Gravity

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:

scan50001scan50002

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:

scan50003

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:

scan50004

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):

scan50005

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):

scan50006

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

scan50007

In a uniform field:scan50008

In a radial field:scan50009

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

scan50010scan50011

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

scan50012

University

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):

scan50015

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

scan50016

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

scan50017

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

scan50018

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:

 scan50013

scan50014

University

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:

scan50019

sincescan50020

Back to Contents: Physics: Mechanics

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s