Gravitation Structured 2015

1

1. This question is about gravitation.

(a) (i) Define gravitational potential at a point in a gravitational field.

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

(ii) Explain why values of gravitational potential have negative values.

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

The Earth and the Moon may be considered to be two isolated point masses. The masses of the Earth and the Moon are 5.98 1024kg and 7.35 1022 kg respectively and their separation is 3.84 108 m, as shown below. The diagram is not to scale.

Earth Moonmass = 5 98 10. kg 24 mass = 7 35 10. kg22

P

3 84 10. m8

(b) (i) Deduce that, at point P, 3.46 108m from Earth, the gravitational field strength is approximately zero.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (3)

2

(ii) The gravitational potential at P is 1.28 106 J kg1. Calculate the minimum speed of a space probe at P so that it can escape from the attraction of the Earth and the Moon.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (3)

(Total 10 marks)

2. This question is about gravitation.

A binary star consists of two stars that each follow circular orbits about a fixed point P as shown below.

star mass P star massM1 M2

R1 R2

The stars have the same orbital period T. Each star may be considered to act as a point mass with its mass concentrated at its centre. The stars, of masses M1 and M2, orbit at distances R1 and R2 respectively from point P.

(a) State the name of the force that provides the centripetal force for the motion of the stars.

................................................................................................................................... (1)

3

(b) By considering the force acting on one of the stars, deduce that the orbital period T is given by the expression

( ) .4 22112

22 RRR

GMT +=

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (3)

(c) The star of mass M1 is closer to the point P than the star of mass M2. Using the answer in (b), state and explain which star has the larger mass.

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (2)

(Total 6 marks)

4

3. This question is about a spacecraft.

A spacecraft above Earths atmosphere is moving away from the Earth. The diagram below shows two positions of the spacecraft. Position A and position B are well above Earths atmosphere.

Earth A B

At position A, the rocket engine is switched off and the spacecraft begins coasting freely. At position A, the speed of the spacecraft is 5.37 103 m s1 and at position B, 5.10 103 m s1. The time to travel from position A to position B is 6.00 102 s.

(a) (i) Explain why the speed is changing between positions A and B.

.........................................................................................................................

......................................................................................................................... (1)

(ii) Calculate the average acceleration of the spacecraft between positions A and B.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

(iii) Estimate the average gravitational field strength between positions A and B. Explain your working.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (3)

5

(b) The diagram below shows the variation with distance from Earth of the kinetic energy Ek of the spacecraft. The radius of Earth is R.

energy Ek

R0

0 distance

On the diagram above, draw the variation with distance from the surface of Earth of the gravitational potential energy Ep of the spacecraft.

(2) (Total 8 marks)

4. This question is about gravitational potential.

(a) Define gravitational potential at a point in a gravitational field.

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (3)

6

(b) A planet has mass M and radius R0. The magnitude g0 of the gravitational field strength at the surface of a planet is

20

0 RMGg =

where G is the gravitational constant.

Use this expression to deduce that the gravitational potential V0 at the surface of the planet is given by

V0 = g0R0.

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (2)

7

(c) The graph below shows the variation with distance R from the centre of the planet of the gravitational potential V. The radius R0 of the planet = 5.0 106 m. Values of V are not shown for R < R0.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

R / 10 m7

V /

10

Jkg

71

Use the graph to determine the magnitude of the gravitational field strength at the surface of the planet.

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (3)

8

(d) A satellite of mass 3.2 103 kg is launched from the surface of the planet. Use the graph to estimate the minimum launch speed that the satellite must have in order to reach a height of 2.0 107 m above the surface of the planet. (You may assume that it reaches its maximum speed immediately after launch.)

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (4)

(Total 12 marks)

5. This question is about the energy of orbiting satellites.

(a) Define the term gravitational potential at a point in a gravitational field.

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (2)

(b) A satellite is in orbit about Earth at a distance R from the centre of Earth. The Earth may be regarded as a point mass situated at its centre.

Deduce that the kinetic energy of the satellite is numerically equal to half the potential energy of the satellite.

