The ISS orbits the Earth at an average distance of 408 km from the surface of the Earth.
The following data are available:
Average distance between the centre of the Earth and the centre of the Moon = 3.80 × 108 m
Mass of the Earth = 5.97 × 1024 kg
Radius of the Earth = 6.37 × 106 m
(b)
Calculate the maximum gravitational field strength experienced by the ISS. You may assume that both the Moon and the ISS can be positioned at any point on their orbital path.
Two planets X and Y have the same mass. Planet X has a radius R and the gravitational field strength on its surface is g. The radius of planet Y is twice that of planet X and the gravitational field strength at the surface of planet Y is one fifth of the value of the gravitational field strength on X.
(d)
Use the equation you derived in part (c) to show that the volume of planet Y is 10 times larger than the volume of planet X.
The gravitational field strength on the surface of a particular moon is 2.5 N kg–1. The moon orbits a planet of similar density, but the diameter of the planet is 50 times greater than the moon.
(a)
Calculate the gravitational field strength at the surface of the planet.
The distance between the Sun and Mercury varies from 4.60 × 1010 m to 6.98 × 1010 m. The gravitational attraction between them is F when they are closest together.
(a)
Show that the minimum gravitational force between the Sun and Mercury is about 43% of F.
Mercury has a mass of 3.30 × 1023 kg and a mean diameter of 4880 km. A rock is projected from its surface vertically upwards with a velocity of 6.0 m s–1.
(
b)
Calculate how long it will take for the rock to return to Mercury's surface.
Venus is approximately 5.00 × 1010 m from Mercury and has a mass of 4.87 × 1024 kg. A satellite of mass 1.50 × 104 kg is momentarily at point P, which is 1.75 × 1010 from Mercury, which itself has a mass of 3.30 × 1023 kg.
(c)
Calculate the magnitude of the resultant gravitational force exerted on the satellite when it is momentarily at point P.
Lead has a density of 11.3 × 103 kg m–3. The larger sphere has a radius of 200 mm and a mass of 170 kg. The smaller sphere has a radius of 55 mm.
The surfaces of two lead spheres are in contact with each other, and a third, iron sphere of mass 20 kg and radius 70 mm is positioned such that the centre of mass of all three spheres lie on the same straight line.
(a)
Calculate the distance between the surface of the iron sphere and the surface of the larger lead sphere which would result in no gravitational force being exerted on the larger sphere.
The smaller lead sphere is removed. The separation distance between the surface of the iron sphere and the large lead sphere is r.
(c)
Sketch a graph on the axes provided showing the variation of gravitational field strength g between the surface of the iron sphere and the surface of the lead sphere.
A kilogram mass rests on the surface of the Earth. A spherical region S, whose centre of mass is underneath the Earth's surface at a distance of 3.5 km, has a radius of 2 km. The density of rock in this region is 2500 kg m–3.
(a)
Determine the size of the force exerted on the kilogram mass by the matter enclosed in S, justifying any approximations.
If the region S consisted of oil of density 900 kg m–3 instead of rock, the force recorded on the kilogram mass would reduce by approximately 2.9 × 10–4 N.
(b)
(i)
Suggest how gravity meters may be used in oil prospecting.
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(ii)
Determine the uncertainty within which the acceleration of free fall needs to be measured if the meters are to detect such a quantity of oil.
A spherical hollow is made in a lead sphere of radius R, such that its surface touches the outside surface of the lead sphere on one side and passes through its centre on the opposite side. The mass of the sphere before it was made hollow is M.
(c)
Show that the magnitude of the force F exerted by the spherical hollow on a small mass m, placed at a distance d from its centre, is given by: