Uniform Circular Motion

Lessons

Senior 3 · Physics

Uniform Circular Motion

4 lessons

Lesson 4 of 4

Centrifugal Effects and Applications of Circular Motion

25 min

Thrown outwards?

On the winding road between Kigali and Musanze, a minibus swings round a sharp bend to the left. The passengers slide to the right, towards the outside of the bend, and one of them says she was "thrown outwards". Many people call this push the centrifugal force. Is there really a force pushing passengers away from the centre of the bend? This lesson answers that question, then looks at where circular motion is used and where it causes accidents.

Experiment 4: coins on a turning disc

You need: a disc of stiff card or hardboard about 35 cm across, a drawing pin or small nail, three identical coins, and a ruler.

Pin the disc through its centre to a wooden table so that it can turn freely.
Put the coins on one radius, 5 cm, 10 cm and 15 cm from the centre.
Turn the disc by hand, slowly at first, then a little faster each turn.
Note which coin slides first, and in which direction it leaves.

Figure 4.1: Friction from the disc is the centripetal force on each coin. The coin farthest out needs the most friction, so it slides first, and it leaves along the tangent.

Observation: the 15 cm coin slides off first, then the 10 cm coin. Each coin leaves along the tangent, not straight out along the radius.

Explanation: all three coins share the same $\omega$ , so each needs a friction force $F = mr\omega^2$ towards the centre. Friction between a coin and the card cannot be larger than a certain value. The needed force is largest for the largest $r$ , so the outer coin reaches the limit first. Once friction is too small, nothing keeps the coin on the circle, and it moves off in a straight line.

Important

Safety. Turn the disc by hand only, never with a motor or drill. Keep your face above and away from the disc so that a flying coin cannot hit your eyes, and do not point the disc edge towards other pupils.

Explanation: inertia, not an outward force

Figure 4.2: Seen from the roadside, the passenger is not pushed outwards. Her body tends to keep moving in a straight line (Newton's first law) while the bus turns under her. The outer side of the bus then pushes her inwards, and that push is her centripetal force.

Seen from the road, only one horizontal force acts on the passenger: the inward push of the seat or the side of the bus. There is no outward force. Inside the bus, though, the passenger turns with the bus, so it feels as if something pushes her outwards. This apparent outward force, felt only by someone who is turning, is called the centrifugal force. It is the effect of inertia, not a real push from any object.

	Centripetal force	Centrifugal force
Direction	towards the centre	away from the centre
Is it real?	yes: a real push or pull (tension, friction, gravity, normal reaction)	no: an apparent force, felt only by an observer who turns with the body
What causes it	another object: a string, the road, the Earth, a wall	the body's inertia, its tendency to go straight on
What it does	changes the direction of motion	nothing seen from the ground: it is only felt

Definition

Centrifugal force: the apparent outward force felt by a body that turns in a circle, when it is described by an observer turning with it. It is not a force exerted by any object; it is the effect of the body's inertia. Example: the "push" that presses you against the side of a turning bus.

A frame of reference is the point of view from which motion is measured: for example, the roadside, or the inside of the moving bus.

Applications of circular motion

Roundabouts and bends. On a flat road, friction between the tyres and the road is the only centripetal force. It has a greatest possible value, which is smaller on a wet road, on sand or on gravel. Since the force needed is $\frac{mv^2}{r}$ , there is a maximum safe speed for each bend. Above it, the vehicle skids off along the tangent, towards the outside of the bend.
Leaning and banking. A cyclist or a moto rider leans into a bend, and racetracks are tilted (banked) so that the push of the road on the tyres has a part pointing towards the centre. Both reduce the friction needed.
The centrifuge. In a health-centre laboratory, a centrifuge spins tubes of blood very fast. The heavier red cells need more centripetal force to stay on their circle than the liquid can supply, so they move to the bottom of the tube and separate from the plasma. The tubes must be placed opposite one another with equal masses, or the spinning rotor shakes.
Spin dryers and honey extractors. The drum wall pushes the clothes round, but water in the fabric is not held by anything at the holes, so it leaves along tangents through them. A beekeeper's honey extractor spins frames of opened honeycomb so that the honey flies out in the same way.
Satellites and the Moon. Gravity is the centripetal force that keeps the Moon and artificial satellites moving round the Earth. You will study orbits in detail at A-level.

Worked example 1: the maximum safe speed on a roundabout

A moto and its rider have a total mass of 180 kg. On a roundabout of radius 15 m, the greatest sideways friction the tyres can get is 1200 N on a dry road and 600 N on a wet road (illustrative values). Find the maximum safe speed in each case.

The friction must supply $\frac{mv^2}{r}$ , so $v_{\max} = \sqrt{\frac{F_{\max}\,r}{m}}$ .

\text{Dry: } v_{\max} = \sqrt{\frac{1200 \times 15}{180}} = \sqrt{100} = 10\ \mathrm{m\,s^{-1}} = 36\ \mathrm{km\,h^{-1}}

\text{Wet: } v_{\max} = \sqrt{\frac{600 \times 15}{180}} = \sqrt{50} = 7.1\ \mathrm{m\,s^{-1}} \approx 25\ \mathrm{km\,h^{-1}}

Halving the grip does not halve the safe speed; it divides it by $\sqrt{2}$ . Even so, a rider who keeps a dry-road speed on a wet road will skid.

