Transcript Our Place in the Cosmos Elective Course Autumn 2006

Our Place in the
Cosmos
Lecture 7
Gravity - Ruler of the Universe
Gravity
• Gravity rules the Universe
• It holds objects like the Sun and Earth together
• Sun’s gravity determines motion of the planets of
the Solar System
• Gravity binds stars into galaxies and galaxies
into clusters
• In this lecture we will follow Newton’s lines of
reasoning in arriving at his law of gravity
What is Gravity?
• Gravity is a force between any two objects
due to their masses
• It is a “force at a distance” - two objects do
not need to come into contact for them to
exert a gravitational force on one another
• As with the law of inertia, our
understanding of gravity begins with
Galileo Galilei
Acceleration due to Gravity
• Galileo observed that all freely falling objects accelerate
towards the Earth at the same rate regardless of their
mass
• A marble and a cannonball dropped at the same time
from the same height will hit the ground simultaneously
• The gravitational acceleration near the Earth’s surface is
usually indicated by the symbol g and has a measured
value of about 10 m/s2
• An object dropped from rest will be moving at 10 m/s
after 1 second, 20 m/s after two and so on (neglecting air
resistance)
Isaac Newton
• Newton realised that if all objects fall with the same
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acceleration, then the gravitational force on an object
must be determined by its mass
Recall that Newton’s 2nd law says
acceleration = Force/mass
Since all objects have the same acceleration, then the
gravitational force divided by mass must be the same for
all objects
A larger mass feels a larger gravitational force:
Fgrav = mg
Note that gravitational mass is the same as inertial mass
- this equivalence is the basis for GR
Weight vs Mass
• Weight is the gravitational force Fg acting on an
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object
An object’s weight thus depends on its location,
whereas its mass does not
On the Earth’s surface, weight is equal to mass
times g, the acceleration due to gravity
It is incorrect (but common) to say that an object
“weighs 2 kg”
A 2 kg mass actually weighs about 2 kg x 10
m/s2 or 20 kg m/s2 or 20 Newtons (20 N)
Gravitational Force
• As with every other force, any gravitational force has an
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equal and opposite force (Newton’s 3rd law)
Drop a 20 kg cannonball and it falls towards the Earth
At the same time Earth falls towards the cannonball!
We do not notice the Earth’s motion in this case because
the Earth is so much more massive than the cannonball
Each object feels an equal and opposite force but
acceleration equals force divided by mass
Gravitational Force
• Newton realised that if doubling the mass of an object
doubles the gravitational force between it and the Earth,
then doubling the mass of the Earth would do the same
• Thus the gravitational force experienced by an object is
proportional to the product of the mass of the object
times the mass of the Earth:
Fg = something x mass of Earth x mass of object
• Since objects fall towards the centre of the Earth, Fg is
an attractive force acting along a line between the two
masses
Gravitational Force
• But why, reasoned Newton, should this law of
gravity apply only to the Earth?
• Surely the gravitational force between any two
masses m1 and m2 should be given by the
product of the masses:
Fg = something x m1 x m2
• Above reasoning follows from Galileo’s
observations of falling objects and Newton’s
laws of motion
• But what is the “something” in the above
equation?
Inverse Square Law
• Kepler had already reasoned that since the Sun
is at the focus of planetary orbits, then it must be
exerting some influence over the planets’ motion
• He also reasoned that this influence weakens
with distance - why else does mercury orbit so
much faster than Jupiter or Saturn?
• The area of a sphere increases with the square
of its radius (A = 4r2)
• Thus Kepler reasoned that the Sun’s influence
should decrease with the square of distance
Inverse Square Law
• Kepler’s proposal was interesting but not a
scientific theory as he lacked a good idea as to
the true source of the influence and also lacked
the mathematical tools to predict how an object
should move under such an influence
• Newton had both - he realised that gravity
should act between the Sun and the planets,
and that the gravitational force was probably
Kepler’s “influence”
• In this case, the “something” in Newton’s
expression for gravity should diminish with the
square of the separation between two objects
Newton’s Universal Law of
Gravitation
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Gravity is a force between any two objects,
and has the following properties
1. It is an attractive force acting along a straight line
between the objects
2. It is proportional to the product of the masses of the
objects m1 x m2
3. It decreases with the square of the separation r
between the objects
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Fg = G x m1 x m2 / r2
universal gravitational constant
Weakness of Gravity
• It is now possible to measure the value of the
gravitational constant G using sensitive
equipment: G = 6.67 x 10-11 N m2 / kg2
• The force between two bowling balls placed 1
foot apart is Fg  4 x 10-8 N, about the same as
the weight of a single bacterium!
