lecture 6: gravity and motion review from last lecture… newton’s universal law of gravitation...
TRANSCRIPT
Lecture 6: Gravity and Motion
Review from Last Lecture…
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?
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
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…
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
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
g = (6.67 x 10-11 m3/(kg s2) ) x 6.0 x 1024 kg / (6.4 x 106 m)2
Mearth = 6.0 x 1024 kg Rearth = 6.4 x 106 m
= 9.8 m/s2
What about on the Moon?
g = G MMoon / (RMoon)2
g = (6.67 x 10-11 m3/(kg s2) ) x 7.4 x 1022 kg / (1.7 x 106 m)2
MMoon = 7.4 x 1022 kg RMoon = 1.7 x 106 m
= 1.7 m/s2
gravity is weaker on the Moon…therefore things gravity is weaker on the Moon…therefore things weighweigh less! less!
Matter and Energy
Energy is what makes matter move
kinetic energy = energy of motion potential energy = stored energy
gravitational chemical electrical
radiative energy = light
Units of Energy
calories kilowatt-hours BTU Joules
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
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
the total amount of energy in the Universe remains the same
energy can change forms but cannot be created or destroyed
Orbital Energy
moving faster largerkinetic energy
moving slower smallerkinetic energy
bound vs. unbound orbits
bound orbits gravitational potential energy balances kinetic energy
unbound orbits kinetic energy greaterthan gravitational potential
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