Topic 9.2 Gravitational field, potential and energy Mechanics Topic 9.2 Gravitational field, potential and energy

Gravitational Force and Field 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 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”

This can be written as F = G m1m2 r2 Where G is Newton´s constant of Universal Gravitation It has a value of 6.667 x 10-11 Nm2kg-2

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 as the ratio of the force the mass m would experience to the mass, m That is the gravitational field strength at a point, it is the force exerted per unit mass on a particle of small mass placed at that point

The force experienced by a mass m placed a distance r from a mass M is F = G Mm r2 And so the gravitational field strength of the mass M is g = G M r2

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

Field Strength at the Surface of a Planet If we replace the particle M with a sphere of mass M and radius R then relying on the fact that the sphere behaves as a point mass situated at its centre the field strength at the surface of the sphere will be given by g = G M R2

If the sphere is the Earth then we have g = G Me Re2 But the field strength is equal to the acceleration that is produced on the mass, hence we have that the acceleration of free fall at the surface of the Earth, g

Gravitational Energy and Potential We know that the graviational potential energy increases as a mass is raised above the Earth The work done in moving a mass between two points is positive when moving away from the Earth By definition the gravitational potential energy is taken as being zero at infinity

The gravitational potential at any point in the Earth´s field is given by the formula V = - G Me r Where r is the distance from the centre of the Earth (providing r >R) The negative sign allows for the fact that all the potentials are negative as they have to increase to zero

Definition The potential is therefore a measure of the amount of work that has to be done to move particles between points in a gravitational field and its units are J kg –1 The work done is independent of the path taken between the two points in the field, as it is the difference between the initial and final potentials that give the value

Equipotential Surfaces Gravitational Potential Equipotential Points Equipotential Surfaces Gravitational Potential

Escape Speed The escape speed is the speed required for a projectile to leave the Earth´s gravitational attraction. i.e. To get to infinity!

If the potential at the Earth´s surface is V = - G Me Re Then the Ep change to get to infinity is G Me x m Re Where m is the mass of the projectile

For this amount of energy to be gained the projectile must have had an equal amount of Ek Therefore ½mv2 = G Me x m Re v = (2GMe Re But using the fact that g = G Me Re2 Then v = (2gRe )