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)
Example
Momentum
Momentum = mass * velocity
Momentum is transferable
Force
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)
Simulation
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?
Answer
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
Figure
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
Universal Law of Gravitation
1.) Every mass attracts every other mass, through force of
gravity
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!
Orbits
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
TIDES
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
