5.4 - Gravitational fields

What is gravity?

Gravity is the universal attractive force which acts between all matter.

What can field lines tell you about a field?

Direction of field lines and strength depending on density of lines.

What is g?

g is the force per unit area in a uniform gravitational field

In a radial field the magnitude of g is the proportionality constant at that point between force and mass

I.e g=GM/r²

What is Newton’s law of gravitation

States that 2 point masses attract each other with a force that is directly proportional to the product of their masses, and inversely proportional to the square of the distances between them.

What is Kepler’s first law?

The orbit of a planet is an ellipse, with the sun at one foci.

The eccentricity of the ellipse is very low, so the motion can be modeled as circular.

What is Kepler’s second law?

A line segment joining a planet and the sun sweeps out equal areas during intervals of equal time.

This is because the speed of the planet is not constant - the planet moves faster when closer to the Sun.

What is Kepler’s third law?

T² ∝ r³. (T₁/T₂)² = (r₁/r₂)³

Square of the orbital period (T) of a planet is directly proportional to the cube of its average distance from the sun ( r ).

Derive T² = 4pi²r³/GM

1) Kepler’s third law, equate formula of centripetal force with formula for gravitational force, mv²/r = GMm/r²

2) Rearrange for v² = GM/r

3) Since velocity in circular motion is 2pir/T, substitute into previous equation to get 4pi²r²/T² = GM/r

4) Rearrange for formula

What are geostationary satellites? What are they used for?

• Orbital period of 24 hours

• Rotate in same direction as Earth’s rotation

• Be in orbit above Earth’s equator

What is gravitational potential?

The potential energy per kilogram, at any point in the field.

0 potential is defined at infinity, so at a point close to a mass, the potential of an object would be negative.

What is gravitational potential difference?

Difference in the gravitational potentials of 2 points in a gravitational field.

What is gravitational potential energy at a point in the field?

The work done per unit mass in moving object from infinity to that point in the field

What is escape velocity?

• The minimum velocity an object requires in order to escape the gravitational field of an object when projected vertically from its surface.

• Vesc derived from equating kinetic energy and gravitational potential energy required to reach infinity.

• Vesc = root of 2GM/r