Download Gravitational field, potential and energy - SJHS-IB

9: Motion in Fields
9.2 Gravitational field, potential
and energy
Gravity
Recap:
Newton’s universal law of gravitation:
F = GMm
r2
Gravitational field strength:
…the force per unit mass experienced by a small
test mass (m) placed in the field.
g = GM
r2
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...
ΔEp = mg∆h
E.g. In many projectile motion questions we assume
the gravitational field strength (g) is constant.
GPE in non-uniform fields
However, as Newton’s 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 Earth’s
surface but at ‘infinity’.
Ep = 0
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 -)
Lots of positive
work must be
done on the
small mass!
Ep = negative
‘Infinity’
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.
Ep = - GMm
r
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 )
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.
Q. What do the indicated properties of these two
graphs represent?
a
b
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 Earth’s 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 Earth’s field.
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.
Thus for a field due to a (point or spherical) mass M:
V = Ep = - GMm
m
rm
So ...
V = - GM
r
V = Gravitational
potential (Jkg-1)
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 )
Gravitational Potential in a uniform field.
For a uniform field…
∆Ep = mg∆h
So…
∆V = ∆Ep = mg∆h
m
m
∆V = g∆h
How far apart are the equipotentials in this
diagram?
r
V
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.
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.
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 = ΔV
Δr
It is related to gravitational field strength...
g = - ΔV
Δr
Escape speed
If a ball is thrown upwards, Earth’s gravitational field does
work against it, slowing it down. To fully escape from Earth’s
field, the ball must be given enough kinetic energy to enable
it to reach infinity.
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).
Loss of KE
= Gain in GPE
½ mv2 = GMm
r
So...
but…
(Note this also = Vm)
so…
Note we could also say...
½ mv2 = GMm = Vm
r
So...
v = √(2V)
Note
we could also say...
½ mv2 = GMm = Vm
r
so...
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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