Year 12 Physics Ark Globe Academy Remote Learning Pack Phase V Monday 22 June- Friday 10 July

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Year 12 Physics

Sessio Title Work to be Resource Outcome On-Line n completed provided Support 1 Uniform - Read 21.1 Oscillations Textbook Complete and mark: - Watch video 3.01 and radial while taking notes. Page(s): Summary questions and 3.02. Takes notes ,which go fields - Complete and mark 338-339 21.1 into your folder. the summary questions

2 Newton’s - Read 21.3 Newton’s Textbook Complete and mark: - Watch video 3.03 law of law of gravitation while Page(s): Summary questions Takes notes ,which go into your folder. gravitatio taking take notes. 343-345 21.3 n, inverse - Complete and mark square the summary questions. Lesson: law 1 - DO NOW Live lesson: 2 - Checking for Uniform and radial Understanding fields Question(s) Live lesson: 3 - Application Newton’s law of Question(s) gravitation and adding fields 3 Gravitatio - Read 21.2 Textbook Complete and mark: - Watch video 3.04 nal Gravitational potential Page(s): Summary questions and 3.05. Takes notes ,which go potential take notes. 340 – 342 21.2 into your folder. 1 - Complete and mark the summary questions 4 Gravitatio - Complete the online Complete online - Diagnostic on nal Gravitational fields diagnostic MS forms potential diagnostic part 1 2 Lesson: Live lesson: 1 - DO NOW Gravitational potential, 2 - Checking for potential difference and Understanding escape velocity Question(s) 3 - Application Question(s) 5 Gravitatio - Read 21.4 Planetary Textbook Complete and mark: - Watch video 3.06 nal fields while taking Page(s): Summary questions Takes notes ,which go into your folder. potential notes. 346 - 350 21.4 - Complete and mark the summary questions

6 Orbits - Read 21.5 Satellite Textbook Complete and mark: - Watch video 3.07 motion Page(s): Summary questions & 3.08 Takes notes ,which go - Complete and mark 351 - 353 21.5 into your folder. the summary questions

7 Orbits 2 Live lesson: Lesson: - Watch video 2.09 Gravitational potential 1 - DO NOW Takes notes ,which go into your folder. graphs and gradient 2 - Checking for Understanding Live lesson: Question(s) Orbits 3 - Application Question(s)

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8 Gravitatio - Complete the online Work pack V Complete online - Diagnostic on nal field Gravitational field Pages(s) 4-12 diagnostic MS forms applied 1 diagnostic part 2 - Complete and mark Complete all MCQ the MCQ exam questions 1 - 23 9 Gravitatio - Complete and mark Work pack V Complete all nal fields the exam questions 24- Pages(s) 12-27 calculation, graphs applied 2 35 and writing questions.

Live lesson: Lesson: Gravitational fields 1 - DO NOW applied 2 - Checking for Understanding Question(s) 3 - Application Question(s) 10 Gravitatio - Complete the exam Textbook Complete exam Get head start y nal field questions on pages 354 Page(s): questions from old watching summary - 357 354- 357 spec videos 4.01 – - Mark answers 4.03 NOTE: After completing each session, check all task/PPQ answers using answers at the back.

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Session 8 - MCQ

Q1. The distance between the Sun and the Earth is 1.5 × 1011 m

What is the gravitational force exerted on the Sun by the Earth?

A 3.5 × 1022 N

B 1.7 × 1026 N

C 5.3 × 1033 N

D 8.9 × 1050 N

(Total 1 mark)

Q2. A planet has a radius half of the Earth’s radius and a mass a quarter of the Earth’s mass. What is the approximate gravitational field strength on the surface of the planet?

A 1.6 N kg–1

B 5.0 N kg–1

C 10 N kg–1

D 20 N kg–1 (Total 1 mark)

Q3. Which one of the following could be a unit of gravitational potential?

