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1May June 2003(a) Define gravitational potential.ya.Shak. [2](b) Explain why values of gravitational potential near to an isolated mass are all negative.dra.Chan. [3](c) The Earth may be assumed to be an isolated sphere of radius 6.4 103 km with its massof 6.0 1024 kg concentrated at its centre. An object is projected vertically from thesurface of the Earth so that it reaches an altitude of 1.3 104 km.Sathe change in gravitational potential,andrearrangedby(i)jitCalculate, for this object,change in potential . J kg–1the speed of projection from the Earth’s surface, assuming air resistance isnegligible.mpiled(ii)Co1speed . m s–1[5]ForExaminer’sUse

2(d) Suggest why the equationyav 2 u 2 2asis not appropriate for the calculation in (c)(ii).Shak.CompiledandrearrangedbySajitChandra. [1]ForExaminer’sUse

3May June 2006The Earth may be considered to be a uniform sphere with its mass M concentrated at itscentre. . GMr [2]bySajitChandrav Shak(a) Show that the linear speed v of the satellite is given by the expressionyaA satellite of mass m orbits the Earth such that the radius of the circular orbit is r.(b) For this satellite, write down expressions, in terms of G, M, m and r, forits kinetic energy,potential energy . [1]its total energy.mp(iii)kinetic energy . [1]its gravitational potential energy,iled(ii)andrearranged(i)Co2total energy . [2]ForExaminer’sUse

4(c) The total energy of the satellite gradually decreases.(i)yaState and explain the effect of this decrease onthe radius r of the orbit,Shak.(ii)dra. [2]the linear speed v of the satellite.Chan.CompiledandrearrangedbySajit. [2]ForExaminer’sUse

5May June 2007(a) Explain what is meant by a gravitational field.ya.Shak. [1]dra(b) A spherical planet has mass M and radius R. The planet may be considered to have allits mass concentrated at its centre.A rocket is launched from the surface of the planet such that the rocket moves radiallyaway from the planet. The rocket engines are stopped when the rocket is at a height Rabove the surface of the planet, as shown in Fig. 1.1.RSaFig. 1.1jitplanetChan2RRShow that, for the rocket to travel from a height R to a height 2R above the planet’ssurface, the change ΔEP in the magnitude of the gravitational potential energy ofthe rocket is given by the expressioned(i)byThe mass of the rocket, after its engines have been stopped, is m.ngGMm.6RmpiledandrearraΔEP Co3[2]ForExaminer’sUse

6During the ascent from a height R to a height 2R, the speed of the rocket changesfrom 7600 m s–1 to 7320 m s–1. Show that, in SI units, the change ΔEK in the kineticenergy of the rocket is given by the expressionya(ii)Use the expressions in (b) to determine a value for the mass M of the planet.(ii)M kg [2]andrearrangedby(i)[1]Sa(c) The planet has a radius of 3.40 106 m.jitChandraShakΔEK (2.09 106)m.State one assumption made in the determination in (i).iled.Comp. [1]ForExaminer’sUse

7October November 2005[3]jitChandra(a) Show that the radius of the geostationary orbit is 4.2 107 m.ShakyaThe Earth may be considered to be a sphere of radius 6.4 106 m with its mass of6.0 1024 kg concentrated at its centre.A satellite of mass 650 kg is to be launched from the Equator and put into geostationaryorbit.andrearrangedbySa(b) Determine the increase in gravitational potential energy of the satellite during its launchfrom the Earth’s surface to the geostationary orbit.energy . J [4](c) Suggest one advantage of launching satellites from the Equator in the direction ofrotation of the Earth.mpiled.[1]Co5ForExaminer’sUse

8October November 20066The definitions of electric potential and of gravitational potential at a point have somesimilarity.(a) State one similarity between these two definitions. [1](b) Explain why values of gravitational potential are always negative whereas values ofelectric potential may be positive or negative. [4]ForExaminer’sUse