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (3)

9

(c) The distance between the centre of the Moon and the centre of Earth is about 4.0 108 m. The Moon may also be regarded as a point mass situated at its centre. The orbital period of the Moon about the Earth is 2.4 106 s.

(i) Calculate the orbital speed of the Moon.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

(ii) Use your answer in (b) and (c)(i) to calculate a value for the gravitational potential due to Earth at a distance of 4.0 108 m from its centre.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

(Total 9 marks)

6. This question is about orbital motion.

(a) State Keplers third law (the law of periods).

.....................................................................................................................................

..................................................................................................................................... (1)

10

(b) A satellite of mass m is in orbit of radius r about Earth. The mass of Earth is ME and the orbital period of the satellite is T.

State, for the satellite,

(i) the name of the force that provides the centripetal force.

........................................................................................................................... (1)

(ii) the orbital speed in terms of T and r.

........................................................................................................................... (1)

(c) Keplers third law may be applied to the satellite orbiting the Earth. Use your answers to (b) to deduce that in Keplers third law there is a constant K given by

K = E

24GM

.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

..................................................................................................................................... (3)

(d) State an expression for the gravitational field strength g at the surface of the Earth in terms of ME and the radius of Earth RE.

..................................................................................................................................... (1)

11

(e) For the Earth, the gravitational field strength, g is 10Nkg1 and the radius RE is 6.4106m. Using your answers to (c) and (d), deduce that the orbital period of a satellite that is at a height RE above the surface of Earth is 1.4104s.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

..................................................................................................................................... (3)

(Total 10 marks)

7. This question is about gravitational fields.

(a) Define gravitational field strength.

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (2)

12

(b) The gravitational field strength at the surface of Jupiter is 25 N kg1 and the radius of Jupiter is 7.1 107 m.

(i) Derive an expression for the gravitational field strength at the surface of a planet in terms of its mass M, its radius R and the gravitational constant G.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

(ii) Use your expression in (b)(i) above to estimate the mass of Jupiter.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

(Total 6 marks)

8. This question is about gravitational fields.

(a) Define gravitational field strength.

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (2)

13

(b) The gravitational field strength at the surface of Jupiter is 25 N kg1 and the radius of Jupiter is 7.1 107 m.

(i) Derive an expression for the gravitational field strength at the surface of a planet in terms of its mass M, its radius R and the gravitational constant G.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

(ii) Use your expression in (b)(i) above to estimate the mass of Jupiter.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

(Total 6 marks)

9. This question is about gravitational field strength and gravitational potential.

(a) Define gravitational field strength.

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (2)

14

(b) State an expression for the magnitude of the gravitational field strength gh at a point height h above a planet in terms of the mass of the planet M, its radius R and the gravitational constant G.

...................................................................................................................................

................................................................................................................................... (1)

(c) The radius of Mars is 3.4 106 m and the magnitude of the gravitational field strength at a height of 1.2 106 m above its surface is 1.8 N kg1. Use your answer to (b) to deduce that the magnitude of the gravitational potential at height of 1.2 106 m above the surface of Mars is 8.3 106 J kg1.

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (3)

(d) A lump of rock is moving with speed v, towards Mars. Its closest distance of approach to Mars is at distance 1.8 106 m above the surface of Mars. Deduce that the lump of rock will go into circular orbit about Mars if the speed v is less than 3.0 103 m s1.

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (2)

(Total 8 marks)

15

10. This question is about gravitational potential.

(a) Define gravitational potential at a point.

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (2)

(b) A meteorite moves towards the Moon from a long distance away.

(i) On the axes below, sketch a graph to show the variation with distance from the centre of the Moon of the gravitational potential of the meteorite as it approaches the Moon. The radius of the Moon is r.

gravitationalpotential

+ve

ve

r0

distance from centre of Moon

(2)

(ii) The radius r of the Moon is 1.7 106 m and its mass is 7.3 1022 kg.

Estimate the impact speed with which the meteorite hits the surface of the Moon.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (3)

16

(iii) Suggest one factor that will make the impact speed greater than your estimate.

.........................................................................................................................

