Gravitation
1 Show that the force of attraction between two large ships of mass 50000 tonnes and separated by a distance of 20 m is 417 N. (1 tonne = 1000 kg)
2 Calculate the gravitational force of attraction between the proton and the electron in a hydrogen atom. Assume the electron is describing a circular orbit with a radius of 5.3 x 10-11 m.
(mass of proton = 1.67 x 10-27 kg; mass of electron = 9.11 x 10-31 kg).
3 A satellite, of mass 1500 kg, is moving at constant speed in a circular orbit 160 km above the Earth’s surface.
(a) Calculate the period of rotation of the satellite.
(b) Calculate the total energy of the satellite in this orbit.
(c) Calculate the minimum amount of extra energy required to boost this satellite into a geostationary orbit which is at a distance of 36 000 km above the Earth’s surface.
4 The planet Mars has a mean radius of 3.4 x 106 m. The Earth’s mean radius is 6.4 x 106 m. The mass of Mars is 0.11 times the mass of the Earth.
(a) How does the mean density of Mars compare with that of the Earth?
(b) Calculate the value of “g” on the surface of Mars.
(c) Calculate the escape velocity on Mars.
5 Determine the potential energy between the planet Saturn and its rings.
The mass of Saturn is 5.72 x 1026 kg. The rings have a mass of 3.5 x 1018 kg and are concentrated at an average distance of 1.1 x 108 m from the centre of Saturn.
6 During trial firing of Pioneer Moon rockets, one rocket reached an altitude of 125 000 km.
Neglecting the effect of the Moon, estimate the velocity with which this rocket struck the atmosphere of the Earth on its return. (Assume that the rocket’s path is entirely radial and that the atmosphere extends to a height of 130 km above the Earth’s surface).
7 (a) Sketch the gravitational field pattern between the Earth and Moon.
(b) Gravity only exerts attractive forces. There should therefore be a position between the Earth and Moon where there is no gravitational field – a so-called ‘null’ point.
By considering the forces acting on a mass m placed at this point, calculate how far this position is from the centre of the Earth.
8 Mars has two satellites named Phobos and Deimos. Phobos has an orbital radius of 9.4 x 106 m and an orbital period of 2.8 x 104 s.
Using Kepler’s third law ( r³/T² = constant ), calculate the orbital period of Deimos which has an orbital radius of 2.4 x 107 m.
9 When the Apollo 11 satellite took the first men to the Moon in 1969 its trajectory was very closely monitored.
The satellite had a velocity of 5374 ms-1 when 26306 km from the centre of the Earth and this had dropped to 3560 ms-1 when it was 54368 km from the centre of the Earth. The rocket motors had not been used during this period.
Calculate the gravitational potential difference between the two points. Remember that the unit of gravitational potential is Jkg-1.
10 Show that an alternative expression for the escape velocity from a planet may be given by: v = √(2gR), where g = the planet’s surface gravitational attraction and R = the radius of the planet.
11 The Escape Velocity for the Earth v = √(2gr), or v = √(2GM/r)
Using data on the Earth, show that the escape velocity equals 1.1 x 104 ms-1.
12 Show that a satellite orbiting the Earth at a height of 400 km has an orbital period of 93 minutes. Note that a height of 400 km is equal to a radius of R + 400 km.
13 (a) A geostationary orbit has a period of approximately 24 hours. Find the orbital radius for a satellite in such an orbit.
(b) Hence find the height of this satellite above the Earth.
(c) Show on a sketch of the Earth the minimum number of geostationary satellites needed for world-wide communication.
