Gravitational Potential Energy © Simon Porter 2007

Gravitational Potential Energy© Simon Porter 2007

How much GPE?

How much GPE?

GPE = mgh?

How much GPE?

GPE = mgh?

How much GPE?

GPE = mgh?

How much GPE?

GPE = mgh?

We do know that the GPE must be

decreasing. But where is the GPE zero?

How much GPE?

GPE = mgh?

We do a little physicists trick. We take the GPE at infinity to be zero! That

means that it has negative GPE at distance closer than

infinity!

Gravitational potential energy

Gravitational potential energy at a point is defined as the work done to move a mass from infinity to that point.

© Simon Porter 2007




M

m

I’ve come from infinity!

R




M

m


R

Work done = force x distance

The force however is changing as the mass gets closer

Gravitational potential energy© Simon Porter 2007

M

m


R

W =

R

Fdr

R

GMmdr

r2

= = [ ]GMm

r

R

=GMm

R

- -



Ep = -GMm

r

Ep is always negative


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

rNote the difference between gravitational potential energy (J) and Gravitational potential (J.kg-1)

Ep = mV


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

V2

(independent of path)

Equipotential surfaces/lines


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

maps give us information about geographical heights.

Field strength = potential gradient

In fact it can be shown from calculus that the gravitational field is given by the potential gradient (the closer the equipotential lines are together, the stronger the field)

g = -dVdr

Let’s stop and read!

Pages 127 to 130Pages142 to 151

Escape speed

Imagine throwing a ball into the air


Escape speed

It falls to the ground (Doh!)


Escape speed

What happens if you throw harder?


Escape speed

It goes higher and takes longer to return.


Escape speed

It goes higher and takes longer to return.

Ouch!


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.


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)


Escape speed

At “infinity”, it gravitational energy is given by Ep = -GMm/r

= zero when r is infinite


Escape speed

Energy conservation tells us that it must therefore have zero energy to start with if it is to escape the earth’s gravity.

i.e. KE + GPE = 0


Escape speed

i.e. KE + GPE = 0

½mv2 + -GMem/Re = 0

(where Re is the radius of the earth)

½mv2 = GMem/Re

v = √2GMe/Re


Escape speed

v = √2GM/Re

v = √(2 x 6.67 x 10-11 x 5.98 x 1024)/6.38 x 106

v = 12000 m.s-1

I can’t throw that fast!

In reality the escape

velocity of the earth is bigger

than this. WHY?


Let’s try some questions!Page 153 Q7, 13

Hold on!

Isn’t electricity similiar?


Gravitational Potential

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

rNote the difference between gravitational potential energy (J) and Gravitational potential (J.kg-1)

Ep = mV


Electrical Potential

The Electrical potential at a point is the work done per unit charge on a small positive test charge moving from infinity to that point. It is given by

V = W

qNote the difference between electrical potential energy (J) and Electrical potential (J.C-1)

Uel = qV


Scalar quantity

Moving charges in potentials

If a charge is moved from a position with potential V1 to a position with potential V2, work = q(V2 – V1) = qΔV

V1

V2

(independent of path)



Ep = -GMm

r

Ep is always negative


Electrical potential energy

Electrical potential energy at a point is defined as the work done to move a positive charge from infinity to that point.

Uel = kQq

r



Ep = -GMmr


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

maps give us information about geographical heights.

Field strength = potential gradient

In fact it can be shown from calculus that the gravitational field is given by the potential gradient (the closer the equipotential lines are together, the stronger the field)

E = dVdr

From “Physics for the IB Diploma”K.A.Tsokos (Cambridge University Press)

Gravitation ElectricityActs on Mass (always +?) Charge (+ or -)

Force F = GM1M2/r2

Attractive only, infinite range

F = kQ1Q2/r2

Attractive or repulsive, infinite range

Relative strength 1 1042

Field g = GM/r2 E = kQ/r2

Potential V = -GM/r V = kQ/r

Potential energy Ep = -GMm/r Ep = kQq/r


Let’s try some questions

Pages 307 Questions 2, 4, 5, 6, 11, 12.

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Gravitational Potential Energy © Simon Porter 2007