Gravitational Potential

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# Gravitational Potential - PowerPoint PPT Presentation

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.

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Presentation Transcript

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 = mgh

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

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:

• The work done by an equal & opposite force F moving through r
• W = Fr

m

r

Planet

V + V

V

V = W/m (substitute for W)

V = Fr/m (rearrange)

F = mV /r Which is equal & opposite to Fgrav

Fgrav = -mV /r

Remember gravitational field strength g = Fgrav/m

 g = - V /r

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

At this stage we “pluck”* the following from thin air.... The gravitational potential is given by:

• V = -GM/r
• What figure do we get for V for the Earth?Mass = 6x1024kg and radius = 6.4x106m
• * Proof is not required for our exam

### 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

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
• Where M is the mass of the planet and m is our 1kg in this case.
• If we move our 1kg mass by a small step r then the work done is given by
• W = Fr = GMmr/r2

### Gravitational Potential Near the Earth

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

• i.e. The gravitational potential is inversely proportional to the distance from the centre of of the planet

### 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.