Topic 2: Circular motion and gravity

Name the force that supplies the centripetal force first, then set the net force towards the centre equal to mv²/r.

Short response and problem-solving questions

A banked velodrome curveShort response · 6 marks

A curve on a velodrome track has a radius of 25.0 m and is banked at 40.0° to the horizontal. A cyclist and bicycle have a combined mass of 80.0 kg. Use g = 9.80 m s⁻².

(a) Calculate the speed at which the cyclist can ride around the curve with no sideways friction on the tyres. [2]

m s⁻¹

(b) Calculate the size of the normal force on the cyclist and bicycle at this speed. [2]

0 words · a sentence or two

A roller coaster loopProblem-solving · 8 marks

A roller coaster car starts from rest at the top of a slope 32.0 m above the bottom of a vertical loop of radius 9.00 m. A 58.0 kg rider sits in the car. Treat the track as frictionless and use g = 9.80 m s⁻².

(a) Calculate the speed of the car at the top of the loop. [3]

m s⁻¹

(b) Calculate the size of the force the seat exerts on the rider at the top of the loop. [3]

0 words · a sentence or two

An Earth observation satelliteProblem-solving · 7 marks

An Earth observation satellite moves in a circular orbit 6.50 × 10⁵ m above Earth's surface. A crewed spacecraft follows the same orbit. Use G = 6.67 × 10⁻¹¹ N m² kg⁻²; Earth's mass is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m.

(a) Calculate the gravitational field strength at the height of the orbit. [2]

N kg⁻¹

(b) Calculate the orbital period of the satellite. [3]

0 words · a sentence or two

Moons of a distant planetProblem-solving · 7 marks

A student astronomy club tracks four moons of a distant planet and finds each moon's orbital radius r and period T. The table gives r³ and T² for each moon. Each period was measured with an uncertainty of ±1.0%, and the radii are taken as exact. Use G = 6.67 × 10⁻¹¹ N m² kg⁻².

Orbital radius cubed and period squared for four moons of a distant planet.
Moon	r (× 10⁸ m)	r³ (× 10²⁴ m³)	T² (× 10⁹ s²)
W	2.00	8.00	3.20
X	3.00	27.0	10.8
Y	4.00	64.0	25.6
Z	5.00	125	50.0

(a) Use the data in the table to determine the gradient of a graph of T² against r³, in s² m⁻³. [2]

s² m⁻³

(b) Calculate the mass of the planet from the gradient. [2]

(d) Explain why the club can find the planet's mass without knowing the masses of the moons. [2]

0 words · a sentence or two

A ball thrown from a sand duneShort response · 6 marks

At a beach near Geraldton, a student throws a ball from the top of a sand dune 12.0 m above the flat beach below. The ball leaves her hand at 15.0 m s⁻¹ at 25.0° above the horizontal. Ignore air resistance and use g = 9.80 m s⁻².

(a) Calculate the time the ball takes to reach the beach. [3]

(b) Calculate the horizontal distance from the point of release to where the ball lands. [1]

0 words · a sentence or two

Weight on a rotating space stationProblem-solving · 7 marks

Designers of a future space station propose a ring of radius 120 m that spins about its centre, far from any planet, so gravity on the station can be ignored. Astronauts stand on the inside of the ring's outer wall, which acts as their floor. As the ring turns, the floor pushes each astronaut towards the centre, and the astronaut feels this push as weight. A ladder runs from the floor to the hub at the centre of the ring.

(a) Calculate the period of rotation that would give the astronauts at the floor an apparent weight equal to their weight on Earth, where g = 9.80 m s⁻². [2]

(b) Explain why an astronaut standing on the floor feels a push from it, even though no gravity acts on her. [3]

0 words · about 50 to 80 words

0 words · a sentence or two

Loading a truck up a rampShort response · 6 marks

A worker pushes a 40.0 kg crate at a steady speed up a ramp 5.00 m long onto the back of a truck. The ramp is inclined at 15.0° to the horizontal, the push is parallel to the ramp, and friction is small enough to ignore. Use g = 9.80 m s⁻².

(a) Calculate the size of the push needed. [2]

(b) Calculate the work done by the worker in pushing the crate up the full length of the ramp. [2]

(c) Explain why using the ramp does not reduce the work needed to raise the crate, compared with lifting it straight up onto the truck. [2]

0 words · a sentence or two

Quick checks: four-option questions

The ATAR examination has no multiple-choice questions. These quick checks practise the same physics.

Question 1

A satellite dish on a turntable at a ground station turns at a constant rate, carrying a bolt on its rim around a horizontal circle at constant speed. Why is the bolt accelerating?

Question 2

A child sits 2.40 m from the centre of a playground roundabout that makes one complete turn every 6.00 s. What is the child's speed?

Question 3

The tyres of a 1350 kg car can provide a sideways friction force of at most 6.50 × 10³ N on a flat, horizontal road. What is the greatest speed at which the car can round a bend of radius 40.0 m? Use g = 9.80 m s⁻².

