Transcript special cases

Lecture 6: Gravity and Motion
Review from Last Lecture…
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Newton’s Universal Law of Gravitation
Kepler’s Laws are special cases of
Newton’s Laws
bound and unbound orbits
tides and tidal friction
Kepler or Newton?
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find the mass of the Earth using the fact
that the Moon’s orbit has a period of 29 ½
days
find the average orbital distance for an
asteroid that orbits the Sun with a period
of 8 years
find the period of a binary star system with
a mean orbital distance of 10 pc
Tides
The Moon’s Tidal Forces on the Earth
Galactic Tidal Forces
Tidal Friction
Synchronous Rotation
Tidal friction and the Moon
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Tidal friction from the Moon acting on the
Earth causes the Earth’s rotation to slow
down.
As a result, the Moon also moves further
and further away from Earth (due to
conservation of angular momentum).
Implications…
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was the Moon’s angular size larger or
smaller in the past?
was the length of a lunar month longer or
shorter in the past?
were eclipses (both solar and lunar) more
or less frequent in the past?
The acceleration of gravity
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the universal law of gravitation allows us
to understand why the acceleration due to
gravity is independent of the mass of the
object
and why our weight is different on other
planets
Why g is independent of mass
Imagine dropping a rock near the surface of the Earth.
The force on the rock is:
Fg = G MEarth Mrock / d2 = G MEarth Mrock / (REarth)2
Newton’s Second Law of Motion says that the force is also:
Fg = Mrock arock = G MEarth Mrock / (REarth)2
arock = g = G MEarth / (REarth)2
Finding the value of g
g = G MEarth / (REarth)2
Mearth = 6.0 x 1024 kg
Rearth = 6.4 x 106 m
g = (6.67 x 10-11 m3/(kg s2) ) x 6.0 x 1024 kg / (6.4 x 106 m)2
= 9.8 m/s2
What about on the Moon?
g = G MMoon / (RMoon)2
MMoon = 7.4 x 1022 kg
RMoon = 1.7 x 106 m
g = (6.67 x 10-11 m3/(kg s2) ) x 7.4 x 1022 kg / (1.7 x 106 m)2
= 1.7 m/s2
gravity is weaker on the Moon…therefore things weigh less!
Matter and Energy
Energy is what makes matter move
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kinetic energy = energy of motion
potential energy = stored energy
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gravitational
chemical
electrical
radiative energy = light
Units of Energy
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calories
kilowatt-hours
BTU
Joules
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1 Joule = 0.00024 Calories
Quantifying Energy
kinetic energy = ½ m v2
where m = mass (in kg)
and v = velocity (in m/s)
answer will be in Joules
(1 J = kg x m2/s2)
Gravitational Potential Energy
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the amount of gravitational potential
energy is proportional to the mass, the
force of gravity, and the distance
for example, for an object suspended
above the earth, the gravitational potential
energy is W = G m MEarth/r = m x g x r
Conservation of Energy
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the total amount of energy in the Universe
remains the same
energy can change forms but cannot be
created or destroyed
Orbital Energy
moving slower 
smaller
kinetic energy
moving faster 
larger
kinetic energy
bound vs. unbound orbits
unbound orbits 
kinetic energy greater
than gravitational potential
bound orbits 
gravitational potential energy
balances kinetic energy
gravitational encounters
escape velocity
We can now derive the escape velocity
by setting the kinetic energy equal to
the gravitational potential energy:
½ m v2 = Gm MEarth/REarth
 vescape = (2GMEarth / REarth)½
The Escape Velocity from Earth
vescape = (2GMEarth / REarth)½
= (2 x 6.67x10-11 m3/(kg s2) x 6.0x1024 kg/6.4x106m)½
vescape = 11 km/s
The End