Centripetal Force

When an object moves in a circle, its direction is constantly changing — even if its speed stays the same. Because velocity is a vector (it has direction), a changing direction means a changing velocity, which means acceleration is occurring. By Newton's second law, acceleration requires a net force. The force responsible for keeping an object on a circular path is called the centripetal force. The word 'centripetal' comes from Latin meaning 'centre-seeking', and that is precisely what this force does: it always points inward, towards the centre of the circular path. It is crucial to understand that centripetal force is not a new type of force — it is simply the name we give to whatever net force happens to be acting inward in a given situation. When a car rounds a bend, friction from the road provides the centripetal force. When the Moon orbits Earth, gravity provides it. When a ball on a string is swung in a circle, tension in the string provides it. In every case, the role is the same: redirecting the object continuously towards the centre, causing it to follow a curved path rather than flying off in a straight line. Without a centripetal force, Newton's first law tells us the object would travel in a straight line at constant velocity — it would simply continue in whatever direction it was moving at that instant. Centripetal force prevents this, continuously 'steering' the object around the circle. Understanding this force is the foundation for analysing planetary orbits, fairground rides, banked roads, and satellite motion.

Key takeaways
  • Circular motion requires inward acceleration
  • Applying Newton's Second Law
  • Centripetal force is a net force, not an extra force
  • The 'centrifugal' myth

Circular motion requires inward acceleration

Even at constant speed, an object in circular motion is accelerating because its direction changes. This acceleration, called centripetal acceleration, always points towards the centre of the circle. Its magnitude is given by a = v²/r, where v is the speed and r is the radius of the circle. Think of a ball on a string: as you swing it, the string constantly pulls the ball inward — without that pull, the ball would leave the circle immediately.

Applying Newton's Second Law

Newton's second law states F = ma. Substituting the expression for centripetal acceleration, the net inward force required for circular motion becomes F_c = mv²/r. This can also be written in terms of angular velocity ω as F_c = mω²r, since v = ωr. Both forms are equivalent; which to use depends on what information you are given. The direction of this force is always radially inward — towards the centre — never tangential (along the circle).

Centripetal force is a net force, not an extra force

A common misconception is to add 'centripetal force' to a free-body diagram as if it were a separate, new force. It is not. It is the label for the resultant of real forces (tension, gravity, friction, normal reaction, etc.) acting towards the centre. When you draw a free-body diagram for circular motion, identify the real forces present, find their net inward component, and set that equal to mv²/r. For example, at the top of a loop-the-loop, both gravity and the normal force point inward, so their sum equals mv²/r.

The 'centrifugal' myth

You may have felt 'pushed outward' in a turning car. This sensation arises because you are in an accelerating (non-inertial) reference frame. In the inertial (ground) frame, there is no outward force — your body's inertia simply tends to continue in a straight line while the car turns beneath you. The so-called 'centrifugal force' is a fictitious force that only appears when you analyse motion from a rotating reference frame. In standard physics problems at this level, always work in the inertial frame.

The Key Equation

Fc=mv2r=mω2rF_c = \frac{mv^2}{r} = m\omega^2 r
  • F_c — Centripetal force (Newtons (N)): The net inward force directed towards the centre of the circular path.
  • m — Mass (Kilograms (kg)): The mass of the object undergoing circular motion.
  • v — Speed (Metres per second (m s⁻¹)): The instantaneous speed of the object along the circular path.
  • r — Radius (Metres (m)): The radius of the circular path — the distance from the object to the centre of the circle.
  • \omega — Angular velocity (Radians per second (rad s⁻¹)): The rate at which the object sweeps out angle; related to speed by v = ωr.

Worked Example

Problem. A car of mass 1200 kg travels at a constant speed of 15 m s⁻¹ around a roundabout of radius 30 m. Calculate the centripetal force required and identify which real force provides it.

Fc=1200×15230=9000 NF_c = \frac{1200 \times 15^2}{30} = 9000 \text{ N}

Answer. The required centripetal force is 9000 N, provided by friction between the tyres and the road.

Figure
Diagram
CentreObjectr = 30 mv (velocity, tangential)F_c (inward)Centripetal force always points toward the centre;velocity is always tangential (perpendicular to radius).
Origins · 1 minChristiaan Huygens and how this was discovered
CH
Christiaan Huygens
1629–1695 · Dutch

Huygens was the first to derive the correct mathematical formula for centripetal acceleration, publishing his results in 'Horologium Oscillatorium' (1673) and 'De Vi Centrifuga' (written 1659, published posthumously). He showed that the inward acceleration for circular motion is v²/r, predating Newton's more general treatment in the Principia. His work on circular motion was essential for Newton's later development of universal gravitation. Huygens also invented the pendulum clock, discovered Saturn's moon Titan, and correctly described the rings of Saturn — all before the age of 30.

  • Newton's Second Law — Centripetal force is derived directly by applying F = ma to circular motion; understanding the second law is essential before tackling centripetal force.
  • Centripetal Acceleration — Centripetal force and centripetal acceleration are linked by F = ma; understanding that a = v²/r points inward underpins the force equation.
  • Gravitational Force & Orbital Motion — In orbital mechanics, gravity acts as the centripetal force; this concept directly applies centripetal force to planetary and satellite orbits.
  • Angular Velocity & Period — The alternative form F_c = mω²r uses angular velocity; understanding ω and how it relates to period T is needed to solve many circular motion problems.