Question 4

Engineers design a banked curve of radius 120 m on a highway near Albany so that a car at 22.0 m s⁻¹ needs no sideways friction. At what angle to the horizontal must the road be banked? Use g = 9.80 m s⁻².

Question 5

A light aircraft flies at a steady 70.0 m s⁻¹ in a horizontal circle with its wings banked at 30.0° to the horizontal. The lift force acts at right angles to the wings, and no other horizontal force acts. What is the radius of the turn? Use g = 9.80 m s⁻².

Question 6

A car rounds a banked curve at exactly the design speed, so no sideways friction acts. What provides the centripetal force on the car?

Question 7

In a school laboratory in Geraldton, a student whirls a stopper in a horizontal circle of fixed radius 0.800 m and measures the string's tension F at several speeds v. The graph shows F against v². What is the stopper's mass?

Question 8

A car approaches a flat, horizontal bend on a road covered in black ice, which provides almost no friction. Why can the car not follow the bend?

Question 9

A roller coaster car passes the top of a vertical loop at the minimum speed that keeps it on the track. Which force or forces provide the centripetal force on the car at that point?

Question 10

A stunt cyclist rides around the inside of a vertical loop of radius 6.50 m. What is the least speed at the top of the loop for which the cyclist stays in contact with the track? Use g = 9.80 m s⁻².

Question 11

A 42.0 kg rider passes through the bottom of a dip on a roller coaster at 11.0 m s⁻¹. The dip is part of a vertical circle of radius 10.0 m. What is the rider's apparent weight at the bottom? Use g = 9.80 m s⁻².

Question 12

A cyclist and bicycle of total mass 85.0 kg ride over a hump in the road at 11.0 m s⁻¹. The top of the hump is part of a vertical circle of radius 25.0 m. What is the size of the normal force from the road at the top of the hump? Use g = 9.80 m s⁻².

Question 13

A roller coaster car starts from rest at a point 25.0 m above the bottom of a vertical loop of radius 8.00 m and runs without friction. What is the size of the force the seat exerts on a 55.0 kg rider at the top of the loop? Use g = 9.80 m s⁻².

Question 14

A 0.400 kg ball on a string 0.750 m long is swung in a vertical circle. At the lowest point it moves at 5.00 m s⁻¹. What is the tension in the string at that point? Use g = 9.80 m s⁻².

Question 15

A bucket of water is swung on a rope in a vertical circle. At which point is the tension in the rope greatest?

Question 16

A ball on a light string of length 0.900 m moves in a vertical circle with no energy losses. Its speed at the top is 3.50 m s⁻¹. What is its speed at the bottom? Use g = 9.80 m s⁻².

Question 17

A small cart runs without friction down a slope and then around the inside of a vertical loop of radius 4.20 m. From what least height above the bottom of the loop must it start from rest to stay on the track at the top? Use g = 9.80 m s⁻².

Question 18

A pilot pulls out of a dive along the bottom of a vertical circle. Why does she feel heavier than usual at the bottom?

Question 19

In a torsion balance experiment, a 160 kg lead sphere and a 0.730 kg lead sphere have their centres 0.230 m apart. What is the size of the gravitational force between them? Use G = 6.67 × 10⁻¹¹ N m² kg⁻².

Question 20

A space probe feels a gravitational force F from a planet. The probe moves out until its distance from the planet's centre is three times as large. What is the force on it now?

Question 21

What is the gravitational field strength at a height of 1.50 × 10⁶ m above Earth's surface? Use G = 6.67 × 10⁻¹¹ N m² kg⁻². Earth's mass is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m.

Question 22

A rocky planet has twice the mass of Earth and twice its radius. What is the gravitational field strength at its surface? Take the field strength at Earth's surface as 9.80 m s⁻².

Question 23

Which statement defines the gravitational field strength at a point?

Question 24

A crane at Fremantle Port lifts a 2.40 × 10³ kg shipping container at constant speed through a height of 18.0 m. How much work, in kilojoules, does the crane do on the container? Use g = 9.80 m s⁻².

Question 25

A spanner falls from rest from the cab of a tower crane, 60.0 m above the ground. Ignore air resistance. How fast is it moving just before it reaches the ground? Use g = 9.80 m s⁻².

Question 26

A lander measures the gravitational field strength at the surface of a small rocky planet as 3.71 m s⁻². The planet's diameter is 6.78 × 10⁶ m. What is the planet's mass? Use G = 6.67 × 10⁻¹¹ N m² kg⁻².

Question 27

A 110 kg refrigerator rests on a loading ramp inclined at 30.0° to the horizontal. What is the component of its weight acting down the slope? Use g = 9.80 m s⁻².

