Circular Motion

There are many examples of bodies moving in circular paths – rides at a fun fair, clothes being spun dry in a washing machine, the planets going round the Sun and the Moon circling the Earth. When a car turns a corner it may follow an arc of a circle. ‘Throwing the hammer’ is a sport practiced at Highland Games in Scotland, Figure 28.1, in which the hammer is whirled round and round before it is released

Centripetal Force

In Figure 28.2 a ball attached to a string is being whirled round in a horizontal circle. Its direction of motion is constantly changing. At A it is along the tangent at A; shortly afterwards, at B, it is along the tangent at B; and so on.

Velocity has both size and direction speed has only size. Velocity is speed in a stated direction and if the direction of a moving body changes, even if its speed does not, then its velocity has changed. A change of velocity is an acceleration and so during its whirling motion the ball is accelerating.

It follows from Newton’s first law of motion that if we consider a body moving in a circle to be accelerating then there must be a force acting on it to cause the acceleration. In the case of the whirling ball it is reasonable to say the force is provided by the string pulling inwards on the ball. Like the acceleration, the force acts towards the centre of the circle and keeps the body at a fixed distance from the centre.

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A larger force is needed if:(i) The speed v of the ball is increased,(ii) The radius r of the circle is decreased,(iii) The mass m of the ball is increased.

The rate of change of direction, and so the acceleration a, is increased by (i) and (ii). It can be shown that a=v2/r and so, from F=m/a, we can write

F=m v2

r

The force, which acts towards the centre and keeps a body moving in a circular path, is called the centripetal force (centre-seeking force).

Should the force be greater than the string can bear, the string breaks and the ball flies off with steady speed in a straight line along the tangent, i.e. in the direction of travel when the string broke (as Newton’s first law of motion predicts). It is not thrown outwards.

Whenever a body moves in a circle (or circular arc) there must be a centripetal force acting on it. In throwing the hammer it is the pull of the athlete’s arms acting on the hammer towards the centre of the whirling path. When a car rounds a bend a frictional force is exerted inwards by the road on the car’s tyres.

Practical Works

A. Investigating Circular Motion

Use the apparatus in Figure 28.3 to investigate the various factors which affect circular motion. Make sure the rubber bung is tied securely to the string and that the area around you is clear of other students. The paper clip acts as an indicator to aid keeping the radius of the circular motion constant.

Spin the rubber bung at the constant speed while adding more weights to the holder; it will be found that the radius of the orbit decreases. Show that if the rubber bung is spun faster, more weights must be added to the holder to keep the radius constant. Are the findings in agreement with the formula given above for the centripetal force F=mv2/r?

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B. Rounding A Bend

When a car rounds a bend a frictional force exerted inwards by the road on the car’s tyres, so providing the centripetal force needed to keep it in its curved path, Figure 28.4a. Here friction acts as an accelerating force (towards the centre of the circle) rather than a retarding force (p.82, p.112). The successful negotiation af a bend on a flat road therefore depends on the tyres and the road surface being in a condition that enables them to provide a sufficiently large frictional force – otherwise skidding occurs.

Safe cornering that does not rely entirely on friction is achieved by ‘banking’ the road as in Figure 28.4b. Some of the centripetal force is then supplied by the part of the contact force N from the road surface on the car that acts horizontally. A bend in a railway track is banked so that the outer rail is not strained by having to supply the centripetal force, by pashing inwards on the wheel flanges.

C. Looping the Loop

A pilot who is not strapped into his aircraft can loop the loop without falling downwards at the top of the loop. A bucket of water can be swung round in a vertical circle without spilling. Some amusement park rides, Figure 28.5, give similar effects. Can you suggest what provides the centripetal force for each of the above three cases (i) at the top of the loop and (ii) at the bottom of the loop?

D. Satellites

For a satellite of mass m orbiting the Earth at radius r with orbital speed v, the centripetal force F=mv2/r is provided by gravity.

To put an artificial satellite in orbit at a certain height above the Earth it must enter the orbit at the correct speed. If it does not, the force of gravity, which decreases with height, will not be equal to the centripetal force needed for the orbit.

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This can be seen by imagining a shell fired horizontally from the top of a very high mountain, Figure 28.6. If gravity did not pull it towards the centre of the Earth it would continue to travel horizontally, taking path A. In practice it might take path B. A second shell fired faster might take path C and travel farther. If a third shell is fired even faster, it might never catch up with the rate at which the Earth’s surface is falling away. It would remain at the same height above the Earth (path D) and return to the mountain top, behaving like a satellite.

