Centripetal Acceleration

When an object moves in a circle at constant speed, it is tempting to say it is not accelerating — after all, the speedometer reading never changes. However, acceleration is the rate of change of velocity, and velocity is a vector: it has both magnitude and direction. Because the direction of motion changes continuously around the circular path, the velocity vector changes continuously too. This means there must be a net acceleration acting on the object at every instant, even though its speed stays the same. This acceleration always points toward the centre of the circle — it is called centripetal acceleration, from the Latin centrum (centre) and petere (to seek). Think of a ball on a string being swung overhead: the string constantly pulls the ball inward, preventing it from flying off in a straight line. Remove the string and the ball immediately travels in a straight line — Newton's first law in action. The string's pull is what continuously redirects the velocity, and the resulting acceleration is centripetal acceleration. Centripetal acceleration is not a new type of force; it is simply the name given to the inward-directed acceleration that any net centripetal force produces. Its magnitude depends on the object's speed and the radius of the circular path, and it always acts perpendicular to the velocity — changing direction but never speed.

Key takeaways
  • Velocity is always tangential
  • Change in velocity points inward
  • Deriving the magnitude
  • Perpendicular to velocity — no change in speed
  • Newton's second law connects force and acceleration

Velocity is always tangential

At every point on a circular path, the velocity vector points along the tangent to the circle — at 90° to the radius. Picture a car on a roundabout: the car always 'wants' to go straight ahead, but the road curves it inward. The direction of 'straight ahead' rotates continuously, which means the velocity vector rotates at the same rate as the object moves around the circle.

Change in velocity points inward

Consider two closely spaced positions on the circle, A and B. The velocity vectors v_A and v_B have the same length (same speed) but slightly different directions. The vector difference Δv = v_B − v_A, when drawn, points toward the centre of the circle. As the time interval Δt → 0, this becomes the instantaneous acceleration vector, always directed radially inward. This is a geometric result — it follows purely from the fact that the velocity vector rotates at the same angular rate as the position vector.

Deriving the magnitude

Using similar triangles (or calculus), one can show that the magnitude of centripetal acceleration is a = v²/r, where v is the linear speed and r is the radius. Equivalently, since v = rω (where ω is the angular velocity in rad/s), we can write a = ω²r. Both forms are useful: the first when you know linear speed, the second when you know how fast the object is rotating. The SI unit of centripetal acceleration is m/s², identical to any other acceleration.

Perpendicular to velocity — no change in speed

A crucial point: centripetal acceleration is always perpendicular to the instantaneous velocity. Work done by a force equals force · displacement, and power equals force · velocity. Because the centripetal force is perpendicular to the velocity at every instant, it does zero work on the object and transfers no kinetic energy. This is why speed remains constant even though the object is accelerating — a beautifully consistent result confirmed by both geometry and energy considerations.

Newton's second law connects force and acceleration

By Newton's second law, F = ma, so the net force required to maintain circular motion is F_c = mv²/r. This centripetal force is not a separate 'type' of force — it is whatever real force (gravity, tension, friction, normal reaction) acts toward the centre. For a satellite, it is gravity; for a car on a bend, it is friction; for the ball on the string, it is tension. If that inward force disappears, the object moves off in a straight line tangent to the circle.

The Key Equation

ac=v2r=ω2ra_c = \frac{v^2}{r} = \omega^2 r
  • a_c — Centripetal acceleration (m s⁻²): The magnitude of the inward-directed acceleration; always points toward the centre of the circular path.
  • v — Linear (tangential) speed (m s⁻¹): The constant speed of the object along the circumference of the circle.
  • r — Radius (m): The radius of the circular path — the distance from the object to the centre.
  • \omega — Angular velocity (rad s⁻¹): The rate at which the object sweeps out angle; related to linear speed by v = ωr.

Worked Example

Problem. A car travels around a roundabout of radius 25 m at a constant speed of 10 m/s. Calculate (a) the centripetal acceleration of the car, and (b) the minimum friction force required to maintain this motion if the car has a mass of 1200 kg.

ac=4 m s−2,Fc=4800 Na_c = 4 \text{ m s}^{-2}, \quad F_c = 4800 \text{ N}

Answer. Centripetal acceleration = 4 m/s²; minimum friction force = 4800 N

Figure
Diagram
O (centre)objectrv (tangential)aₙ (centripetal)vaₙtop of circleAt every point: v is tangential (blue), aₙ points toward centre O (red)
Origins · 1 minChristiaan Huygens and how this was discovered
CH
Christiaan Huygens
1629–1695 · Dutch

Huygens was the first to derive the correct formula for centripetal acceleration, publishing the result a = v²/r in his landmark work Horologium Oscillatorium (1673). Working from careful geometric arguments about circular motion, he established that the inward acceleration depends on the square of the speed and inversely on the radius — a result that Newton later incorporated into his own dynamics. Huygens also invented the pendulum clock and made major contributions to optics and the wave theory of light. Huygens derived his centripetal acceleration formula roughly 15 years before Newton published the Principia, yet the two men corresponded and influenced each other's thinking on the laws of motion. Huygens initially kept the result secret, encoding it as a Latin anagram in a letter — a common practice among 17th-century scientists to establish priority without revealing the discovery.

  • Uniform Circular Motion — Centripetal acceleration is the defining characteristic of uniform circular motion; understanding the kinematics of circular paths is the essential foundation.
  • Centripetal Force — By Newton's second law, centripetal acceleration is produced by a centripetal force F = mv²/r. Understanding the link between these two concepts is essential for solving circular motion problems.
  • Angular Velocity — Angular velocity ω is directly related to centripetal acceleration via a = ω²r, and connects the rotational description of circular motion to its linear description.
  • Newton's Second Law — Newton's second law F = ma is used to relate centripetal acceleration to the forces acting on an object in circular motion, such as gravity, tension, or friction.