Topic 6: Fields and Forces 6.1 Gravitational force and field

Newton ’ s universal law of gravitation Newton proposed that a force of attraction exists between any two masses. This force law applies to point masses not extended masses However the interaction between two spherical masses is the same as if the masses were concentrated at the centres of the spheres.

Newton´s Law of Universal Gravitation Newton proposed that “every particle of matter in the universe attracts every other particle with a force which is directly proportional to the product of their masses, and inversely proportional to the square of their distance apart”

i.e. & Where m 1 and m 2 are the masses of the 2 objects and r is the distance between them

This can be written as F = G m 1 m 2 r 2 Where G is Newton´s constant of Universal Gravitation It has a value of 6.67 x Nm 2 kg -2

Define gravitational field strength. A mass M creates a gravitational field in space around it. If a mass m is placed at some point in space around the mass m it will experience the existance of the field in the form of a gravitational force

We define the gravitational field strength at a point in a gravitational field as the force acting on a 1kg mass placed at that point.

The force experienced by a mass m placed a distance r from a mass M is F = G Mm r 2 And so the gravitational field strength of the mass M is given by dividing both sides by m g = G M r 2

The units of gravitational field strength are N kg -1 The gravitational field strength is a vector quantity whose direction is given by the direction of the force a mass would experience if placed at the point of interest

Variation of g with distance from a point mass, M Gravitational field strength, g = GM/r²

Variation of g with distance from the centre of a uniform spherical mass of radius, R

Variation of g on a line joining the centres of two point masses s

In order to calculate the overall gravitational field strength at any point, we must use vector addition. The overall gravitational field strength at any point b/w the Earth and the Moon must be a result of both pulls. g = F resultant /m

There will be a single point somewhere b/w the Earth and the Moon where the total gravitational field due to these two masses is zero.

Radius R Planet of mass M Relying on the fact that the sphere behaves as a point mass situated at its centre, from Newton’s Law of gravitation, For an object of mass m on the surface of the planet, the gravitational force is F = G Mm R 2

Field Strength at the Surface of a Planet Since gravitational field strength is defined as the gravitational force acting on 1kg of mass, The field strength on the surface of the planet is g = G M R 2

If the planet is the Earth then we have the gravitational field strength on the surface of Earth as g = G M e R e 2 where M e and R e are the mass and radius of the earth respectively. M e = 6.0 x kg R e = 6.4 x 10 6 m

What is the relation between the Acceleration due to Gravity and the Universal Constant of Gravitation? From newton’s 2nd law, F = mg We also know F = G M e m R e 2 hence we have that the acceleration of free fall at the surface of the Earth, g g = G M e R e 2