The orbital period T (the time for one orbit) of a satellite = distance/velocity. So far a circular orbit

T=2 πrv

Satellites in high orbits have longer periods than those in low orbits.The moon is kept in a circular orbit round the Earth by the force of gravity between it

and the Earth. It has an orbital period of 27 days.

a) Communication Satellites

These circle the Earth in orbits above the equator. Geostationary satellites have an orbit high above the equator (36.000 km); they travel with the same speed as the Earth rotates, so appear to be stationary at a particular point above the Earth’s surface – their orbital periods is 24 hours. They are used for transmitting television, intercontinental telephone and data signals. Geostationary satellites need to be well separated so that they do not interfere with each other; there is room for about 400.

Mobile phone networks use many satellites in much lower equatorial orbits; they are slowed by the Earth’s atmosphere and their orbit has to be regularly adjusted by firing a rocket engine. Eventually they run out of fuel and burn up in the atmosphere as they fall to Earth.

b) Monitoring Satellites

These circle the Earth rapidly in low polar orbits, i.e. passing over both poles; at a height of 850 km the orbital period is only 100 minutes. The Earth rotates below them so they scan the whole surface at short range in a 24 hour period and can be used to map or monitor regions of the Earth’s surface which may be inaccessible by other means. They are

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widely used in weather forecasting to transmit infrared pictures of cloud patterns continuously down to Earth, Figure 28.7, which are picked up in turn by receiving stations around the world.

c) Orbital Relationships

For a geostationary satellite at height 36.000 km above the Earth’s surface, r = (36000 + 6000) km = 42000 km = 42 × 106 m (taking the Earth’s radius as 6000 km) and T = 24 hours = 24 × 3600 s. Hence the orbital velocity v is

v=2 πrT

=2 π × 42×106 m24 × 3600 s

If F is the gravitational force per kg to keep the satellite in orbit, then

F=m v2

r= 1.0 kg

42 ×106 ×(3.1× 103)2 m2

s2 =9.642

kgm /s2=0.23 N

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Question

1. An apple is whirled round in a horizontal circle on the end of a string which is tied to the stalk. It is whirled faster and faster and at a certain speed the apple is torn from the stalk. Why?

2. A car rounding a bend travels in an arc of a circle.a. What provides the centripetal force?b. Is a larger or smaller centripetal force required if

(i) The car travels faster,(ii) The bend is less curved,(iii) The car has more passengers?

3. Racing cars are fitted with tyres called ‘slicks’ (which have no tread pattern) for dry tracks and ‘treads’ for wet tracks. Why?

4. A satellite close to the Earth (at a height of about 200 km) has an orbital speed of 8 km/s. Taking the radius of the orbit as approximately equal to the Earth’s radius of 6400 km, calculate the time it takes to make one orbit

5. Figure 28.8 shows a model railway track.

Figure 28.8The power is switched on and the locomotive starts at A. It reaches its maximum speed at B, then movies at a constant speed to D. At D, the power is switched off and the locomotive continues until it hits the spring-loaded buffers at E, where it rebounds.a. Although the locomotive travels around the circular track at a constant speed from B

to C, there is a constant net force acting on the locomotive.(i) Sketch the diagram of the track and draw an arrow to show the direction of this

force, when the locomotive is at P.(ii) State where this force is applied to the locomotive and how it is produced.(iii) The mass of the locomotive is 0.40 kg and the constant force acting on it is

0.032 N. Calculate the acceleration of the locomotive.(iv) Explain why the locomotive is accelerating, even though its speed is constant.

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b. Between D and E the speed of the locomotive, of mass 0.40 kg, decreases from 0.20 m/s to 0.15 m/s. It takes 0.30 s to travel from D to E.(i) For this locomotive travelling from D to E, calculate

1. The change in momentum,2. The average force acting,3. The average power loss.

(ii) Explain why the locomotive rebounds from the used.

c. Figure 28.9 shows the design of the spring buffers used.

Figure 28.9The limit of proportionally of the springs that used was found to be too low.(i) With the aid of a labeled sketch graph, explain what limit of proportionally

means.(ii) In a laboratory test to find a replacement spring, that has a suitable limit of

proportionally, describe how you would1. Apply the force and accurately measure its size,2. Work out the value of the limit of proportionally.

