
Mechanics Questions
Orbital Motion
A "geostationary" orbit is one which has the same angular velocity as the rotation of the planet on its axis.
Question 1
Considering the orbit of the earth to be a circle of radius 1·5×10^{8 }km and time period 365·25 days, 
a) 
calculate the angular velocity of the earth 
b) 
calculate the mass of the sun. 
Question 2
A planet has a radius R = 5000km. A satellite is in orbit 10000km above the surface of the planet. The time period of the satellite's motion is 12·39 hours. Calculate the average density of the planet. 
Question 3
A satellite has a mass m = 300kg and is in a geostationary orbit around a planet of mass M = 8×10^{24 }kg. One "day" on this planet is 30 of your earthly hours (I speak as an inhabitant of the planet, which is called Yob and is many light years from here; but that is another story…) 
a) 
calculate the radius of the orbit 
b) 
calculate the total energy possessed by the satellite 
c) 
calculate the kinetic energy possessed by the satellite 
d) 
calculate the speed of the satellite, in kmh^{1}. 
Question 4
The satellite in the previous question now falls to an orbit of radius 10^{7}m. 
a) 
Calculate the total energy possessed by the satellite in its new orbit, and 
b) 
the amount of work done on the satellite to cause it to fall to the new orbit. 
Question 5
a) 
State Kepler’s laws of planetary motion. 
b) 
At aphelion, the distance of the earth from the sun is 1·51×10^{8}km. At perihelion, it is 1·46×10^{8}km. For a short time near perihelion (e.g. a day) let the kinetic energy of the earth be K_{p} and for the same time near aphelion let it be K_{a}.
Calculate the ratio K_{p}/K_{a}. 
Question 6
A star has two planets in orbit around it. Planet 1 has a time period of two (Earth) years and the average radius of its orbit is 2×10^{8 }km. Planet 2 is in an orbit of average radius 6×10^{8 }km. 
a) 
Calculate the time period of planet 2. 
b) 
Calculate the mass of the star. 
Question 7
a) 
The distance from the centre of the earth to the centre of the moon is approximately 3·84×10^{8}m. The time period of the moon’s orbital motion is 27·3days. Calculate the magnitude of the centripetal acceleration of the moon as it moves round the earth. 
b) 
If the radius of the earth is R then the distance from the centre of the earth to the centre of the moon is about 60R (see diagram below). 

Using this information, show that your answer to part a) can be used as evidence for Newton’s inverse square law of gravitation. 