Question 28

A 105 kg motorbike is parked on a smooth ramp inclined at 20.0° to the horizontal, held by a strap parallel to the ramp. What is the size of the normal force from the ramp on the motorbike? Use g = 9.80 m s⁻².

Question 29

A block is released from rest and slides 3.50 m down a frictionless slope inclined at 18.0° to the horizontal. How fast is it moving at the bottom? Use g = 9.80 m s⁻².

Question 30

A marble rolls off a horizontal bench 1.20 m high at 2.50 m s⁻¹. Ignore air resistance. How far from the foot of the bench does it land? Use g = 9.80 m s⁻².

Question 31

A football is kicked at Optus Stadium at 24.0 m s⁻¹ at 40.0° above the horizontal. Ignore air resistance. What is the greatest height it reaches above the point where it was kicked? Use g = 9.80 m s⁻².

Question 32

A ball is kicked from level ground at 22.0 m s⁻¹ at 35.0° above the horizontal and lands on the same level. Ignore air resistance. How far away does it land? Use g = 9.80 m s⁻².

Question 33

A netball is thrown at an angle above the horizontal. Ignore air resistance. Which statement describes the ball at the top of its path?

Question 34

A stone is thrown from the top of a cliff 15.0 m above the sea at 12.0 m s⁻¹, 30.0° above the horizontal. Ignore air resistance. How far from the foot of the cliff does it land in the water? Use g = 9.80 m s⁻².

Question 35

A ball is thrown from a balcony 20.0 m above the ground at 8.00 m s⁻¹. The thrower does not say at what angle. Ignore air resistance. How fast is the ball moving just before it hits the ground? Use g = 9.80 m s⁻².

Question 36

Astronauts aboard a space station in a low circular orbit float freely in the cabin. Which statement explains this?

Question 37

A satellite moves in a circular orbit of radius 7.00 × 10⁶ m, measured from Earth's centre. What is its orbital speed? Use G = 6.67 × 10⁻¹¹ N m² kg⁻². Earth's mass is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m.

Question 38

A weather satellite orbits in a circle 8.00 × 10⁵ m above Earth's surface. What is its orbital period in minutes? Use G = 6.67 × 10⁻¹¹ N m² kg⁻². Earth's mass is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m.

Question 39

A communications satellite stays above the same point on the equator, so its orbital period is 24.0 h. What is the radius of its orbit, measured from Earth's centre? Use G = 6.67 × 10⁻¹¹ N m² kg⁻². Earth's mass is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m.

Question 40

Two moons orbit the same planet in circular orbits. Moon X has orbital radius r and period T. Moon Y has orbital radius 4r. What is Moon Y's period?

Question 41

A moon orbits a gas giant planet in another planetary system in a circle of radius 1.88 × 10⁹ m, with a period of 16.7 days. What is the planet's mass? Use G = 6.67 × 10⁻¹¹ N m² kg⁻².

Question 42

A student astronomy club measures the orbital radius r and period T of four moons of a distant planet. The graph shows T² against r³. What is the planet's mass? Use G = 6.67 × 10⁻¹¹ N m² kg⁻².

Question 43

Two satellites move in the same circular orbit around Earth. Satellite A has twice the mass of satellite B. How do their orbital periods compare?

Question 44

A section of a racing track near Perth is a curve of radius 250 m banked at 25.0° to the horizontal. At what speed can a car take the curve with no sideways friction at all? Use g = 9.80 m s⁻².

Question 45

A roller coaster car passes over the crest of a hill, where a rider's apparent weight is half her true weight. The crest is part of a vertical circle of radius 16.0 m. The car then descends 24.0 m to the bottom of a dip that is part of a vertical circle of radius 12.0 m. Ignore friction and air resistance. What is the rider's apparent weight at the bottom of the dip? Use g = 9.80 m s⁻².

Question 46

A curve of radius 90.0 m on a freeway exit ramp is banked so that a car travelling at 15.0 m s⁻¹ needs no sideways friction. A 1250 kg car takes the curve at 25.0 m s⁻¹ without slipping. What is the size of the friction force the road exerts on the car's tyres? Use g = 9.80 m s⁻².

Question 47

A stone is thrown from a fishing platform 12.0 m above the water at 16.0 m s⁻¹, 30.0° above the horizontal. Ignore air resistance. At what angle to the horizontal is the stone moving just before it reaches the water? Use g = 9.80 m s⁻².

Question 48

An Earth-imaging satellite moves in a circular orbit and completes exactly 15 orbits every 24.0 h. What is its altitude above Earth's surface, in kilometres? Use G = 6.67 × 10⁻¹¹ N m² kg⁻². Earth's mass is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m.

Question 49

A satellite moves in a circular orbit around Earth at a constant speed of 6.00 × 10³ m s⁻¹. What is the ratio of the gravitational field strength at the satellite's orbit to the field strength at Earth's surface? Use G = 6.67 × 10⁻¹¹ N m² kg⁻². Earth's mass is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m.