(UCLES IGCSE Physics Supplementary, May 1997)

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Motion and Energy

Additional Questions

Speed, velocity and acceleration; graphs and equations1. The approximate velocity-time graph for a car on a 5-hour journey is shown below.

(There is a very quick driver change midway to prevent driving fatigue!)a. State in which of the regions OA, AB, BC, CD, DE the car is (i) accelerating, (ii)

decelerating, (iii) traveling with uniform velocity.b. Calculate the value of the acceleration, deceleration or constant velocity in each

region.c. What is the distance travelled over each region?d. What is the total distance travelled?e. Calculate the average velocity for the whole journey.

2. The distance-time graph for a motor cyclist riding off from rest is shown below.a. Describe the motion.b. How far does the motorbike move in 30 seconds?c. Calculate the speed.

3. The speeds of a car travelling on a straight road are given below at successive intervals of 1 second.

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The car travels

1 with an average velocity of 4 m/s2 16 m in 4 s3 with a uniform acceleration of 2 m/s2

Which statement(s) is (are) correct?A 1, 2, 3 C 2, 3 E 3B 1, 2 D 1

4. If a train travelling at 10 m/s starts to accelerate at 1 m/s2 for 15 s on a straight track, its final velocity in m/s isA 5 B 10 C 15 D 20 E 25

5. Two men, Mr. A and Mr. B, live in the same apartment block Hometown, and both work at the same factory in Worktown.

HINT: For this question, work in km and hours.

a. Mr. A travels to work by car, but it is a poor road and he can only average a speed of 50 km/hour. The distance by road is 30 km. How long does Mr. A take to get to work?

b. Mr. B leaves home at the same time as Mr. A, but he1) Walks the 0.5 km to Hometown station,2) Waits 6 minutes (0.1 hour) for the train,3) Travels on the train to Worktown station (journey distance 30 km),4) Walks the 1.0 km from Worktown station to the factory.The train averages 100 km/hour and Mr. B walks at 5 km/.hour. Does Mr. B arrive at work before or after Mr. A, and by how much?

c. Apart from a short time accelerating out of Hometown station and a short time slowing down whilst approaching Worktown station, the train travels at a constant speed.

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On axes like those below, sketch a speed-time graph for Mr. B from home to factory. On your graph, clearly label the four stages of the journey.

(UCLES IGCSE Physics Core, Nov 1998)

6. A firework leaves the ground with an initial velocity of 45 m/s, travelling vertically upwards. It reaches a maximum height of 100 m.At this point the firework fails to explode and falls back down the same vertical path to the ground.At any point on its path, the firework has both a velocity and a speed.a. Using the terms vector and scalar, explain the difference between velocity and speed.b. The graph shows the height of the firework above the ground during the first 5 s of its

journey.

(i) Use the information on the graph to1) Find the time taken for the firework to reach its maximum height above

the ground,2) Describe how the motion of the firework changes over the first 5 s of its

journey.(ii) The acceleration of free fall is 10 m/s2 and air resistance on the firework is

negligible. State1) The deceleration of the firework as it is rising,2) The total time taken for the firework to rise 100 m and then to fall back to

the ground.

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(iii) State the velocity with which the falling firework hits the ground.(UCLES IGCSE Physics Supplementary, May/June 2000)

7. A small pebble is thrown horizontally at 20 m/s from the top of a cliff 80 m high. Ignoring air resistance and taking g = 10 m/s2, calculatea. How long the pebble takes to reach the ground,b. The distance from the foot of the cliff to where the pebble strikes the ground,c. The vertical velocity of the pebble as it hits the ground, andd. The horizontal velocity of the pebble as it hits the ground.

Force and acceleration8. a. Two skydivers jump from a plane. Each holds a different position in the air

Complete the following sentenceSkydiver … will fall faster because …The diagram below shows the direction of the forces acting on one of the skydivers.

b. In the following sentences, two lines in each box are wrong. Complete each sentence using the correct line.(i) Force X is caused by

(ii) Force Y is caused by

(iii) When force X is bigger than force Y, the speed of the skydiver will

(iv) After the parachute opens, force X

c. How does the area of an opened parachute affect the size of force Y?

