Describe uniform circular motion using centripetal acceleration and force, relating them to speed, radius and period.

What this dot point is asking

You need to explain why circular motion is accelerated even at constant speed, identify the centripetal force in a given situation, and calculate centripetal acceleration and force using speed, radius and period.

Why constant speed still means accelerating

Velocity is a vector. In circular motion the speed (the size of the velocity) is constant, but the direction is continuously changing, always tangent to the circle. A changing velocity is an acceleration, so the object is accelerating even though it is not speeding up or slowing down.

This acceleration points toward the centre of the circle. It is called centripetal ("centre-seeking") acceleration.

Period, frequency and speed

For an object completing circles, the period $T$ is the time for one revolution and the frequency $f = 1/T$. The speed is the circumference divided by the period:

Substituting into $a = v^2/r$ gives an equivalent form in terms of period:

Identifying the centripetal force

The centripetal force is whatever real force (or combination) provides the inward net force:

• A ball on a string: the tension points inward.
• A car turning on a flat road: friction between tyres and road.
• A satellite orbiting Earth: gravity.
• A passenger on a fairground ride: the normal force from the seat or wall.

If the required inward force is not available (e.g. not enough friction), the object cannot follow the circle and moves off tangentially in a straight line.

Banked tracks

On a banked curve the road is tilted so the normal force gains a horizontal component pointing toward the centre. This horizontal component supplies part or all of the centripetal force, reducing the reliance on friction. At the design speed, the horizontal component of the normal force alone supplies the entire $mv^2/r$, so the curve can be taken safely even on a frictionless (icy) surface.

The "centrifugal" misconception

There is no outward force on the object. In the rotating frame of a passenger it feels as if they are pushed outward, but in an inertial frame the only forces are real and the net force is inward. What the passenger feels is their own inertia (tendency to go straight) resisting the inward push of the seat.

How SACE assesses this

SACE Stage 2 circular-motion questions usually give two of speed, radius and period and ask for the centripetal acceleration or force, often in a real context such as a turbine blade, a fairground ride or a car on a bend. The dependable approach is to identify which real force provides the centripetal force (friction, tension, gravity or the normal force), then equate it to $\dfrac{mv^2}{r}$. Where the period is involved, convert between $v$ and $T$ with $v = \dfrac{2\pi r}{T}$ first. Several "show that" parts ask you to verify a period or acceleration, so quote the formula, substitute with units, and arrive visibly at the printed value rather than just stating it.