• Gravity is only noticeable in everyday life
because the Earth is so massive
Acceleration due to Gravity
• For an object of mass m, Newton’s 2nd law of motion
says Fg = mg
• Universal law of gravitation says
Fg = G Mm/R2
• Equating these two expressions gives
mg = G Mm/R2
• The mass m appears on both sides and so may be
divided out to give
g = G M/R2
• Thus the acceleration g due to gravity is independent of
the mass of the object - as observed by Galileo!
Mass of the Earth
• Rearranging the last expression for g, we find
M = g R2/G
• Everything on RHS may be measured
• g by acceleration of falling objects
• R by altitude of celestial pole with latitude
• G via lab experiments
• We find M  6 x 1024 kg
• Newton inverted this argument to estimate a value for G
by assuming that Earth has the same density as typical
rocks
Gravity and Orbits
• Newton speculated that Kepler’s solar “influence” on the
planets’ orbits is gravity, but a good physical theory
should be testable
• He lacked the sensitive apparatus to measure
gravitational forces directly, but he was able to show that
his law of gravity predicted that the planets should orbit
the Sun just as Kepler’s empirical laws described
• Newton was thus able to explain Kepler’s laws
• Gravity is just one example of a physical law that was
first tested by astronomical observations
Predicting Orbits
• A full prediction of planets’ orbits requires use of
the branch of mathematics known as calculus
that Newton invented for the purpose
• However, we can still gain a conceptual
understanding of how orbits come about by a
series of thought experiments
• These are experiments that are not executable
in practice, but that still give us a good
conceptual grasp of a physical problem
Falling around the Earth
• In this thought experiment we fire a cannonball
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horizontally from a height of a few metres and neglect air
resistance
As cannonball travels horizontally, it also falls towards
the ground
The faster we fire the ball, the further it travels before
hitting the ground
As the ball travels further and further, the ground starts
to curve away from underneath it
If we fire the ball fast enough, it will maintain a constant
height above the ground and complete an orbit of the
Earth
Captions
Falling around the Earth
• An object orbiting the Earth is literally “falling
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around it” - it is always falling towards the
Earth’s centre
First man-made satellite to orbit the Earth was
the Sputnik I satellite launched in 1957
Astronauts float around the cabin of an orbiting
spacecraft for the same reason: both the
spacecraft and the astronaut are in free fall according to Newton’s law of gravity both
accelerate towards the Earth at the same rate
Astronaut falling freely
around the Earth
Shuttle and astronaut
experience same
gravitational acceleration
Both are independent
satellites sharing the same
orbit
Orbital Velocity
• How fast must Newton’s cannonball move to
orbit the Earth?
• An object moving round in a circle requires a
centripetal force to prevent it from flying off in a
straight line (Newton’s 1st law)
• For a ball on a string, the string provides the
force, for an object in orbit it is gravity
• For a satellite on a circular orbit, force required
for uniform circular motion = force provided by
gravity
Circular Velocity
• One can show that centripetal acceleration for
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an object moving in circle of radius r at velocity v
is given by a = v2/r
By Newton’s 2nd law of motion, the centripetal
force is given by F = ma = mv2/r
If object of mass m is orbiting a body of much
larger mass M, centripetal force is provided by
gravity
Fg = G Mm/r2
Equating these forces, mv2/r = GMm/r2
Mass m cancels out, leaving v2circ = GM/r
Sun’s Mass
• Any satellite moving on a stable circular orbit must be
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travelling at the circular velocity vcirc
Circular velocity at Earth’s surface is about 8 km/s
Earth’s orbit about the Sun is almost circular with a
speed of about 30 km/s [determined from stellar
aberration]
We also know radius of Earth’s orbit
[1 AU  1.5 x 1011 m]
We can then invert the formula for circular velocity to
estimate the Sun’s mass: M  2 x 1030 kg
“Harmony of the Worlds”
• We can now predict the period for a circular orbit
• Period P = circumference of orbit/circular velocity
P = 2r/[G M/r]
• Square each side and rearrange to give
P2 = 42/(G M) x r3
• Newton was thus able to predict Kepler’s 3rd law for
circular orbits
• Kepler’s laws provide an empirical test of Newton’s
theory  Newton’s theory helps us understand Kepler’s
laws
Mass Estimates
• Newton’s form of Kepler’s 3rd law can be
rearranged to read
M = 42/G x (A3/P2)
• This formula is used throughout astronomy to make
mass estimates
• It still holds when mass of orbiting object is comparable
to central mass
• In this case each object orbits about their common
centre of mass and M above is the total mass of the
system
Summary
• Starting with Galileo’s observation that objects
fall at the same rate, Newton predicted a
gravitational force between all masses that was
proportional to the product of the masses and
inversely proportional to the square of their
separation
• He showed that this simple model could explain
Kepler’s three laws of planetary motion and
could be used to estimate masses of
astronomical objects
• However, we still don’t really know exactly what
gravity is