Chapter 4: Motion, Energy, and Gravity

Introduction to Motion

         Differences between speed, velocity, & acceleration
                  Speed = distance travelled / time required
                  speed = d / t
                  Velocity = speed, with a specified direction
                  Acceleration = (change in velocity) / time period
                  a = (change in v) / t
                  Examples of acceleration

         Gravitational Acceleration
                  Gravity pulls objects toward Earth
                  Causes objects to move faster and faster, thus accelerate
                  velocity increases by ~10 m/s, for every second of fall
                  thus, acceleration of gravity (g) ~10 m/s2 (actually, 9.8 m/s2)

                  Momentum = mass * velocity
                  Momentum is transferable
                  Force = (change in momentum) / time period
                  F = (change in (m * v)) / t
                  ''= m * ((change in v) / t), if m = constant
                  ''= m * a

                  Force, acceleration, and velocity have direction
                  Net force = sum of all forces (consider their directions!)
         Mass & Weight
                  Mass =/= weight, technically
                  Mass corresponds to amount of material
                          unit = kilogram

                  Weight is a force
                          unit = kg m / s2 (or pound in English system)
                  Weight corresponds to force associated with gravity
                  A box of candy (mass = 1 kg) sits on a table on Earth.
                  So, its weight is m*g = (1 kg)*(9.8 m / s2) = 9.8 kg m / s2

         Question: Does this statement make sense, yes or no, and why?
                  #1.) If you could buy a 9.8 kg m / s2 box of candy on the moon,
                  you would get a lot more candy than in a 9.8 kg m / s2 box of
                  candy bought on earth. Yes or no?
        How your body feels about velocity and acceleration
                  Feel weightless when you:
                          fall ('free fall')
                          jump from diving board
                          if you were the space shuttle or fly in it
                  Objects orbiting the Earth 'feel' like are in free-fall

         Leaving the Earth
                  Imagine shooting off 3 cannon balls at different speeds:
                  Slow object falls back to earth
                  Faster object goes into orbit (8 km/s to 11 km/s)
                  Even faster object escapes Earth's gravity (12 km/s)

         The spoiler: friction

         Question: Does this statement make sense, yes or no, and why?
                  #2.) Suppose you can go into a vacuum chamber on Earth,
                  or standing on the moon (i.e. no air friction).
                  You simultaneously drop a hammer and a feather.
                  Both would hit the floor simultaneously. Yes or no?

Newton's Universal Laws of Motion

         They are "universal" = they work everywhere
         1st Law: If there is no (net) force on an object, the object's
                  velocity will not change. Think of inertia
         2nd Law: Force = (change of momentum) / time
                  = mass * acceleration, if m = constant        
                  Remember, this can happen because of changing direction!
         3rd Law: For any force, there is an equal and opposite
                  "reaction" force         
                  Ex: As you sit in a chair, you feel the chair holding you up and
                  the chair feels compressed by you
                  Ex: When you hit a ball with a baseball bat (or ping pong
                  paddle) the ball feels a force (forward) and the bat (or paddle)
                  feels a force (backward)
                  Ex: Sun pulls Earth toward itself, and Earth pulls Sun
                  toward itself

         Newton's Laws --> Conservation of Momentum
                  Momentum = m * v = constant, unless a force is applied

                  Can apply momentum conservation idea to a system of 2
                  objects. Unless a net force is applied to the system, the
                  momentum of the system remains the same
                           Reason: If one object applies a force on another,
                  then the second applies an equivalent force to the first,
                  so that the momentum of the system remains the same

         Rotational Version of Conservation of Momentum
                  For orbiting or spinning objects
                  Analog of momentum -> angular momentum
                  Angular momentum of object moving in a circle = m x v x r

                  Analog of force -> torque
                  Torque = (change in angular momentum) / time

                  If there is no torque (torque = 0), then
                  angular momentum = constant         
                  In astronomy, we see many examples
                  Case with torque =/= 0:
                  Torque = (change in (m x v x r)) / time, so radius is important

         Conservation of Energy
                  Kinetic Energy
                  Thermal Energy, note thermal energy, =/= temperature
                  Potential Energy, ex: waterfall and water wheel
                  Mass Energy
                  Can convert from one type of energy to another, cannot destroy energy

