Gravitational Force
chapter 2 #Studying
#Mechs
2.1 Representing a Gravitational Field
We can represent the Earth's gravitational field by drawing field lines that show two things;
1) arrows that show the direction of the gravitational force on a mass placed in the field
2) the spacing between field lines which indicate the strength of the gravitational field (the further, the
weaker)
The Earth's gravitational field shows that all objects are attracted towards the centre of the Earth.
The great separation between the field lines shows that the gravitational force gets weaker as you
get further away
The gravitational field of the Earth is as if its entire mass was concentrated at its centre; this is known
as its centre of mass
When looking at the field lines at the surface of the Earth, it seems as if the lines are nearly parallel
and evenly spaced. The strength of the field is virtually the same. The field lines looks as if they're
going straight downwards.
Your weight does not become much less if you go up stairs.
The gravitational field of the Earth is said to be radial since the field lines diverge radially from the
centre of the Earth.
On the scale of a building, however, it appear to be uniform since they are equally spaces.
Newton's Law of Gravitation
Newton used his ideas about mass and gravity to suggest a law of gravitation for two point masses
Each point attracts the other with force F, according to Newton's third law of motion ( he point
masses interact with each other and therefore exert equal but opposite forces on each other.)
Newton's Law of Gravitation suggests that F is directly proportional to the product of their masses
and inversely proportional to the square of their r.
- When talking about "point masses" we can use particles alternatively as each particle of a body
attracts each particle of another body.
- It is more complicated to think of solid bodies with volume of space
Mm
F ∝
r
2
therefore:
GM m
F =
r
2
or
F =
Gm 1 m 2
r2
F is in the opposite direction of r because it is attractive
G = 6.67×10
N m kg
, the universal gravitational constant
−11
2
−2
Explanation:
The gravitational force on you (your weight) depends on your mass and the mass of the planet
you're standing on
The further you are, the weaker the pull (inverse law rule).
We measure r from the centre of mass of one body to the centre of mass of the other. The two
bodies attract each other with equal and opposite forces (Newton's 3rd Law).
2.2 Gravitational Field Strength g
Gravitational Field Strength: how strong or weak a gravitational field is.
Gravitational field (g) = 9.8ms
−2
, the weight of a body if mass m is mg.
It is clearer to write g as 9.8N kg
−1
because each 1kg experiences 9.8N .
Gravitational field strength at a point is the gravitational force exerted per unit mass on a small object
at said point.
g = F /m
where F is the gravitational force on the object and m is the mass of the object. Gravitational field strength has units of N kg
equivalent to ms
−2
−1
. This is
.
We can integrate this into the Law of Gravitation equation, so the gravitational field strength g due to
the mass M at the distance r from its centre is;
g =
F
m
GM m
g =
r2 m
GM
g =
r
2
Gravitational field strength is a vector quantity, like force.
g
decreases if r increases, however, close to the Earth's surface
When close to the Earth's surface, the r barely changes. Even if you go up Mount Everest, it only
decreases by 0.3%
Gravitational field strength is also the acceleration of free fall.
2.3 Energy in a Gravitational Field
If you lift an object from the ground, you increase its gravitational potential energy. The higher you lift
it, the more work is done, the more the g.p.e is.
g. p. e = mgΔh
Δh =
change in height
This equation works in case were relative to the Earth's surface. it cannot be used if:
were considering an object moving across space. g.p.e assumes that the object has no
potential energy on the Earth's surface but the Earth's surface is irrelevant in this case.
if the object is lifted a great distance so the g decreases which affect g.p.e.
Therefore, we must picture the mass at infinity, an infinite distance from all other masses, and say
that here the mass's g.p.e is zero.
relative to that point, we calculate the work done after the mass has been lifted which is equal to
g.p.e.
2.4 Gravitational Potential
Gravitational Potential: (at a point) the work done per unit mass bringing a unit mass from infinity to a
point.
It tells us about the g.p.e per unit mass at the point. Its symbol is φ
GM
φ = −
r
GM m
g. p. e = −
r
for objects with a mass of m
The minus sign is present because gravitational potential is always negative as a mass is brought
towards another mass and g.p.e is zero at infinity.
