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Gravitational Potential

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Learning Objectives

Book Reference : Pages 56-68

Book Reference : Pages 63-65

Gravitational Potential

To continue to explore the concept of Gravitational potential

To examine gravitational potential near a spherical planet

- Compare with contour lines on a map....

Equipotentials 1

MJkg-1

Note in keeping with the inverse square law, the gravitational field becomes weaker further away from the planet

i.e. Equal increments in equipotential are spaced further apart

-40

-60

-80

Field Lines

-100

Planet

However, near the surface of a planet we consider the gravitational field to be uniform and we consider the equipotentials to be horizontal & parallel to the ground

Equipotentials 2

A 1kg mass raised from the Earth’s surface by 1m gains 9.81J of G.P.E. It gains another 9.8J1 for the next metre etc

Ep = mgh

Can only be applied where h is small compared to the radius of the planet

Planet

Definition : The potential gradient at a particular point in a gravitational field is the change in potential per metre

Potential Gradient 1

Near the Earth’s surface this is 9.81Jkg-1m-1 However, further away this reduces rapidly

m

r

In general for a change in potential V over a small distance r then

the potential gradient = V / r

Planet

V + V

V

For a small mass m being moved from a planet by r against the gravitational force Fgrav then its gravitational potential is increased by:

Potential Gradient 2

- The work done by an equal & opposite force F moving through r
- W = Fr

m

r

Planet

V + V

V

For amass m, then the change in potential (remember W = mV)

V = W/m (substitute for W)

V = Fr/m (rearrange)

F = mV /r Which is equal & opposite to Fgrav

Fgrav = -mV /r

Remember gravitational field strength g = Fgrav/m

g = - V /r

Potential Gradient 3

g is the negative of the potential gradient

Meaning that g acts in the opposite direction of the potential gradient.

The gradient is always at right angles to the equipotentials

Potential Gradient 4

- At this stage we “pluck”* the following from thin air.... The gravitational potential is given by:
- V = -GM/r

Gravitational Potential Near the Earth

When calculated the previous equation give us a value of -63MJkg-1This means that 63MJ of work must be done to remove each 1kg from the Earth’s surface to infinity.

Gravitational Potential Near the Earth

If we redraw this diagram with representative numbers for Earth....

Gravitational Field Strength

Gravitational Potential Near the Earth

g

g/4

g/9

0

R

2R

3R

4R

Distance from centre of planet with Radius R

Gravitational Potential Near the Earth

Each square represents a 1N force acting for a distance of 2.5x106 m (and since W.D. = f x d each square represents 2.5MJ)

- The area under the curve is the application of “work done = force x distance moved”....
- But for a force which varies with
- F = GMm/r2

- If we move our 1kg mass by a small step r then the work done is given by
- W = Fr = GMmr/r2

Gravitational Potential Near the Earth

- We have seen....V = -GM/r

Potential Gradients Near a Spherical Planet

- You are advised to know how to draw g 1/r2 & V -1/r & hence be able to comment upon how g changes more sharply than V with increasing r

Potential Gradients Near a Spherical Planet

- Moreover, the gradient of the potential (V) curve at any point is –g where g is the field strength.
- This can be found by drawing a tangent on the V curve.