Notes - Potential and potential energy in Gravity Fields

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Universal Gravitation & Field Strength

• Newton’s Law of Universal Gravitation is:

• And we learned last year that the force of gravity on the Earth can be calculated by:

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Example Problem

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Now what if it was dropped on the Moon?

• Moon’s mass is 7.35x1022_{kg & radius is}

1.74x103_km

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Gravitational Field Strength

• That acceleration due to the gravitational field is also known as the field strength • The gravitational field strength at a

certain point is the force per unit mass

experienced by a small point mass, m, at that point.

• It has units of N•kg-1

2

r

M

G

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Gravitational Field Lines

• The field lines show the direction of the force. For a spherical mass they become more dense closer to the mass (and

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Gravitational Field close to the Earth

• Close to the Earth’s surface, we can

assume that the field lines are parallel and the field is uniform.

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Adding Field Forces

• Since the field is a vector, we can add the field strengths of multiple masses

• What is the field strength at point “A” • Calculate the field strength at point “B”

• Calculate the field strength at “A” if the big mass were changed to a 100kg mass

B A

2.5m 2.5m

1m

1000kg

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Gravitational Potential Energy

• If you lift a mass up from the ground, you increase its potential energy: PE = mgh • If PE = mgh, the “gh” part just describes

the location of the object: which gravity field and how far from it

• We call that location in the gravity field its “gravitational potential” : (V)

• V tells how much PE a mass would have there

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Moving masses in potentials

If a mass is moved from a position with

potential V₁ to a position with potential V₂, work = m(V₂ – V₁) = mΔV

V₁

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Examples: • What is the potential at “A”?

• If a body is moved to point “B”, what is the change in potential?

• How much work is done moving 2kg A to B? • Potential difference

between C & D?

• Potential difference between A & E?

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Field and equipotentials

• Equipotentials are always perpendicular to field lines.

Diagrams of equipotential lines give us information about the gravitational field in much the same way as contour

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Equipotential surfaces/lines

• Work done = ∆V x mass

• Work done also equals force x distance ∆Vm=mgh

h

V

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Contour Lines on Maps

The contour lines on a map show how steep the slope is.

When they are closer together, the slope is steeper.

In our equation, that would mean the ∆h is smaller and the gradient (slope) is greater.

h

V

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How did we calculate Potential

Energy?

• GPE = mgh

• So higher means more potential energy, right?

• What if we went really high?

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GPE in space?

• In space, h→∞.

– Do we therefore have infinite potential energy? – I think not!

• The old way we calculated GPE is really only a way to simply determine the change

in GPE near the surface of the Earth • We need a way to state an absolute

potential energy.

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Gravitational potential energy

Gravitational potential energy at a point is defined as the work done to move a

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Gravitational potential energy

mass from infinity to that point.

M

m

I’ve come from infinity!

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Gravitational potential energy

M

m

R

Work done = force x distance

The force however is changing as the

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Gravitational potential energy

M

m

R

W =

R

Fdr

R

GMmdr

r2

= ₌

[ ]

GMm

r

R

= GMm

R

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Gravitational potential energy

E

_p

= -GMm

r

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Gravitational Potential

It follows that the Gravitational potential

at a point is the work done per unit mass

on a small point mass moving from infinity to that point. It is given by

V = -GM

r

Note the difference between gravitational potential energy (J) and

Gravitational potential (J.kg-1₎

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Let’s do some examples:

• The moon’s mass is 7.4x1022 kg and the Earth’s

mass is 6.0x1024 kg. The average distance to the moon is 3.8x108_{m. If you travel in a 2000kg}

rocket:

• Calculate the gravitational potential when you’re 1.0x107 m from the moon.

• Calculate the rocket’s PE at that point.

1 6 7 22 11 8 24 11 10 6 . 1 10 1 10 4 . 7 10 67 . 6 10 7 . 3 10 6 10 67 .

6   _

        kg J x x x x x x x V r GM r GM V A m m E E A J x x x m V

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Escape speed

Imagine throwing a ball into the air

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Escape speed

It falls to the ground (Doh!)

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Escape speed

What happens if you throw harder?

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Escape speed

It goes higher and takes longer to return.

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Escape speed

It goes higher and takes longer to return.

Ouch!

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Escape speed

The kinetic energy of the ball changes to gravitational potential energy as the ball rises. This in turn turns back into kinetic energy as the ball falls again.

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Escape speed

How fast would you have to throw the ball so that it doesn’t come back? (i.e. goes to “infinity” or escapes the gravitational field of the earth)

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Escape speed

At “infinity”, its gravitational energy is given by :

and PE=0 when r=∞

Loss of KE = gain in PE

r GMm PE  

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Escape speed

For the Earth:

I can’t throw that fast!

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Energy of an orbiting body • As a body orbits, it has KE from its

movement and PE due to its position • We know:

• Setting gravity force = centripetal force:

2

2 1

& KE mv

r GMm

PE   

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Energy of an orbiting body • Total energy = PE + KE so:

• The orbit of a high satellite is less negative, so requires more energy