Examstyle Questions: Circular Motion

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 10^{4} m s^{1} at an orbital radius, r, of 4.2 x 10^{8}m.
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 10^{27} kg.
(4 Marks)
c) Show that the gravitational potential at the top of Jupiter's atmosphere, 7.1 x 10^{7} m from the centre of the planet, is about 2 x 10^{9} Jkg^{1}.
Assume that Jupiter is a sphere.
(2 Marks)
d) In July 1994, comet ShoemakerLevy 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 10^{12} 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 10^{20} 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 outline and marking scheme for question:

a) Time = 2ð r/1.7 x 10^{4}
= 155230s
= 43.1 hours
(2 Marks)
b) i) GMm/r^{2}
= mv^{2}/r
So: Gm/r^{2} = v^{2}/r
Therefore M = v^{2}r/G
ii) M = (1.7 x 10^{4} )^{2} x 4.2 x 10^{8}/ 6.67 x 10^{11}
= 1.82 x 10^{27 }kg
(4 Marks)
c) Vg = GM/r = 6.67 x 10^{11} x 1.9 x 10^{27}/7.1 x 10^{7}
=  1.79 x 10^{9} K kg^{1}
(2 Marks)
d) i) ½ mv^{2 }= ½ x 4 x 10^{12 }x 10000^{2} = 2 x 10^{20} 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 = v^{2} = 8^{2} = 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)