A N

B J

C N kg–1

D J kg–1 (Total 1 mark)

Q4. A mass of 5 kg is moved in a gravitational field from a point X at which the gravitational potential is –20 J kg–1 to a point Y where it is –10 J kg–1. The change in potential energy of

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the mass, in J, between X and Y is

A –50

B –10

C +10

D +50

(Total 1 mark)

Q5. The diagram shows two points, P and Q, at distances r and 2r from the centre of a planet.

The gravitational potential at P is −16 kJ kg−1. What is the work done on a 10 kg mass when it is taken from P to Q?

A – 120 kJ

B – 80 kJ

C + 80 kJ

D + 120 kJ (Total 1 mark)

Q6. A projectile moves in a gravitational field. Which one of the following is a correct statement for the gravitational force acting on the projectile?

A The force is in the direction of the field.

B The force is in the opposite direction to that of the field.

C The force is at right angles to the field.

D The force is at an angle between 0° and 90° to the field. (Total 1 mark)

Q7. The diagram shows two positions, X and Y, at different heights on the surface of the Earth.

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Which line, A to D, in the table gives correct comparisons at X and Y for gravitational potential and angular velocity?

gravitational potential at X angular velocity at X compared with Y compared with Y

A greater greater

B greater same

C greater smaller

D same same

(Total 1 mark)

Q8. The gravitational potential difference between the surface of a planet and a point P, 10 m above the surface, is 8.0 J kgࢤ 1 . Assuming a uniform field, what is the value of the gravitational field strength in the region between the planet’s surface and P?

A 0.80 N kgࢤ 1

B 1.25 N kgࢤ 1

C 8.0 N kgࢤ 1

D 80 N kgࢤ 1 (Total 1 mark)

Q9. The graph shows how the gravitational potential, V, varies with the distance, r , from the centre of the Earth.

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What does the gradient of the graph at any point represent?

A the magnitude of the gravitational field strength at that point

B the magnitude of the gravitational constant

C the mass of the Earth

D the potential energy at the point where the gradient is measured (Total 1 mark)

Q10. What is the angular speed of a satellite in a geo-synchronous orbit around the Earth?

A 7.3 × 10–5 rad s–1

B 2.6 × 10–1 rad s–1

C 24 rad s–1

D 5.0 × 106 rad s–1 (Total 1 mark)

Q11. An artificial satellite of mass m is in a stable circular orbit of radius r around a planet of mass M. Which one of the following expressions gives the speed of the satellite? G is the universal gravitational constant.

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D (Total 1 mark)

Q12. Satellites N and F have the same mass and move in circular orbits about the same planet. N is the nearer satellite and F is the more distant. Which one of the following is smaller for N than for F?

A gravitational force on the satellite

B speed

C kinetic energy

D time for one orbit (Total 1 mark)

Q13. Which one of the following has different units to the other three?

A gravitational potential

B gravitational field strength

C force per unit mass

D gravitational potential gradient (Total 1 mark)

Q14. As a comet orbits the Sun the distance between the comet and the Sun continually changes. As the comet moves towards the Sun this distance reaches a minimum value. Which one of the following statements is incorrect as the comet approaches this minimum distance?

A The potential energy of the comet increases.

B The gravitational force acting on the comet increases.

C The direction of the gravitational force acting on the comet changes.

D The kinetic energy of the comet increases. (Total 1 mark)

Q15. When two similar spherical objects of radius R are touching, the gravitational force of

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attraction between them is F. When the gravitational force between them is F/4, the distance between the surfaces of the spheres is

A R

B 2R

C 4R

D 6R

(Total 1 mark)

Q16. A satellite is in orbit at a height h above the surface of a planet of mass M and radius R. What is the velocity of the satellite?

D (Total 1 mark)

Q17. The diagram shows two point masses each of mass m separated by a distance 2r.

What is the value of the gravitational field strength at the mid-point, P, between the two masses?

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D zero (Total 1 mark)

Q18. Which one of the following graphs correctly shows the relationship between the gravitational force, F, between two masses and the distance, r, between them?

