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