Worked example 2: how hard does a centrifuge pull?

A laboratory centrifuge holds its tubes 0.10 m from the axis and spins at 3000 rev/min. Find the centripetal acceleration and compare it with $g = 9.8\ \mathrm{m\,s^{-2}}$ .

\omega = 2\pi \times \frac{3000}{60} = 314\ \mathrm{rad\,s^{-1}}

a = r\omega^2 = 0.10 \times 314.2^2 = 9.9 \times 10^{3}\ \mathrm{m\,s^{-2}}

\frac{a}{g} = \frac{9870}{9.8} \approx 1000

The contents of the tube behave as if gravity were about 1000 times stronger, so cells settle in minutes instead of hours.

Warning

Things fly off along the tangent, not straight outwards. A car that skids on a bend, mud leaving a tyre and a coin leaving a disc all go straight on from the point where the grip failed. And never write "the centrifugal force balances the centripetal force": seen from the ground there is no outward force, and the centripetal force is unbalanced. That is why the body accelerates.

Tip

Ask "who is watching?" From the roadside: one real force, inwards (centripetal). From inside the turning bus: it feels like an outward push (centrifugal). Exam answers should use the roadside view.

Try it: in PhET: Gravity and Orbits (version 1.6.17), choose the Earth and Moon, tick the gravity force and velocity arrows, and watch: the force always points to the Earth, at right angles to the velocity. Then switch gravity off and see the Moon fly off along a straight line.

Wait, Centrifugal Force Isn't Real?! (NOVA PBS, What the Physics?!). Watch for how the same motion looks from inside a turning object and from outside it.

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Practice

A bus turns sharply to the left. Explain, using Newton's first law, why the passengers slide to the right.
Give two differences between centripetal force and centrifugal force.
A 1000 kg car goes round a flat bend of radius 30 m. The greatest friction force the road can give is 3000 N. Find the maximum speed at which the car can take the bend.
Explain why a moto rider leans into a bend, and why some bends and racetracks are banked.
A centrifuge turns at 4000 rev/min with its tubes 8.0 cm from the axis. Find the centripetal acceleration, and how many times $g$ this is ( $g = 9.8\ \mathrm{m\,s^{-2}}$ ).
Why must the tubes in a centrifuge be placed opposite one another with equal masses?
Explain how a spin dryer removes water from wet clothes.
On a turning disc, a coin 12 cm from the centre starts to slide when $\omega = 5.0\ \mathrm{rad\,s^{-1}}$ . At what angular velocity will an identical coin 3.0 cm from the centre slide?

Show answersHide answers

The passengers tend to keep moving in a straight line (inertia) while the bus turns left beneath them, so relative to the bus they move towards the right, the outside of the bend, until a seat or the side of the bus pushes them round with it.
Centripetal force is real and points to the centre; centrifugal force is apparent, points away from the centre and is felt only by an observer turning with the body. Centripetal force is exerted by another object; centrifugal "force" is the effect of inertia.
$v_{\max} = \sqrt{\frac{3000 \times 30}{1000}} = \sqrt{90} = 9.5\ \mathrm{m\,s^{-1}}$ (about $34\ \mathrm{km\,h^{-1}}$ ).
Leaning, or a banked road, tilts the push of the road on the tyres so that part of it points towards the centre of the bend. That part helps to provide the centripetal force, so less friction is needed and skidding is less likely.
$\omega = 2\pi \times \frac{4000}{60} = 419\ \mathrm{rad\,s^{-1}}$ ; $a = 0.080 \times 419^2 = 1.4 \times 10^{4}\ \mathrm{m\,s^{-2}}$ , about 1400 times $g$ .
If one side is heavier, the rotor's axle must supply a large, constantly changing sideways force to keep that side's mass on its circle. The machine shakes and can be damaged or tip over.
The drum wall pushes the clothes towards the centre, keeping them moving in a circle. Water at the holes has nothing pushing it inwards, so it moves off along the tangent, out through the holes.
Maximum friction is the same, so $r\omega^2$ is the same: $\omega^2 = \frac{0.12 \times 5.0^2}{0.030} = 100$ , so $\omega = 10\ \mathrm{rad\,s^{-1}}$ .

Key points

Centrifugal force is an apparent outward force felt in a turning frame; it is the effect of inertia, not a real push.
Seen from the ground, a turning body has only an unbalanced inward (centripetal) force on it.
On a flat bend, friction limits the safe speed: $v_{\max} = \sqrt{\frac{F_{\max}\,r}{m}}$ . Wet roads lower it.
Centrifuges, spin dryers and honey extractors separate materials because the parts not held on the circle move off along tangents.

1Thrown outwards?

2Experiment 4: coins on a turning disc

3Explanation: inertia, not an outward force

4Applications of circular motion

Thrown outwards?

Experiment 4: coins on a turning disc

Explanation: inertia, not an outward force

Applications of circular motion