# Gravity

Gravitational fields

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All masses exert a force on each other due to gravity. Since the masses are not in contact with each other, the force is conveyed by a gravitational field. These can be represented by field lines, which show the direction that a test mass (a mass small enough not to affect the shape of the field with its own gravity) would move in the field. There are two main types of field: radial (all the field lines converge at a point) and uniform (the field lines are parallel and evenly spaced:

The spacing of the field lines represents the gravitation field strength, g. This is the force per unit mass on a small test mass placed in the field, measured in Nkg-1:

This means that in a uniform field, where the lines are evenly spaced, the field strength is equal at all points, and in a radial field, the field strength increases nearer to the mass.

Newton defined the force due to a gravitational field in terms of a universal constant of gravitation, G:

This can be substituted into our original equation for g to get g for a spherical mass of radius R when r is greater than R (the mass can then be considered to all be at the centre):

Gravitational potential

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Gravitational potential energy, E, at a point in space is the work done to move a small object from infinity (where the field strength is zero) to that point (measured in joules, J):

Gravitational potential, V, is gravitational potential energy per unit mass. It is measured in Jkg-1

In a uniform field:

In a radial field:

Lines of equal potential, or equipotentials, can be drawn onto field diagrams. Equipotentials cross field lines at right angles:

The potential gradient is the change of potential per unit distance at that point:

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Potential energy can also be given the symbol U or W. The change in potential energy is given using integration (we take r1 to be infinity):

We can also calculate the escape velocity (velocity needed to have enough energy to escape) using conservation of energy:

If the speed is the speed of light, the Schwarzchild radius, rs (the radius of a black hole) is found:

The total energy of an object in a circular orbit is given by (using the escape velocity):

Orbits

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A satellite is a body that orbits another body. Two bodies in orbit are actually orbiting each other, but one is often much more massive, so moves a very small amount in comparison with the other mass. We can use the equations from circular motion to derive:

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This equation gives the proof for Kepler’s third law, that T2 is proportional to r3 (we can see this as the left hand side is constant). Kepler has three laws:

1. Planets move in elliptical orbits with the sun at one focus
2. A line from the sun to the planet sweeps out equal areas in equal time
3. T2 α r3

We can prove Kepler’s second law using conservation of angular momentum:

since

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