9May June 2004R2CM2M1ChandraR1ShakyaA binary star consists of two stars that orbit about a fixed point C, as shown in Fig. 3.1.jitFig. 3.1SaThe star of mass M1 has a circular orbit of radius R1 and the star of mass M2 has a circularorbit of radius R2. Both stars have the same angular speed ω, about C.the gravitational force between the two stars,ed(i)by(a) State the formula, in terms of G, M1, M2, R1, R2 and ω for.ng(ii) the centripetal force on the star of mass M1.rra.[2]mpiledandrea(b) The stars orbit each other in a time of 1.26 108 s (4.0 years). Calculate the angularspeed ω for each star.Co7angular speed . rad s–1 [2]ForExaminer’sUse

10(c) (i)Show that the ratio of the masses of the stars is given by the expression(ii)The ratioChandraShakyaM1R 2.M2R1[2]M1is equal to 3.0 and the separation of the stars is 3.2 1011 m.M2R1 . mR2 . m[2]By equating the expressions you have given in (a) and using the data calculated in(b) and (c), determine the mass of one of the stars.mass of star . kgCompiledandrea(d) (i)rrangedbySajitCalculate the radii R1 and R2.(ii)State whether the answer in (i) is for the more massive or for the less massive star.[4]ForExaminer’sUse

11October November 2002By equating the kinetic energy of the object at the planet’s surface to its total gainof potential energy in going to infinity, show that the escape speed v is given byv2 Shak(a) (i)yaIf an object is projected vertically upwards from the surface of a planet at a fast enoughspeed, it can escape the planet’s gravitational field. This means that the object can arrive atinfinity where it has zero kinetic energy. The speed that is just enough for this to happen isknown as the escape speed.2GM,RHence show thatby(ii)SajitChandrawhere R is the radius of the planet and M is its mass.edv 2 2Rg,mpiledandrearrangwhere g is the acceleration of free fall at the planet’s surface.Co8[3]ForExaminer’sUse

12(b) The mean kinetic energy Ek of an atom of an ideal gas is given byyaEk 32 kT,where k is the Boltzmann constant and T is the thermodynamic temperature.ShakUsing the equation in (a)(ii), estimate the temperature at the Earth’s surface such thathelium atoms of mass 6.6 10–27 kg could escape to infinity.jitChandraYou may assume that helium gas behaves as an ideal gas and that the radius of Earth is6.4 106 m.bySatemperature . K [4]ForExaminer’sUse

13October November 2006A rocket is launched from the surface of the Earth.h1 19.9 106v1 5370h2 22.7 106v2 5090Shakspeed / m s–1draheight / myaFig. 4.1 gives data for the speed of the rocket at two heights above the Earth’s surface, afterthe rocket engine has been switched off.ChanFig. 4.1The Earth may be assumed to be a uniform sphere of radius R 6.38 106 m, with its massM concentrated at its centre. The rocket, after the engine has been switched off, hasmass m.G, M, m, h1, h2 and R for the change in gravitational potential energy of the rocket,Sa(i)jit(a) Write down an expression in terms ofm, v1and v2 for the change in kinetic energy of the rocket.ed(ii)by. [1]. [1]mpiledandrearrang(b) Using the expressions in (a), determine a value for the mass M of the Earth.Co9M kg [3]ForExaminer’sUse

14akyaForExaminer’sUsendraSh10 A spherical planet has mass M and radius R.The planet may be assumed to be isolated in space and to have its mass concentrated at itscentre.The planet spins on its axis with angular speed ω, as illustrated in Fig. 1.1.ajitCRhamass mpole ofplanetdbySequator ofplanetFig. 1.1rrangeA small object of mass m rests on the equator of the planet. The surface of the planet exertsa normal reaction force on the mass.(a) State formulae, in terms of M, m, R and ω, for(i)the gravitational force between the planet and the object,(ii)rea. [1]the centripetal force required for circular motion of the small mass,(iii)dand. [1]the normal reaction exerted by the planet on the mass.Explain why the normal reaction on the mass will have different values at theequator and at the poles.mp(b) (i)ile. [1].Co. [2] UCLES 20089702/04/O/N/08