......................................................................................................................... (1)

(c) A similar meteorite moves towards the Earth from a long distance away.

Suggest how the total energy of the meteorite varies with distance when the meteorite is

(i) outside the Earths atmosphere;

.........................................................................................................................

......................................................................................................................... (1)

(ii) inside the Earths atmosphere.

.........................................................................................................................

......................................................................................................................... (1)

(Total 10 marks)

11. This question is about gravitational field strength near the surface of a planet.

(a) (i) Define gravitational field strength.

.........................................................................................................................

......................................................................................................................... (2)

(ii) State why gravitational field strength at a point is numerically equal to the acceleration of free fall at that point.

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (1)

17

(b) A certain planet is a uniform sphere of mass M and radius R of 5.1 106 m.

(i) State an expression, in terms of M and R, for the gravitational field strength at the surface of the planet. State the name of any other symbol you may use.

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (1)

(ii) A mountain on the surface of the planet has a height of 2000 m. Suggest why the value of the gravitational field strength at the base of the mountain and at the top of the mountain are almost equal.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

18

(c) A small sphere is projected horizontally near the surface of the planet in (b). Photographs of the sphere are taken at time intervals of 0.20 s. The images of the sphere are placed on a grid and the result is shown below.

point ofrelease

1.00 cm represents 1.00 m

The first photograph is taken at time t = 0. Each 1.00 cm on the grid represents a distance of 1.00 m in both the horizontal and the vertical directions.

Use the diagram to

(i) explain why air resistance on the planet is negligible;

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (2)

(ii) calculate a value for the acceleration of free fall at the surface of the planet.

.........................................................................................................................

.........................................................................................................................

.........................................................................................................................

......................................................................................................................... (3)

19

(d) Use your answer to (c)(ii) and data from (b) to calculate the mass of the planet.

...................................................................................................................................

...................................................................................................................................

...................................................................................................................................

................................................................................................................................... (2)

(Total 13 marks)

12. This question is about the possibility of generating electrical power using a satellite orbiting the

Earth.

(a) Define gravitational field strength.

.....................................................................................................................................

.....................................................................................................................................

..................................................................................................................................... (2)

(b) Use the definition of gravitational field strength to deduce that

GM = g0R2

where M is the mass of the Earth, R its radius and g0 is the gravitational field strength at the surface of the Earth. (You may assume that the Earth is a uniform sphere with its mass concentrated at its centre.)

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

..................................................................................................................................... (2)

20

A space shuttle orbits the Earth and a small satellite is launched from the shuttle. The satellite carries a conducting cable connecting the satellite to the shuttle. When the satellite is a distance L from the shuttle, the cable is held straight by motors on the satellite.

Diagram 1

L

conducting cable

Earths magnetic field

shuttledirection of orbitspeed v

EARTH

satellite

As the shuttle orbits the Earth with speed v, the conducting cable is moving at right angles to the Earths magnetic field. The magnetic field vector B makes an angle to a line perpendicular to the conducting cable as shown in diagram 2. The velocity vector of the shuttle is directed out of the plane of the paper.

Diagram 2

conducting cable

B

(c) On diagram 2, draw an arrow to show the direction of the magnetic force on an electron in the conducting cable. Label the arrow F.

(1)

21

(d) State an expression for the force F on the electron in terms of B, v, e and , where B is the magnitude of the magnetic field strength and e is the electron charge.

..................................................................................................................................... (1)

(e) Hence deduce an expression for the emf E induced in the conducting wire.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

..................................................................................................................................... (3)

(f) The shuttle is in an orbit that is 300 km above the surface of the Earth. Using the expression

GM = g0R2

and given that R = 6.4 106 m and g0 = 10 N kg1, deduce that the orbital speed v of the satellite is 7.8 103 m s1.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

..................................................................................................................................... (3)

22

(g) The magnitude of the magnetic field strength is 6.3 106 T and the angle = 20. Estimate the length L of the cable required in order to generate an emf of 1 kV.

.....................................................................................................................................

.....................................................................................................................................

..................................................................................................................................... (2)

(Total 14 marks)