Radial field

A field in which the field lines are straight and converge or diverge as if from a single point

Uniform field

A region where the field strength is the same in magnitude and direction at every point in the field

Field line

It’s direction indicates the direction of the force

It represents the direction of the gravitational force is acting on an object in that field

Strength of a gravitational field

The force per unit mass on a small test mass placed in the field - measures in Nkg -1 g=f/m

Also given as the negative of the potential gradient

g=-V/r

From newtons law of gravitation we get:

g=GM/r^2

Gravitational potential energy

The energy of an object due to its position in a gravitational field

The work done to move an object from infinity to that point

Gravitational potential

The work done per unit mass to move a small object from infinity to that point

V = W/m

Measured in J kg^-1

Also considering newtons law of gravitation: V = -GM/r

Gravitational potential is zero at infinity

Equipotentials

A line or surface in a field along which the gravitational potential is constant

Potential gradient

The change in potential per metre at that point

They are like contour lines on a map - the closer the equipotentials, the greater the potential gradient

Given by dV/dr

Gravitational field strength is the negative of the potential gradient -dV/dr

Kepler’s third law

For any planet, the cube of its mean radius of orbit r is directly proportional to the square of its time period T.

Using Newton’s law of gravitation it can be shown that:

r^3/T^2 = GM/4pi^2

Newton’s law of gravitation

Assumes that the gravitational force between any two point objects is:

Always an attractive force

Proportional to the mass of each object

Proportional to 1/r^2 where r is their distance apart

Universal attractive force acting between all matter

F = GMm/r^2

G is the universal constant of gravitational

M and m are the two masses involved

r is the distance between the centres of the two masses (m)

Low orbit satellites

Defined as a satellite which orbits between 180-2000km above the Earth’s surface

Cheaper to launch and require less powerful transmitters as they’re closer to the Earth

Useful for communications but their proximity to Earth and relatively high orbital speed means you need multiple satellites working together to maintain constant coverage

Close enough to see the Earth’s surface in a high level of detail

E.g. Imaging satellites for monitoring the weather

Usually lie in a plane including the North and South Pole

Each orbit is over a new part of the Earth’s surface as the Earth rotates underneath - so the whole Earth can be scanned

Geostationary satellites

Orbit the Earth once every 24 hours - always above the same point on Earth

Synchronous orbit - orbital period is the same as the rotational period of the orbited object

Must always be directly above the equator

Orbital radius ~ 42000km and 36000km above the Earth’s surface

Useful for sending TV and phone signals - you don’t have to alter the angle of the receiver

Energy of an orbiting satellite

Remains constant

Circular orbit - speed and distance above mass are constant so Ek and Ep are constant

Elliptical orbit - speed up as its height decreases and slow down as its height increases. Ek and Ep are exchanged but total energy remains constant

Force field

A region in which an object will experience a non-contact force

Gravitational potential difference

The energy needed to move a unit mass

2 points at different distances from a mass will have different gravitational potentials - therefore there is a potential difference

Given by: dW=m dV

dW - work done in J

m - mass in kg

dV - gravitational potential difference in Jkg^-1

Kepler’s observations

In the early 1600s Johannes Keplar made some observations and concluded the following:

1 - planets move around the sun in elliptical orbits with the sun at one focus

2 - the line joining a planet to the sun sweeps out equal areas in equal intervals of time

3 - the square of the time period of the planet is directly proportional to the cube of the average radius of its orbit

Only need to know the third law

Additional equations that aren’t given in the equation sheet

Energy of the satellite:

E(total) = -GMm/2r Ek = GMm/2r Ep = -GMm/r

Escape velocity:

V esc = (2GM/r)^1/2

V esc = (2gR)^1/2

Where R is radius of planet

And r is distance from the centre of the planet

Deriving the equation for work done

-dV/dr = F/m

m dV = -F dr

W = Fd

Hence W = m dV

Deriving the equation for orbital speed

mv^2/r = GMm/r^2

v = square root GM/r

Hence as orbital radius increases, orbital speed decreases

Deriving the equation for escape velocity

conservation of energy:

kinetic energy lost = gravitational potential gained

1/2 mv^2 = GMm/r

v = square root 2GM/r

Deriving Kepler’s third law

mw^2 r = GMm/r^2

r^3 = GM/w^2

r^3 = GM/4pi^2 T^2

Draw a graph of gravitational potential against distance from the centre of an object

Draw a graph of gravitational field stength against distance from the centre of an object

See CGP book

How does increasing distance from the moon to the Earth impact the moon’s orbital period?

Gravitational force on the moon is reduced

Therefore v and w decrease

Hence orbital period increases

Also Kepler’s third law can be used to explain this

Why is a point with a more negative potential closer to the centre of a planet?

Potential decreases as distance form the planet decreases

Potential is zero at infinity so the further the potential is form zero the further the point is from the planet