F_c$ — centripetal force (N), always directed toward the centre
• $m$ — mass of the object (kg)
• $v$ — speed around the circle (m s⁻¹)
• $r$ — radius of the circle (m)

Centripetal force is a role, not a new force

This is the single most important idea on the page, and it is examined constantly.

• There is no such thing as "a centripetal force" that exists on its own. The name describes the job being done by a force you can already name.
• In every situation, ask: which real force is pointing toward the centre?

Situation	The real force providing $F_c$
Ball whirled on a string	Tension in the string
Car rounding a flat corner	Friction between tyres and road
Moon orbiting the Earth	Gravity
Clothes in a spin dryer	Normal force from the drum wall
Rider on a fairground wall-of-death	Normal force from the wall

• At Level 2 the standard limits circular motion to cases where one force only provides the centripetal force. You will not be asked to combine two.
• Never write "centrifugal force" on a free-body diagram or in an explanation. There is no outward force; the only force is inward.

What happens when the force is not enough

• If the available force is smaller than $mv^2/r$, the object cannot follow that circle.
• It does not fly outward — it simply follows a wider, straighter path than the circle, moving off along the tangent as Newton's first law requires.
• Real examples to use in answers:
  • a car on an icy corner: friction is too small, so the car understeers and slides toward the outside of the bend,
  • a string that snaps: tension vanishes and the ball flies off along the tangent.

How speed and radius change the force needed

$$F_c = \frac{mv^2}{r}$$

• Double the speed → four times the force (the $v^2$ term).
• Halve the radius → double the force (a tighter corner is harder).
• Double the mass → double the force.
• This is why:
  • advisory speed signs on tight bends specify low speeds — the force needed rises with the square of the speed,
  • heavy trucks need to corner more slowly than cars for the same available friction,
  • a motorway curve is built with a very large radius so that the friction needed at $100$ km h⁻¹ stays small.

Change the speed and radius below and watch the required centripetal force respond: