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(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.
Use the graph to calculate
(i) the gravitational potential at a distance 2 R from the centre of the Earth,
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(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 3 R.
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(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
2 R from the centre of the Earth.
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(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) (a) (i) Explain what is meant by gravitational field strength.
(1)
(ii) Describe how you would measure the gravitational field strength close to the surface of the Earth. Draw a diagram of the apparatus that you would use. (6)
(b) (i) The Earth’s gravitational field strength ( g ) at a distance ( r ) of 2.0 × 10^7 m from its
centre is 1.0 N kg–1. Complete the table with the 3 further values of g.
g /N kg–1^ 1.
r /10^7 m 2.0^ 4.0^ 6.0^ 8.
(2)
(b) A space vehicle has a mass of 16 800 kg and is in orbit 900 km above the surface of the Earth. mass of the Earth = 5.97 × 10^24 kg radius of the Earth = 6.38 × 10^6 m (i) Show that the orbital speed of the vehicle is approximately 7400 m s–1. (4) (ii) The space vehicle moves from the orbit 900 km above the Earth’s surface to an orbit 400 km above the Earth’s surface where the orbital speed is 7700 m s –1. Calculate the total change that occurs in the energy of the space vehicle. Assume that the vehicle remains outside the atmosphere after the change of orbit. Use the value of 7400 m s–1^ for the speed in the initial orbit. change in energy ____________________ J (4) (Total 10 marks)
Figure 1 shows (not to scale) three students, each of mass 50.0 kg, standing at different points on the Earth’s surface. Student A is standing at the North Pole and student B is standing at the equator. Figure 1 Figure 2 The radius of the Earth is 6370 km. The mass of the Earth is 5.98 × 10^22 kg. In this question assume that the Earth is a perfect sphere. (a) (i) Use Newton’s gravitational law to calculate the gravitational force exerted by the Earth on a student. force ____________________ N (3)
(ii) Figure 2 shows a closer view of student A. Draw, on Figure 2 , vector arrows that represent the forces acting on student A. (2)
The graph below shows how the gravitational potential energy, E p, 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. (a) Explain why the values of E p are negative.
(2)
(b) Use data from the graph to determine the mass of Mars. mass of Mars ____________________ kg (3)
(c) Calculate the escape velocity for an object on the surface of Mars. escape velocity ____________________ m s– (3) (d) Show that the graph data agree with (3) (Total 11 marks) (a) (i) State the relationship between the gravitational potential energy , E p, 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 E p and V if the mass m is doubled? value of E p _____________________________________________________ value of V _____________________________________________________ (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. (1) (c) Explain why it is not possible to use the equation ∆ E p = mg ∆ h when determining the change in the gravitational potential energy of a satellite as it moves between these orbits.
(1) (Total 10 marks) The gravitational field strength at the surface of a planet, X, is 19 N kg–1. (a) (i) Calculate the gravitational potential difference between the surface of X and a point 10 m above the surface, if the gravitational field can be considered to be uniform over such a small distance.
(ii) Calculate the minimum amount of energy required to lift a 9.0 kg rock a vertical distance of 10 m from the surface of X.
(iii) State whether the minimum amount of energy you have found in part (ii) would be different if the 9.0 kg mass were lifted a vertical distance of 10 m from a point near the top of the highest mountain of planet X. Explain your answer.
(3)
(b) Calculate the gravitational field strength at the surface of another planet, Y, that has the same mass as planet X, but twice the diameter of X.
(2) (Total 5 marks) (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 × 10^3 kg m– (2) (ii) Calculate the gravitational force between the spheres when their surfaces were in contact. answer = ______________________ N (2)
Centripetal acceleration a Speed v
A
B
C
D
(Total 1 mark) A satellite orbiting the Earth moves to an orbit which is closer to the Earth. Which line, A to D , in the table shows correctly what happens to the speed of the satellite and to the time it takes for one orbit of the Earth? Speed of satellite Time For One Orbit Of Earth A decreases decreases B decreases increases C increases decreases D increases increases (Total 1 mark)
A spacecraft of mass m is at the mid-point between the centres of a planet of mass M 1 and its
moon of mass M 2. If the distance between the spacecraft and the centre of the planet is d , what
is the magnitude of the resultant gravitational force on the spacecraft? A B C D (Total 1 mark)
Which one of the following statements about gravitational potential is correct? A gravitational potential can have a positive value B the gravitational potential at the surface of the Earth is zero C the gravitational potential gradient at a point has the same numerical value as the gravitational field strength at that point D the unit of gravitational potential is N kg– (Total 1 mark)
When a space shuttle is in a low orbit around the Earth it experiences gravitational forces F E due to the Earth, F M due to the Moon and F S due to the Sun. Which one of the following correctly shows how the magnitudes of these forces are related to each other? mass of Sun = 1.99 × 10^30 kg mass of Moon = 7.35 × 10^22 kg mean distance from Earth to Sun = 1.50 × 10^11 m mean distance from Earth to Moon = 3.84 × 10^8 m A F E > F S > F M B F S > F E > F M C F E > F M > F S D F M > F E > F S (Total 1 mark)
Mark schemes
(a) (i) –31 MJ kg–1^ (1)
(ii) increase in potential energy = m Δ V (1)
= 1200 × (62 – 21) × 10^6 (1)
= 4.9 × 10^10 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] (a) (i) force per unit mass (allow equation with defined terms) B (1)
(ii) diagram of method that will work (pendulum / light gates / solenoid and mechanical gate / strobe photography / video) B pair of measurements (eg length of pendulum and (periodic) time / distance and time of fall – could be shown on diagram) M instruments to measure named quantities (may be on diagram) A correct procedure (eg calculate period for range of lengths, measure the time of fall for range of heights) B good practice – series of values and averages / use of gradient of graph B
appropriate formula and how g calculated
B (6)
(b) (i) evidence of gr^2 being used
C values of 0.25, 0.11, 0.06(25) no s.f. penalty here unless values given as fractions A (2) (ii) points correctly plotted on grid (e.c.f.) B
smooth curve of high quality at least to 10 × 10^7 m, no intercept on r axis
B (2) (iii) attempt to use area under curve B evidence of × 800 kg B (4.3 – 5.3) × 10^9 J B or
use of equation for potential Δ EG = m ( g 1 r 1 – g 2 r 2 )
B evidence of × 800 kg B (4.7 – 4.9) × 10^9 J B
max 2 if assumed values of G and M used
allow calculation of GM from graph followed by substitution into Δ EG = MG(m /
r 1 – m / r 2 ) for 3 marks
(3) [14] (a) Idea that both astronaut and vehicle are travelling at same (orbital) speed or have the same (centripetal) acceleration / are in freefall Not falling at the same speed B No (normal) reaction (between astronaut and vehicle) B 2
(ii) ΔPE = 6.67 × 10−11^ × 5.97 × 10^24 × 1.68 × 10^4 (1 / (7.28 × 106 ) − 1 / (6.78 × 10^6 ) ) C −6.8 × 10^10 J C ΔKE =0.5 × 1.68 × 10^4 ×(7700^2 −7400^2 ) = 3.81 × 10^10 J C ΔKE − ΔPE = (−) 2.99 × 10^10 (J) A OR Total energy in original orbit shown to be (−) GMm / 2r or mv^2 / 2 − GMm / r C Initial energy = − 6.67 × 10−11^ × 5.97 × 10^24 × 1.68 × 10^4 / (2 × 7.28 × 10^6 ) = 4.59 × 10^11 C Final energy = − 6.67 × 10−11^ × 5.97 × 10^24 × 1.68 × 10^4 / (2 × 6.78 × 10^6 ) = 4.93 × 10^11 3.4 × 10^10 (J) Condone power of 10 error for C marks A 4 [10]
(a) (i) Use of F – GMm/r^2 C Allow 1 for -correct formula quoted but forgetting square in substitution Correct substitution of data M -missing m in substitution 491 (490)N A -substutution with incorrect powers of 10 Condone 492 N, (ii) Up and down vectors shown (arrows at end) with labels B allow W, mg (not gravity); R allow if slightly out of line / two vectors shown at feet up and down arrows of equal lengths B condone if colinear but not shown acting on body In relation to surface W ≤ R (by eye) to allow for weight vector starting in middle of the body Must be colinear unless two arrows shown in which case R vectors ½ W vector(by eye) (b) (i) Speed = 2 πr / T B Max 2 if not easy to follow 2π6370000 / (24 × 60 × 60) B 463 m s− B Must be 3sf or more (ii) Use of F = mv^2 /r C Allow 1 for use of F = mrω^2 with ω= 460