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9. The graph shows the speed of a small very dense object which is falling vertically from an aeroplane, up to the point at which it hits the ground. The air resistance on the object is negligibly small for the first 5 s of its fall. The object is fitted with a parachute which springs open after a certain time of fall.

a. State the type of motion(i) Between 0 and 5 s,(ii) Between 42 s and 47 s.

b. Estimate the time at which the parachute opens.c. On copies of the diagrams below, indicate by labeled arrows the vertical forces acting

on the falling object(i) After 3 s of fall,

(ii) After 45 s of fall.

d. State whether of not there is a resultant vertical force acting on the falling object(i) After 3 s of fall,(ii) After 45 s of fall.

e. Calculate the distance fallen in the first 5 s of fall.(UCLES IGCSE Physics Supplementary, May 2001)

Momentum; kinetic and potential energy10. The speed of the rocket shown, of mass

20.000 kg, has to be increase by 5.0 m/s. If it burns 10 kg of fuel and oxygen in doing this, what is the speed of the exhaust gases produced?

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11. A car of mass 1000 kg has a head-on collision in a 50 km/h zone with a truck of mass 2500 kg, in which both vehicles stop dead. If the truck’s speed, as detected by police radar, was 30 km/h, what was the car’s speed and was the law being broken? (Neglect the momentum of the truck’s load and of the passengers after the collision.)

12. a. The diagram shows two laboratory trolleys which collide and stick together.

(i) What is the momentum of trolley D before the collision? Give a reason for your answer.

(ii) In the sentence below, two lines in the box are not correct. Complete the sentence using the correct line.

The momentum of trolley D before the collision is its momentum after the collision.

(iii) Copy the second diagram above and draw an arrow to show the direction in which both C and D move after the collision.

b. In each of the pictures below decide if the labeled object gains or loses momentum.(i) A sprinter just leaving the starting block.

(ii) A cricket ball just hitting a cricket stump.

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(iii) A cannon and cannon ball, just after the cannon has been fired.

(SEG Foundation, Summer 1998)

13. A car’s speed was recorded at intervals as it travelled along a straight, level track. See the table below.

a. Plot a speed-time graph for the car’s motion.b. (i) Describe the motion of the car

(ii) Use the graph to calculate the distance the car travelled. Show your working.c. After 8.0 s, the brakes were applied. Choose words to complete the description below of the

energy change that took place after the brakes were applied.……… energy → ……… energy

(UCLES IGCSE Physics Core, Nov 1999)

14. The diagram shows the outline of a machine for driving steel pillars (called piles) into the grounds.

The steel mass is raised by an electric motor and then falls under gravity. The falling steel has a mass of 200 kg and falls a distance of 6.0 m.a. The acceleration of free fall is 10 m/s2. Calculate

(i) The potential energy gained by the mass each time it is raised,(ii) The maximum speed at which the mass hits the pile.

b. When the mass hits the pile, it has kinetic energy. This energy is transformed into other forms of energy as the speed of the falling mass rapidly reduces to zero. As this happens, the pile is forced a small distance into the ground.

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(i) State the energy conversions which take place, starting from the kinetic energy of the falling mass.

(ii) Explain how a large force is produced when the pile is driven a short distance into the ground

c. In raising the steel mass 6.0 m, the electric motor uses more energy than that calculated in a (i). Write down and explain two causes of this higher energy requirement.

d. The equipment design is changed so that when the mass falls once, the pile is driven further into the ground than before the design was changed. Suggest three changes that could be made to do this.

(UCLES IGCSE Physics Supplementary, Nov 1999)

15. The data in the table below shows how the stopping distance of a car depends on its speed.

a. Write down two factors, apart from speed, that affect stopping distance of a car.b. Use d=the data to draw a graph of stopping distance against speed.c. The speed limit in a supermarket car park is 7.5 m/s. Use your graph to estimate the

stopping distance of a car travelling at this speed.d. Describe how the stopping distance changes as the speed of a car increases.e. Explain why the stopping distance is reduced if a driver made aware of possible

hazards.f. The speed limit on roads in towns is 15 m/s. Some road safety campaigners are asking

the government to change this to 10 m/s. Suggest why this may be a good idea.(London Higher, June 1999)

16. The diagram shows a moving car on a level road.

a. Calculate the momentum of the car.b. The brakes of the car are applied for 4 s, which reduces the speed of the car to 5 m/s.

(i) Calculate the average force of the brakes.(ii) Calculate the average deceleration of the car.

(UCLES IGCSE Physics Supplementary, May 2001)

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