Imagine a spacecraft visiting the solar system. Here are three parts of its journey:
1. The spacecraft approaches the Earth and it attracts it. It's g.p.e lowers therefore gravitational
potential lowers
2. The spacecraft leaves the Earth and its attraction. It's doing work against the Earth's gravity (in the
direct of g.p.e), g.p.e increases therefore gravitational potential. The Earth's field creates a giant
"potential well" in space.
3. The spacecraft approaches the Sun, it is attracted to it. Since the Sun's mass is much bigger than
Earth's, the decrease in g.p.e is greater so the decrease in gravitational potential is also greater. The
Sun's potential well is much stronger than the Earth's.
Gravitational Potential difference
The equation for the change in potential, φ = −GM /r , can be used twice, once to find the potential
at the surface and once to find the potential at the orbital height. However, it is much easier to
combine the two operations and use the equation:
1
Δφ = GM (
1
−
r1
)
r2
Fields: terminology
Field strength tells us about the force on unit mass at a point.
Potential tells us about potential energy of unit mass at a point.
2.5 Orbiting Under Gravity
When an object is orbiting a planet, for example a satellite orbiting the Earth, gravity provides the
centripetal force that keeps the satellite in orbit.
The one force acting on the satellite is the gravitational attraction of the earth, a very simple situation.
The satellite travels in a circular path because the gravitational force is perpendicular to the velocity.
We can work with the equation of centripetal force and gravitational force to give:
mv
2
F =
r
GM m
r
v
mv
=
2
2
2
r
GM
=
r
= gravitational / centripetal force
F
= mass of satellite
= velocity of satellite
m
v
= mass of planet (earth)
= distance between satellite and centre of planet
M
r
This equation allows us to calculate the velocity the satellite must travel in to stay in circular orbit.
The mass of the satellite, m, does not affect it
2.6 The Orbital Period
Orbital Period, T : the time taken to complete a full orbit.
We can say the distance around an orbit is 2πr so:
2πr
v =
T
We can expand this even further:
2
4π r
2
T 2
T
2
GM
=
r
4π
2
= (
)r
3
GM
2.7 Orbiting the Earth
The earth has many satellites, the Moon included, that use its gravitational field to provide them with
centripetal force to stay in orbit. They must travel a certain speed, to stay in that orbit, which is given
by the equation V
2
= GM /r
If they are too slow, they will fall towards the earth and If they're too fast they will move into a higher
orbit.
Geostationary Orbits
Geostationary orbit: when a satellite travels from west to east with the same angular speed of the
earth so it remains above a fixed point on the earth's equator.
There are over 300 satellites in that orbit, they are used for telecommunication and satellite television
transmissions.
Geostationary satellites have a lifetime of about 10 years, they gradually drift out of their correct orbit
so they need to be fuelled again to return to the geostationary position. But eventually they run out of
fuel and need to be replaced.
We can determine the distance of a geostationary satellite from earth using T
2
2
= (4π /GM )r
3
:
This means an object needs to be 42300km from the centre of the earth to be in geostationary orbit.
Summary
The force of gravity is an attractive force between any two objects due to their masses.
The gravitational field strength g at a point is the gravitational force exerted per unit mass on a small object placed at that point:
GM
g =
r
2
The external field of a uniform spherical mass is the same as that of an equal point mass at the centre of the sphere.
Newton’s law of gravitation states that:
Any two point masses attract each other with a force that is directly proportional to the product
of their masses and inversely proportional to the square of their separation.
The equation for Newton’s law of gravitation is:
GM m
F =
r
2
The gravitational field strength at a point is the gravitational force exerted per unit mass on a small object placed at that point:
g =
GM
r
2
On or near the surface of the Earth, the gravitational field is uniform, so the value of g is approximately constant. Its value is equal to the
acceleration of free fall.
The gravitational potential at a point is the work done in bringing unit mass from infinity to that point.
The gravitational potential of a point mass is given by:
GM
φ = −
r
The orbital period of a satellite is the time taken for one orbit.
The orbital period can be found by equating the gravitational force F =
GM m
r
2
to the centripetal force F =
mv
r
2
.
The orbital speed of a planet or satellite can be determined using the equation:
v
2
GM
=
r
Geostationary satellites have an orbital period of 24 hours and are used for telecommunications transmissions and for television broadcasting.