15(ii)The radius of the planet is 6.4 106 m. It completes one revolution in 8.6 104 s.Calculate the magnitude of the centripetal acceleration atndraShakya1. the equator,haacceleration .m s–2 [2]rrangedbySajitC2. one of the poles.acceleration .m s–2 [1](c) Suggest two factors that could, in the case of a real planet, cause variations in theacceleration of free fall at its surface.rea1. .dand2. .Compile.[2] UCLES 20089702/04/O/N/08ForExaminer’sUse

1611ShakyaForExaminer’sUse(a) Define gravitational field strength. [1]bySajitChandra(b) A spherical planet has diameter 1.2 104 km. The gravitational field strength at thesurface of the planet is 8.6 N kg–1.The planet may be assumed to be isolated in space and to have its mass concentratedat its centre.Calculate the mass of the planet.edmass . kg [3]State, with a reason, whether point X or point Y is nearer to the planet.ea(i)rrang(c) The gravitational potential at a point X above the surface of the planet in (b) is– 5.3 107 J kg–1.For point Y above the surface of the planet, the gravitational potential is– 6.8 107 J kg–1.dr.A rock falls radially from rest towards the planet from one point to the other.Calculate the final speed of the rock.Compiled(ii)an. [2]speed . m s–1 [2] UCLES 20099702/04/M/J/09

1712ShakyaForExaminer’sUse(a) State Newton’s law of gravitation.Chandra. [2](b) The Earth may be considered to be a uniform sphere of radius R equal to 6.4 106 m.A satellite is in a geostationary orbit.(i)Describe what is meant by a geostationary orbit.jit.Sa.(ii)by. [3]Show that the radius x of the geostationary orbit is given by the expressionedgR 2 x 3ω 2andrearrangwhere g is the acceleration of free fall at the Earth’s surface and ω is the angularspeed of the satellite about the centre of the Earth.iledDetermine the radius x of the geostationary orbit.Comp(iii)[3]radius . m [3] UCLES 20099702/41/O/N/09

1813ShakyaForExaminer’sUse(a) The Earth may be considered to be a uniform sphere of radius 6.38 103 km, with itsmass concentrated at its centre.(i)Define gravitational field strength.Chandra. [1]By considering the gravitational field strength at the surface of the Earth, show thatthe mass of the Earth is 5.99 1024 kg.[2]ngedbySajit(ii)Use data from (a) to calculate the angular speed of a GPS satellite in its orbit.angular speed . rad s–1 [3]Compiledandrea(i)rra(b) The Global Positioning System (GPS) is a navigation system that can be used anywhereon Earth. It uses a number of satellites that orbit the Earth in circular orbits at a distanceof 2.22 104 km above its surface. UCLES 20099702/42/O/N/09

ForExaminer’sUse14(a) Define gravitational potential at a point.Shakya. [2]Chandra(b) The Earth may be considered to be an isolated sphere of radius R with its massconcentrated at its centre.The variation of the gravitational potential φ with distance x from the centre of the Earthis shown in Fig. 1.1.distance x0R2R3R4R5RSajit0by–2.0ed/ 107 J kg–1rang–4.0ndrear–6.0ileda–8.0Fig. 1.1The radius R of the Earth is 6.4 106 m.By considering the gravitational potential at the Earth’s surface, determine a valuefor the mass of the Earth.Comp(i)mass . kg [3] UCLES 20109702/43/M/J/10

(ii)A meteorite is at rest at infinity. The meteorite travels from infinity towards theEarth.ForExaminer’sUseraShakyaCalculate the speed of the meteorite when it is at a distance of 2R above the Earth’ssurface. Explain your working.In practice, the Earth is not an isolated sphere because it is orbited by the Moon, asillustrated in Fig. 1.2.jit(iii)Chandspeed . m s–1 [4]MoonEarthngedbySainitial pathof meteoriteraFig. 1.2 (not to scale)earThe initial path of the meteorite is also shown.ndrSuggest two changes to the motion

The planet may be assumed to be isolated in space and to have its mass concentrated at its centre. The planet spins on its axis with angular speed ω, as illustrated in Fig. 1.1. R mass m pole of planet equator of planet Fig. 1.1 A small object of mass m rests on the equator of the planet. The s

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