**Gravitational fields**

__A2__

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**

__A2__

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:

__University__

Potential energy can also be given the symbol *U* or *W*. The change in potential energy is given using integration (we take *r _{1}* 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, *r _{s} *(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**

__A2__

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:

** **

__University__

This equation gives the proof for Kepler’s third law, that *T*^{2} is proportional to *r*^{3} (we can see this as the left hand side is constant). Kepler has three laws:

- Planets move in elliptical orbits with the sun at one focus
- A line from the sun to the planet sweeps out equal areas in equal time
*T*^{2}α*r*^{3}

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

since

Back to Contents: Physics: Mechanics