A B

C D

(Total 1 mark)

Q19. Masses of M and 2M exert a gravitational force F on each other when the distance between their centres is r. What is the gravitational force between masses of 2M and 4M when the distance between their centres is 4r?

A 0.25 F

B 0.50 F

C 0.75 F

D 1.00 F (Total 1 mark)

Q20. Two stars of mass M and 4M are at a distance d between their centres.

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The resultant gravitational field strength is zero along the line between their centres at a distance y from the centre of the star of mass M.

What is the value of the ratio ?

D (Total 1 mark)

Q21. The following data refer to two planets.

radius / km density / kg m−3

planet P 8000 6000

planet Q 16000 3000

The gravitational field strength at the surface of P is 13.4 N kg−1. What is the gravitational field strength at the surface of Q?

A 3.4 N kg−1

B 13.4 N kg−1

C 53.6 N kg−1

D 80.4 N kg−1 (Total 1 mark)

Q22. g is the strength of the gravitational field at the surface of the Earth; R is the radius of the Earth. The potential energy lost by a satellite of mass m falling to the Earth’s surface from

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a height R above the surface is

A 4mgR

B 2mgR

D (Total 1 mark)

Q23. What would the period of rotation of the Earth need to be if objects at the equator were to appear weightless?

radius of Earth = 6.4 × 106 m

A 4.5 × 10–2 hours

B 1.4 hours

C 24 hours

D 160 hours (Total 1 mark) Session 9 – Calculation & written

Q24. (a) Define gravitational field strength at a point in a gravitational field.

______

______(1)

(b) Tides vary in height with the relative positions of the Earth, the Sun and the moon which change as the Earth and the Moon move in their orbits. Two possible configurations are shown in Figure 1.

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Configuration A

Configuration B

Figure 1

Consider a 1 kg mass of sea water at position P. This mass experiences forces FE, FM and FS due to its position in the gravitational fields of the Earth, the Moon and the Sun respectively.

(i) Draw labelled arrows on both diagrams in Figure 1 to indicate the three forces experienced by the mass of sea water at P. (3)

(ii) State and explain which configuration, A or B, of the Sun, the Moon and the Earth will produce the higher tide at position P.

______

______(2)

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(c) Calculate the magnitude of the gravitational force experienced by 1 kg of sea water on the Earth’s surface at P, due to the Sun’s gravitational field.

radius of the Earth’s orbit = 1.5 × 1011 m

mass of the Sun = 2.0 × 1030 kg

universal gravitational constant, G = 6.7 × 10−11 Nm2 kg−2

(3) (Total 9 marks)

Q25. For an object, such as a space rocket, to escape from the gravitational attraction of the Earth it must be given an amount of energy equal to the gravitational potential energy that it has on the Earth’s surface. The minimum initial vertical velocity at the surface of the Earth that it requires to achieve this is known as the escape velocity.

(a) (i) Write down the equation for the gravitational potential energy of a rocket when it is on the Earth’s surface. Take the mass of the Earth to be M, that of the rocket to be m and the radius of the Earth to be R.

(1)

(ii) Show that the escape velocity, v, of the rocket is given by the equation

(2)

(b) The nominal escape velocity from the Earth is 11.2 km s–1. Calculate a value for the escape velocity from a planet of mass four times that of the Earth and radius twice

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that of the Earth.

(2)

(c) Explain why the actual escape velocity from the Earth would be greater than the nominal value calculated from the equation given in part (a)(ii).

______

______(2) (Total 7 marks)

Q26. (a) State Newton’s law of gravitation.

______

______(2)

(b) In 1798 Cavendish investigated Newton’s law by measuring the gravitational force between two unequal uniform lead spheres. The radius of the larger sphere was 100 mm and that of the smaller sphere was 25 mm.

(i) The mass of the smaller sphere was 0.74 kg. Show that the mass of the larger sphere was about 47 kg.

density of lead = 11.3 × 103 kg m–3

(2)

(ii) Calculate the gravitational force between the spheres when their surfaces were in contact.

