This question is about the planet Jupiter and one of the moons that orbits it, called lo. lo orbits Jupiter at a speed, v of 1.7 x 104 m s-1 at an orbital radius, r, of 4.2 x 108m. a) Show that lo takes approximately 43 hours to orbit Jupiter once. (2 Marks) b) lo is held in its orbit by a centripetal force,, where m is the mass of lo. This force is the gravitational attraction between lo and Jupiter. i) Show thatwhere M is the mass of Jupiter. ii) Show that the mass of Jupiter is about 2.0 x 1027 kg. (4 Marks) c) Show that the gravitational potential at the top of Jupiter' s atmosphere, 7.1 x 107 m from the centre of the planet, is about -2 x 109 Jkg-1. Assume that Jupiter is a sphere. (2 Marks) d) In July 1994, comet Shoemaker-Levy 9 crashed into Jupiter causing dramatic heating of the planet' s atmosphere. During the approach to the planet, the comet broke up. One piece that struck the planet had a mass of about 4 x 1012 kg. This fragment crossed the orbit of lo heading directly towards Jupiter with a velocity of 10 km s-1. i) Show that kinetic energy of the fragment at that moment is 2 x 1020 J. (1 Mark) ii) Explain why the fragment will enter the atmosphere of Jupiter with a velocity greater than 10km s-1. (2 Marks) (Marks available: 11) a) A cyclist goes round a circular track, of radius 10 m, at a constant speed of 8.0 m s-1. What is the acceleration of the cyclist and what is its direction? (3 Marks) b) What is the resultant force on the cycle and the rider if together they have a mass of 90 kg? (2 Marks) (Marks available: 5) Answer Answer outline and marking scheme for question: a) Time = 2ð r/1.7 x 104 = 155230s = 43.1 hours (2 Marks) b) i) -GMm/r2 = -mv2/r So: Gm/r2 = v2/r Therefore M = v2r/G ii) M = (1.7 x 104 )2 x 4.2 x 108/ 6.67 x 10-11 = 1.82 x 1027 kg (4 Marks) c) Vg = -GM/r = -6.67 x 10-11 x 1.9 x 1027/7.1 x 107 = - 1.79 x 109 K kg-1 (2 Marks) d) i) ½ mv2 = ½ x 4 x 1012 x 100002 = 2 x 1020 J (1 Mark) ii) The fragment will have gained kinetic energy (1 Mark) as it lost gravitational potential energy during the approach to the planet (1 Mark). (Or force argument: Attracted by gravity (1 Mark) causes it to accelerate (1 Mark) (2 Marks) (Marks available: 11) a) a = v2 = 82 = 6.4 m s-2 r 10 Direction: towards the centre of the track. (3 Marks) b) F = ma (1 Mark) = 90 x 6.4 = 576 N (1 Mark) (2 Marks) (Marks available: 5)