1.1. To To continue continue to explore the concept of to explore the concept of Gravitational potentialGravitational potential

2.2. To examine gravitational potential near a To examine gravitational potential near a spherical planet spherical planet

Book Reference : Pages 56-68Book Reference : Pages 56-68Book Reference : Pages 63-65Book Reference : Pages 63-65

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

Planet

Field Lines

MJkg-1

-40

-60

-80

-100

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

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

Planet

A 1kg mass raised from the A 1kg mass raised from the Earth’s surface by 1m gains Earth’s surface by 1m gains 9.81J of G.P.E. It gains another 9.81J of G.P.E. It gains another 9.8J1 for the next metre etc9.8J1 for the next metre etc

EEpp = mg = mghh

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

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

PlanetNear the Earth’s surface this is 9.81JkgNear the Earth’s surface this is 9.81Jkg-1-1mm-1-1 However, further away this reduces However, further away this reduces rapidlyrapidly

rm

V

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

the potential gradient = the potential gradient = V / V / r r

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

Planet

rm

V

V + V

The work done by an equal & The work done by an equal & oppositeopposite force force FF moving moving through through rr

W = FW = Frr

For a mass For a mass mm, then the change in potential , then the change in potential (remember (remember W = mW = mV)V)

V = V = W/mW/m (substitute for W) (substitute for W)

V = FV = Fr/m r/m (rearrange)(rearrange)

F = mF = mV /V /r r Which is equal & opposite to Which is equal & opposite to FFgravgrav

FFgravgrav = - m = - mV /V /rr

Remember gravitational field strength g = FRemember gravitational field strength g = Fgravgrav/m/m

g = - g = - V /V /rr

gg is the is the negativenegative of the potential gradient of the potential gradient

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

The gradient is always at right angles to the The gradient is always at right angles to the equipotentialsequipotentials

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

V = -GM/rV = -GM/r

What figure do we get for V for the Earth?What figure do we get for V for the Earth?

Mass = 6x10Mass = 6x102424kg and radius = 6.4x10kg and radius = 6.4x1066mm

* Proof is not required for our exam* Proof is not required for our exam

When calculated the previous equation give us a When calculated the previous equation give us a value of -63MJkgvalue of -63MJkg-1-1

This means that 63MJ of work must be done to This means that 63MJ of work must be done to remove each 1kg from the Earth’s surface to remove each 1kg from the Earth’s surface to infinity.infinity.

R 2R 3R 4R0

g

g/4

g/9

Distance from centre of planet with Radius RDistance from centre of planet with Radius R

Gravitational Field StrengthGravitational Field Strength If we redraw this diagram If we redraw this diagram with representative with representative numbers for Earth....numbers for Earth....

Each square represents a 1N force acting for a Each square represents a 1N force acting for a distance of 2.5x10distance of 2.5x1066 m (and since W.D. = f x d each m (and since W.D. = f x d each square represents 2.5MJ)square represents 2.5MJ)

The area under the curve is the application of The area under the curve is the application of “work done = force x distance moved”....“work done = force x distance moved”....

But for a force which varies withBut for a force which varies with

F = GMm/rF = GMm/r22

Where M is the mass of the planet and m is our Where M is the mass of the planet and m is our 1kg in this case.1kg in this case.

If we move our 1kg mass by a small step If we move our 1kg mass by a small step r then r then the work done is given bythe work done is given by

W = FW = Fr = GMmr = GMmr/rr/r22

We have seen....We have seen....

V = -GM/rV = -GM/r

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

Moreover, the gradient of the potential (V) curve Moreover, the gradient of the potential (V) curve at any point is –g where g is the field strength.at any point is –g where g is the field strength.

This can be found by drawing a tangent on the V This can be found by drawing a tangent on the V curve.curve.

You are advised to know how to draw g You are advised to know how to draw g 1/r 1/r22 & & V V -1/r & hence be able to comment upon how -1/r & hence be able to comment upon how g changes more sharply than V with increasing rg changes more sharply than V with increasing r