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answer = ______N (2)

(c) Modifications, such as increasing the size of each sphere to produce a greater force between them, were considered in order to improve the accuracy of Cavendish’s experiment. Describe and explain the effect on the calculations in part (b) of doubling the radius of both spheres.

______

______(4) (Total 10 marks)

Q27. The gravitational field associated with a planet is radial, as shown in Figure 1, but near the surface it is effectively uniform, as shown in Figure 2.

Alongside each figure, sketch a graph to show how the gravitational potential V associated with the planet varies with distance r (measured outwards from the surface of the planet) in each of these cases.

Figure 1

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Figure 2 (Total 4 marks)

Q28. (a) Explain what is meant by the gravitational potential at a point in a gravitational field.

______

______(2)

(b) Use the following data to calculate the gravitational potential at the surface of the Moon.

mass of Earth = 81 × mass of Moon radius of Earth = 3.7 × radius of Moon gravitational potential at surface of the Earth = –63 MJ kg–1

______

______(3)

(c) Sketch a graph on the axes below to indicate how the gravitational potential varies with distance along a line outwards from the surface of the Earth to the surface of the Moon.

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(3) (Total 8 marks)

Q29. The graph below shows how the gravitational potential energy, Ep, of a 1.0 kg mass varies with distance, r, from the centre of Mars. The graph is plotted for positions above the surface of Mars.

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(a) Explain why the values of Ep are negative.

______

______(2)

(b) Use data from the graph to determine the mass of Mars.

mass of Mars ______kg (3)

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escape velocity ______m s–1 (3)

(d) Show that the graph data agree with

(3) (Total 11 marks)

Q30. (a) Explain why the mass of an object is constant but its weight may change.

______

______(3)

(b) The table gives the gravitational potentials, V, at three different distances, r, from the centre of the Earth.

distance from centre of Earth gravitational potential r / km V / 107 J kg–1

7500 –5.36

12500 –3.22

22500 –1.79

(i) Explain why the gravitational potential at a point in a gravitational field is negative.

______

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______

______(2)

(ii) Show that the data in the table are consistent with V r –1.

______

______(3)

(iii) A satellite of mass 450 kg is moved from an orbit of radius 7500 km around the Earth to an orbit of radius 12 500 km.

Use data from the table to show that the potential energy of the satellite increases, by about 10 GJ.

______

______(2)

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(i) Calculate the kinetic energy of the 450 kg satellite when it is in an orbit of radius 12 500 km.

mass of the Earth = 6.0 × 1024 kg

______

kinetic energy ______GJ (4)

(ii) Calculate the change in kinetic energy of the satellite when it moves into the higher orbit.

______

change in kinetic energy ______GJ (1)

(iii) Calculate the total energy that has to be supplied to move the 450 kg satellite from an orbit of radius 7500 km to an orbit of radius 12 500 km.

______

total energy ______GJ (1) (Total 16 marks)

Q31. (a) (i) State the relationship between the gravitational potential energy, Ep, and the gravitational potential, V, for a body of mass m placed in a gravitational field.

______

______(1)

(ii) What is the effect, if any, on the values of Ep and V if the mass m is doubled?

value of Ep ______

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value of V ______(2)

(b)

The diagram above shows two of the orbits, A and B, that could be occupied by a satellite in circular orbit around the Earth, E. The gravitational potential due to the Earth of each of these orbits is:

orbit A – 12.0 MJ kg–1 orbit B – 36.0 MJ kg–1.

(i) Calculate the radius, from the centre of the Earth, of orbit A.

answer = ______m (2)

(ii) Show that the radius of orbit B is approximately 1.1 × 104 km.

(1)

(iii) Calculate the centripetal acceleration of a satellite in orbit B.

answer = ______m s–2 (2)

(iv) Show that the gravitational potential energy of a 330 kg satellite decreases by about 8 GJ when it moves from orbit A to orbit B.

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(1)

(c) Explain why it is not possible to use the equation ∆Ep = mg∆h when determining the change in the gravitational potential energy of a satellite as it moves between these orbits.

