Why circular motion is accelerated

Problem

Explain why a body in uniform circular motion is subject to a non-zero acceleration, even though the magnitude of its velocity stays constant. Which real force (not fictitious!) causes it, in the case of a car cornering on a flat road?

Solution

Velocity is a vector, not a number. Acceleration is defined as the rate of change of velocity, $\vec{a} = \dd\vec{v}/\dd t$. Velocity $\vec{v}$ has magnitude and direction: it is enough for just one of these two features to change for there to be acceleration.

In uniform circular motion the magnitude $\abs{\vec{v}}$ stays constant, but the direction of the velocity vector (always tangent to the circle) changes at every instant. So $\vec{v}$ changes, and consequently $\vec{a} = \frac{\dd\vec{v}}{\dd t} \neq \vec{0}$ This acceleration has no component along the trajectory (the magnitude of $v$ does not change): it is entirely directed towards the centre of the circle. It is the centripetal acceleration, of magnitude $a_c = v^2/r$.

It is not a paradox: “accelerating” in the physical sense does not only mean “going faster”, but also “changing direction”. It is the same reason the Moon, which orbits at an almost constant speed, “falls” continually towards the Earth without ever moving away from it.

The real force on the cornering car. Every acceleration requires a real force (Newton’s second law: $\vec{F} = m\vec{a}$). For a car travelling round a flat bend, the force pointing towards the centre — i.e. the centripetal force — is provided by static friction between the tyres and the road. It is static (not kinetic) friction because the tyres, if they do not skid, do not slide over the road surface at the point of contact: they roll. If the available static friction ($F_{a,\max} = \mu_s\,mg$) is not enough to provide the required centripetal force, the car slides outwards and “runs off the road”.

Result. $\ev{\text{The direction of }\vec{v}\text{ changes}\;\Rightarrow\;\vec{a}\neq 0\text{ (centripetal); on a flat bend it is provided by static friction.}}$

Links

Topics: Dynamics · Friction Concepts: Uniform circular motion · Centripetal acceleration · Centripetal force · Static friction