The Force of Gravity

         One of the most important forces in astronomy
         Example: Galaxies attracted to each other and merging

         Universal Law of Gravitation
                  1.) Every mass attracts every other mass, through force of
                  2.) The force is proportional to M1 times M2
                  3.) The force weakens as 1/(distance)2
                  Formula: Fg = G M1 M2 / d2
                  G = Gravitational Constant = 6.67 x 10-11 m3/(kg sec2)

         Newton's work explained and extended Kepler's Laws for Orbits
                  Kepler's first two laws apply directly to all orbiting bodies                  
                  Kepler's 3rd law can be generalized to all orbiting systems
                  Kepler's 3rd Law:
                           (Planet's period in years) 2 = (orbital distance in AU)3
                           1 AU = astronomical distance = distance between Earth & Sun
                  Generalized form:
                           period2 = p2 = a3 [ 4 (pi)2 / G (M1 +M2) ]
                           or: M1 + M2 = a3 4 (pi)2 / (G p2)
                           p = period
                           a = distance = "semi-major axis"
                           M1 = mass of 1st object
                           M2 = mass of 2nd object
                           pi = 3.14
                           G = Gravitational Constant = 6.67 x 10-11 m3/(kg sec2)
                  Can use this equation to "measure" the mass of an object!

                  All objects in the system orbit about the center of mass
                  Example: binary stars
                  "Bound orbit" (object makes full loop around Sun)
                           circular orbits, elliptical orbits
                  "Unbound Orbit" (object comes near Sun, then leaves)

         The Universal Law of Gravitation meets Fgravity = mg
                  Consider this ball,
                  Fgravity = massball * "g"
                  and Fgravity = G Mball MEarth / d2
                  where d = distance from center of Earth to ball (i.e. Earth's radius)

                  therefore, "g" = G MEarth / d2
                           Notice that "g" doesn't depend on Mball
                           Could have used a hammer or a feather (as in Moon film)
                           All objects feel the same gravitational acceleration

         Orbital Energy and Escape Energy
                  Orbital energy of planet/moon/asteroid/comet/whatever
                  = kinetic energy + potential energy
                  Unless you give or take away energy from the planet/etc,
                           it has constant energy, so stays on its orbit
                  How to give or take away energy:
                  "Gravitational Encounters" transfer energy
                           Here, comet loses orbital energy and Jupiter gains it

                  The comet started with an unbound orbit (not destined to
                           make loops) lost energy to Jupiter, and ended up in a
                           bound orbit (destined to make loops)
                           So, Ebound orbit < Eunbound orbit

                  Reaching escape velocity requires getting enough energy to
                           go from a bound orbit to an unbound orbit


So far, we've talked about Sun/Earth/moon/etc as if they were completely rigid
But, what if they have some plasticity (flexibility, deformability)?

Interesting effects. Earth example = tides
         Tides =/= waves
         Tides = ocean level rising & falling twice/day
         Underlying reason
                  Earth (rock + oceans + atm) stretches because of moon's pull
                  Oceans "do most of the stretching"
                  Oceans shift to moon-facing and moon-opposing sides
                  Earth rotates under the distribution of water
                  Figure of Moons tides on the Earth
The Sun gets into the act.
         If the sun is along the Earth-moon axis, then contributes to the
         stretch. Makes a "Spring Tide"
         If the sun is perpendicular to the Earth-moon axis, then counteracts
         stretch. Makes a "Neap Tide"

Tidal Friction
         As Earth's solid part rotates under the oceans, there is Tidal Friction
         Pulls oceans along, so ocean "bulges" ahead of Earth-Moon axis
         Slows down Earth's rotation
         Think conservation of angular momentum of Earth-Moon system
         or, think about the moon being attracted to the deformed Earth
                  Moon gets sped up
                  So, moon slips further away from Earth

Synchronous Rotation
         Tidal forces make the two bodies want to face each other
         => Moon has same side facing Earth
         To do so, moon rotates with same period as it orbits
         Pluto and it's moon, Charon BOTH rotate with same period as orbit
                  Figure of Pluto-Charon Synchronous Rotation