______

______(1) (Total 10 marks)

Q32. Communications satellites are usually placed in a geo-synchronous orbit.

(a) State two features of a geo-synchronous orbit.

______

______(2)

(b) Given that the mass of the Earth is 6.00 × 1024 kg and its mean radius is 6.40 × 106 m,

(i) show that the radius of a geo-synchronous orbit must be 4.23 × 107 m,

______

(ii) calculate the increase in potential energy of a satellite of mass 750 kg when it is raised from the Earth’s surface into a geo-synchronous orbit.

______

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______

______(6) (Total 8 marks)

Q33. The Global Positioning System (GPS) is a system of satellites that transmit radio signals which can be used to locate the position of a receiver anywhere on Earth.

(a) A receiver at sea level detects a signal from a satellite in a circular orbit when it is passing directly overhead as shown in the diagram above.

(i) The microwave signal is received 68 ms after it was transmitted from the satellite. Calculate the height of the satellite.

______

(ii) Show that the gravitational field strength of the Earth at the position of the satellite is 0.56 N kg–1.

mass of the Earth = 6.0 × 1024 kg mean radius of the Earth = 6400 km

______

______(4)

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(b) For the satellite in this orbit, calculate

(i) its speed,

______

(ii) its time period.

______

______(5) (Total 9 marks)

Q34. (a) The graph shows how the gravitational potential varies with distance in the region above the surface of the Earth. R is the radius of the Earth, which is 6400 km. At the surface of the Earth, the gravitational potential is −62.5 MJ kg–1.

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Use the graph to calculate

(i) the gravitational potential at a distance 2R from the centre of the Earth,

______

(ii) the increase in the potential energy of a 1200 kg satellite when it is raised from the surface of the Earth into a circular orbit of radius 3R.

______

______(4)

(b) (i) Write down an equation which relates gravitational field strength and gravitational potential.

______

(ii) By use of the graph in part (a), calculate the gravitational field strength at a distance 2R from the centre of the Earth.

______

(iii) Show that your result for part (b)(ii) is consistent with the fact that the surface gravitational field strength is about 10 N kg–1.

______

______(5) (Total 9 marks)

Q35. (a) State, in words, Newton’s law of gravitation.

______

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______(3)

(b) By considering the centripetal force which acts on a planet in a circular orbit, show that T2 R3, where T is the time taken for one orbit around the Sun and R is the radius of the orbit.

______

______(3)

(i) The mean radius of the orbit of Mercury is 5.79 × 1010 m. Calculate the length of Mercury’s year.

______

(ii) Neptune orbits the Sun once every 165 Earth years.

Calculate the ratio .

______

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______

______(4) (Total 10 marks)

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Mark schemes

Q1. A [1]

Q2. C [1]

Q3. D [1]

Q4. D [1]

Q5. C [1]

Q6. A [1]

Q7. B [1]

Q8. A [1]

Q9. A [1]

Q10. A [1]

Q11. B [1]

Q12. D [1]

Q13. A [1]

Q14. A [1]

Q15. B [1]

Q16. C [1]

Q17. D [1]

Q18. D [1]

Q19. A

[1]

Q20. B [1]

Q21. B [1]

Q22. C [1]

Q23. B [1]

Q24.

(a) force acting per unit mass or g = F / m or g = with terms defined (1)

(b) (i) direction of FE correct in each diagram B1

direction of FM correct in each diagram B1

direction of FS correct in each diagram B1

FS must be distinguished from FM

penalty of 1 mark for any missing labelling (3)

(ii) sun and moon pulling in same direction / resultant of FM and FS is greatest / clear response including summation of FM and FS M1

configuration A A1 (2)

correct substitution C1

(5.95 or 5.96 or 5.9 or 6.0) × 10−3 N kg−1 A1 (3) [9]

Q25.

