6.2 Newton’s Law of Gravitation

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. 

Two planets P and Q are in concentric circular orbits about a star S. 

The radius of P's orbit is R and the radius of Q's orbit is 2R. The gravitational force between P and Q is F when angle SPQ is 90° as shown. 

Planet P is twice the mass of planet Q. 

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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. 

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.

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. 

A student has two unequal, uniform lead spheres. 

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. 

The smaller lead sphere is removed. The separation distance between the surface of the iron sphere and the large lead sphere is r. 

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. 

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. 

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. 

Scientists want to put a satellite in orbit around planet Venus.

The satellite's orbital time, T, and its orbital radius, R, are linked by the equation:

T² = kR³

Venus has a mass of 4.9 × 10²⁴ kg.

One day on Venus is equal to 116 Earth days and 18 Earth hours.