9: Motion in Fields

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9 Motion in Fields

• 9.2 Gravitational field, potential and energy

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Gravity Recap Newtons universal law of
gravitation Gravitational field
strength
F GMm r2
the force per unit mass experienced by a small
test mass (m) placed in the field.
g GM r2
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GPE in a uniform field When we do vertical work
on a book, lifting it onto a shelf, we increase
its gravitational potential energy (Ep). If the
field is uniform (e.g. Only for very short
distances above the surface of the Earth) we can
say... GPE gained (Ep) Work done F x d
Weight x Change in height so... ?Ep
mg?h E.g. In many projectile motion questions we
assume the gravitational field strength (g) is
constant.
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GPE in non-uniform fields However, as Newtons
universal theory of gravity says, the force
between two masses is not constant if their
separation changes significantly. Also, the true
zero of GPE is arbitrarily taken not as Earths
surface but at infinity.
If work must be done to lift a small mass from
near Earth to zero at infinity then at all points
GPE must be negative. (This is not the same as
change in GPE which can be or -)
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The gravitational potential energy of a mass at
any point is defined as the work done in moving
the mass from infinity to that point.
The GPE of any mass will always be due to
another mass (after all, what is attracting it
from infinity?) Strictly speaking, the GPE is
thus a property of the two masses. E.g.
Calculate the potential energy of a 5kg mass at a
point 200km above the surface of Earth. ( G
6.67 ? 10-11 N m2 kg-2 , mE 6.0 ? 1024 kg, rE
6.4 ? 106 m )
Ep - GMm r
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The gravitational potential energy of a mass at
any point is defined as the work done in moving
the mass from infinity to that point.
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Q. What do the indicated properties of these two
graphs represent?
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Gravitational Potential Whereas gravitational
force on an object on Earth depends upon the mass
of the object itself, gravitational field
strength is a measure of the force per unit mass
of an object at a point in Earths
field. Similarly, whereas the GPE of say a
satellite, depends upon both the mass of Earth
and the satellite itself, gravitational potential
is a measure of the energy per unit mass at a
point in Earths field.
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Thus for a field due to a (point or spherical)
mass M So ... E.g. Calculate the potential of
a 5kg mass at a point 200km above the surface of
Earth. What would be the potential of a 10kg
mass at the same point? ( G 6.67 ? 10-11 N m2
kg-2 , mE 6.0 ? 1024 kg, rE 6.4 ? 106 m )
The gravitational potential at a point in a field
is defined as the work done per unit mass in
bringing a point mass from infinity to the point
in the field.
V Ep - GMm m r m
V Gravitational potential (Jkg-1)
V - GM r
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Gravitational Potential in a uniform field. For a
uniform field ?Ep mg?h So
?V ?Ep mg?h m m
?V g?h
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How far apart are the equipotentials in this
diagram?
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Equipotential Surfaces Equipotential surfaces or
lines join points of equal potential together.
Thus if a mass is moved around on an
equipotential surface no work is done.
Thus the force due to the field, and therefore
the direction of the field lines, must be
perpendicular to the equipotential surfaces at
all times.
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• Potential Gradient
• The separation of the equipotential surfaces
  tells you about the field
• Uniform fields have equal separation
• Fields with decreasing field strength have
  increasing separation.

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If the equipotentials are close together, a lot
of work must be done over a relatively short
distance to move a mass from one point to another
against the field i.e. the field is very
strong. This gives rise to the concept of
potential gradient. The potential gradient is
given by the formula... Potential
gradient ?V ?r It is related to
gravitational field strength... g -
?V ?r
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Escape speed If a ball is thrown upwards, Earths
gravitational field does work against it, slowing
it down. To fully escape from Earths field, the
ball must be given enough kinetic energy to
enable it to reach infinity. Loss of KE
Gain in GPE ½ mv2 GMm (Note
this also Vm) r So... but
so
The escape speed is the minimum launch speed
needed for a body to escape from the
gravitational field of a larger body (i.e. to
move to infinity).
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Note we could also say... ½ mv2 GMm
Vm r So... v v(2V)
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• Note we could also say...
• ½ mv2 GMm Vm
• r
• so... v v(2V)
• Assumptions
• Planet is a perfect sphere
• No other forces other than gravitational
  attraction of the planet.
• Note
• Applies only to projectiles
• - Direction of projection is not important if we
  assume that the planet is not rotating

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