(a) (i) g.p.e. = must be equation (condone “V =”)

B1 1

(ii) equate with k.e. must be seen

cancelling correct m must be seen

A1 2

(b) correct ratios taken ( )

v = 15.8(4) km s–1

A1 2

k.e. of rocket → internal energy of rocket and atmosphere/ work is done against air resistance

A1 2 [7]

Q26. (a) force of attraction between two point masses (or particles) (1)

proportional to product of masses (1)

inversely proportional to square of distance between them (1)

[alternatively

quoting an equation, F = with all terms defined (1)

reference to point masses (or particles) or r is distance between centres (1)

F identified as an attractive force (1)] max 2

3 3 3 (b) (i) mass of larger sphere ML (= πr ρ) = π × (0.100) × 11.3 × 10 (1) = 47(.3) (kg) (1)

[alternatively

use of M ∝ r3 gives (1) (= 64)

and ML = 64 × 0.74 = 47(.4) (kg) (1)] 2

(ii) gravitational force F (1)

= 1.5 × 10−7 (N) (1) 2

mass of either sphere would be 8 × greater (378 kg, 5.91 kg) (1)

this would make the force 64 × greater (1)

but separation would be doubled causing force to be 4 × smaller (1)

net effect would be to make the force (64/4) = 16 × greater (1) (ie 2.38 × 10−6 N) max 4

[10]

Q27.

gradient decreases as r increases (1) V increases as r increases (1) only negative values of V shown (1)

constant gradient (1) V increases as r increases (1) [max 4]

Q28. (a) work done/energy change (against the field) per unit mass (1) when moved from infinity to the point (1) 2

(b) VE = – and VM = – (1)

VM = – G × × = VE (1)

= 4.57 × 10–2 × (–63) = –2.9 MJ kg–1 (1) (2.88 MJ kg–1) 3

(c)

limiting values (–63,–VM) on correctly curving line (1) rises to value close to but below zero (1) falls to Moon (1) from point much closer to M than E (1) max 3 [8]

Q29. (a) zero potential at infinity (a long way away)

energy input needed to move to infinity (from the point) work done by the field moving object from infinity potential energy falls as object moves from infinity

B1 2

(b) Any pair of coordinates read correctly

C1 ±1/2 square

Use of

Rearrange for M

6.4 (±0.5) × 1023 kg

A1 3

C1 or reads radius of mars correctly (3.5 × 106)

equates to ½ v2 (condone power of 10 in MJ)

C1 use of v = √(2GM/r) with wrong radius

5000 ± 20 m s−1 (condone 1sf e.g. 5 km s−1)

A1 e.c.f. value of M from (b) may be outside range for other method 6.2 × 10−9x √their M 3

(d) Attempts 1 calculation of Vr

B1 Many values give 4.2.... so allow mark is for reading and using correct coordinates but allow minor differences in readings Ignore powers of 10 but consistent

Two correct calculation of Vr

Three correct calculations with conclusion

B1 3 [11]

Q30. (a) mass depends only on the amount of matter present owtte

weight is force between body and Earth/depends on g/mg/ gravitational field strength or answers in terms of Newton’s gravitational law

g (etc) varies at different points on and above the Earth or is different on different planets etc

B1 3

(b) (i) reference is ‘infinity’ where potential is 0

energy has to be put in/work has to be done to move mass to infinity or a bodies energy/PE decreases as a body moves from infinity towards the Earth

B1 2

(ii) need to show Vr to be constant, clear from algebra or final statement

two sets of data used correctly

all three sets of data used correctly (4.02, 4.025, 4.028)

B1 3

(iii) energy change per kg = (5.36 – 3.22) × 107 (J)

total change = 963 (960) × 107 J

B1 2

v2 = 3.2 × 107m2s–2 or v = 5670 ms–1

use of KE = ½ mv2 using their v

7.2 GJ

A1 4

(ii) KE changes by 4.8 GJ (allow ecf, 12 – their ci)

B1 1

(iii) total energy (supplied) = (4.8) GJ (cnao)

(allow 5.2 GJ using 10 GJ for change in Ep) (allow variations due to rounding off if physics is correct in previous parts)

B1 1 [16]

