Gravitational Fields

Gravitational field lines

These are used in diagrams to show the direction of a force acting on an object in a field, gravitational or otherwise. In the case of gravitational fields, these lines always point to the centre of a point mass and, unless a second large mass is introduced, they are straight. Their spacing from one another indicates the magnitude of the field strength (spaced further out being weaker, and closer together being stronger).

Field lines example question: What would be the difference in the field lines comparing 2 circular objects of equal size, yet 1 has a much greater mass?

They would both have a radial field but the one with a greater mass would have field lines closer together.

Law of gravitation example question: define newtons law of gravitation.

Every point mass attracts to every other point mass with a force that is proportional to the sum of it masses and inversely proportional to the square of the distance between them.

Satellite example question: What height must a geosynchronous satellite of 200kg have from the surface if the Earth?

F=4pi^2 x mr/T^2 = GMm/r^2 r^3=GMT^2/4pi^2 r=42.3x10^6 r - R = (42.3x10^6)-(6.37x10^6) =36000km

Gravitational fields inside a planet

From the surface of a planet, the gravitational field decreases proportional to the square of the distance between the test mass and the planet. However, from the centre of the planet to its surface, the field strength increases proportional to the distance from the centre. This means that the equation g = GM/r^2 does not apply, as r is the distance from the surface. This is due to the fact that at the centre, there is an equal amount of mass surrounding, to create a net grav force of 0. The further out, the more total mass is beneath, and therefor the more grav force.

Gravitational field strength

It is commonly misconstrued as "Gravity", and is what causes a small object to be attracted to a large point of mass. It is proportional to the mass of the object, yet, is inversely proportional to the distance between said mass, and a test mass in it's field. It is defined as the force per unit mass on a "test" mass in a gravitational field. g=GM/r^2.

Gravitational potential

It is the work done per unit mass to move a test mass from infinity to a point in a gravitational field. An object has 0 gravitational potential when its distance from the object is at infinity (r = infinity, V = 0), therefore, an object will always have potential in a gravitational field. The potential of a object is always negative and increases the further away it gets from the centre of the field. However, the change in potential (deltaV) can be positive as well as negative, deltaV = g x deltaR = deltaW/m.

Newtons law of gravitation

Newtons law of gravitation is defined as every point mass attracts to every other point mass by a force that is proportional to the product of the masses and inversely proportional to the square of the distance between them. F=GMm/r^2.

Satellites

The motion of a satellite is heavily related to circular motion, but gravitational fields have a lot to do with it. For a satellite to stay in orbit, it needs a certain velocity relating to its distance from the planet it's orbiting. Its centripetal force must be equal to the gravitational force it's experiencing, otherwise it wont have a circular orbit. Using F=mv^2/r and g=GM/r^2, you can say mv^2/r=GM/r^2 and so, v^2=GM/mr. Geosynchronous satellites are satellites that reside above the Earths equator and they have an orbital period of one day, making them seem stationary, compared to the Earth.

There are different types of fields patterns

The two types of gravitational fields are "Radial" and "Uniform". Radial fields being where all field lines point to a small point, and uniform fields being where all field lines point parallel towards mass. They are not actually very different, because the masses that have major gravitational fields are so large, we can say that, up close, there is such a small change in their angles, that they're basically parallel. A good example of a uniform field would be on the surface of the Earth, where it actually has a radial field, but it's so large, it doesn't matter.

Gravitational potential example question: What is the change in potential for a geosynchronous satellite of mass 200kg.

Trick question (the next one will be harder)! The change in gravitational potential is 0, as it doesn't change height

Gravitational field example question: Calculate the gravitational force for a 20kg object 3200km above the surface.

Using g=GM/r^2 =(6.67x10^-11)(5.98x10^24) / (6.37x10^6 + 3.2x10^6)^2 =4.5 N/kg