Q31. (a) (i) relationship between them is Ep = mV (allow ΔEp = mΔV) [or V is energy per unit mass (or per kg)] (1) 1

(ii) value of Ep is doubled (1)

value of V is unchanged (1) 2

(b) (i) use of V = gives rA = (1)

= 3.3(2) × 107 (m) (1) 2

(ii) since V (1)

(which is ≈ 1.1 × 104 km) 1

(iii) centripetal acceleration gB = (1)

[allow use of 1.1 × 107 m from (b)(ii)]

= 3.2 (m s–2) (1)

[alternatively, since gB = (–) (1)

= 3.2 (m s–2) (1)] 2

6 (iv) use of ΔEp = mΔV gives ΔEp = 330 × (–12.0 – (–36.0)) × 10 (1)

(which is 7.9 × 109 J or ≈ 8 GJ) 1

(or g decreases as height increases or work done per metre decreases as height increases or field is radial and/or not uniform) (1) 1 [10]

Q32. (a) period = 24 hours or equals period of Earth’s rotation (1) remains in fixed position relative to surface of Earth (1) equatorial orbit (1) same angular speed as Earth or equatorial surface (1) max 2

(b) (i) = mω2r (1)

T = (1)

(1)

(gives r = 42.3 × 103 km)

(ii) ΔV = GM (1)

= 6.67 × 10–11 × 6 × 1024 ×

= 5.31 × 107 (J kg–1) (1)

7 10 ΔEp = mΔV (= 750 × 5.31 × 10 ) = 3.98 × 10 J (1)

(allow C.E. for value of ΔV)

[alternatives:

calculation of (6.25 × 107) or (9.46 × 106) (1)

or calculation of (4.69 × 1010) or (7.10× 109) (1)

calculation of both potential energy values (1) subtraction of values or use of mΔV with correct answer (1)] 6 [8]

Q33. (a) (i) h (= ct) (= 3.0 × 108 × 68 × 10–3) = 2.0(4) × 107 m (1)

(ii) g = (–) (1) r (= 6.4 × 106 + 2.04 × 107) = 2.68 × 107 (m) (1) (allow C.E. for value of h from (i) for first two marks, but not 3rd)

g = (1) (= 0.56 N kg–1) 4

(b) (i) g = (1)

v = [0.56 × (2.68 × 107)]½ (1)

= 3.9 × 103m s–1 (1) (3.87 × 103 m s–1)

(allow C.E. for value of r from a(ii)

[or v2 = = (1)

v = (1)

= 3.9 × 103 m s–1 (1)]

(ii) = (1)

= 4.3(5) × 104s (1) (12.(1) hours) (use of v = 3.9 × 103 gives T = 4.3(1) × 104 s = 12.0 hours) (allow C.E. for value of v from (I)

[alternative for (b):

(i) (1)

= 3.8(6) × 103 m s–1 (1)]

(allow C.E. for value of r from (a)(ii) and value of T)

(ii) (1)

= (1.90 × 109 (s2) (1)

T = 4.3(6) × 104 s (1) 5 [9]

Q34. (a) (i) –31 MJ kg–1 (1)

(ii) increase in potential energy = mΔV (1) = 1200 × (62 – 21) × 106 (1) = 4.9 × 1010 J (1) (4)

(b) (i) g = – (1)

(ii) g is the gradient of the graph = (1)

= 2.44 N kg–1 (1)

(iii) g and R is doubled (1)

∝

expect g to be = 2.45 N kg–1 (1)

[alternative (iii)

g and R is halved (1)

expect∝ g to be 2.44 × 4 = 9.76 N kg–1 (1)] (5) [9]

Q35. (a) attractive force between point masses (1) proportional to (product of) the masses (1) inversely proportional to square of separation/distance apart (1) 3

(b) mω2R = (–) (1)

(use of T = gives) (1)

G and M are constants, hence T2 R3 (1) 3

Tm = 87(.5) days (1)

12 (ii) (1) (gives RN = 4.52 × 10 m)

ratio = = 30(.1) (1) 4