Lecture Notes

Arny Chapter 2

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Read Essay II in the text if you haven't already.

Chapter 2 Gravity and Motion

Section 2.2 - Inertia
Galileo did experiments on how things move (physics experiments).
The prevailing laws of physics at the time were those of Aristotle, Galileo proved that most of Aristotle's ideas on physics were wrong!
He proved it by doing experiments.

We don't have time to compare all of Galileo's and Aristotle's ideas.
Aristotle believed moving objects naturally slow to a stop.
Galileo said that moving objects naturally want to continue moving at the same speed in the same direction.

If I push a chair across the floor, Aristotle says it continues a little while due to inertia but then slows to its natural state of rest.
Galileo says the chair wants to continue moving but is being slowed down by a force, friction with the floor in this case.
On ice the chair would move much further, the frictional force is less.
In space, there is no air-resistance (no air) or friction, so the chair would just keep on moving.
Just like planets can move around the Sun without ever slowing down.

Force = any kind of push or pull acting on an object.
Speed = velocity = rate of motion = distance / time.

Isaac Newton extended the work started by Galileo.
Newton summarized his work on motion in three laws.
Not to be confused with Kepler's three laws of planetary motion.

Newton's first law of motion:
1. A body continues in a state of rest or uniform motion in a straight line unless made to change that state by forces acting on it.

This law just summarizes Galileo's results.
Any time a body changes how it is moving, there must be a force causing that change.

Section 2.3 - Orbital Motion and Gravity
Newton claimed there was a force called gravity that causes all bodies to attract all other bodies.

Consider the Moon going around Earth.
First, what holds the Moon up?
If Earth pulls on the Moon with gravity, why doesn't the Moon come crashing down?

If the Moon was just sitting up there,
it would fall straight towards Earth.
But the Moon isn't just sitting there.
It is moving sideways very fast.
[No good pictures in text, Fig. 2.3
on page 78 is the best.]


(It's a long ways away, so it only
shifts its position slowly relative to
the stars.)

If Earth didn't pull on it, the Moon
would fly off in a straight line at constant velocity.
We have a balance between the natural straight motion and the force from Earth.
Gravity is always bending the path towards Earth, the sideways speed keeps it from crashing into Earth.
The Moon "falls into an orbit."

The same sort of thing happens for planets orbiting the Sun.
With just the right speed, you can get circular orbits.
If the speed is less (or more), you get elliptical orbits (Newton proved this).
Too much speed, and gravity will not hold the object into an orbit, it will escape.
No speed, falls into the Earth or Sun, see figure 2.4 on page 79.

Section 2.4 - Newton's Second Law of Motion
Objects move with some speed in some direction.
If either the speed or the direction changes, the body is said to be accelerating.
Acceleration is defined as the rate of change of velocity,
acceleration a = (change in velocity) / (time)
Note: a = acceleration and a = semimajor axis, can be confusing.

The chair sliding across the floor is accelerating because its speed is decreasing.
The Moon orbiting Earth is accelerating because its direction of motion is changing.
Newton's first law of motion tells us that accelerations (changes to motion) are always caused by forces.

Newton's second law of motion tells us the exact relationship between forces and accelerations.
2. The amount of acceleration (a) that force (F) can produce, depends on the mass (m) of the object being accelerated.
Simpler if stated in mathematical form:
F = ma or a = F/m
F = net (total) force in newtons (N)
m = mass of object in kilograms (kg)
a = acceleration of object (in m/s^2).
This is a special formula, as long as you use force values in newton units, masses in kilogram units, and acceleration in (m/s^2 or m/s/s) units, you can just plug in numbers to solve for an unknown.


Example: A 10-kg bike is pushed with a force of 40 N (that's 9 pounds of force), how fast will it accelerate?

Solution: a = F/m = (40) / (10) = 4 = 4 m/s^2 = 4 (m/s)/s
This means the bike will increase in speed by 4 m/s (= 9 mph) every second for as long as it is pushed with this force.



Mass measures the inertia (resistance to change) of an object.
Objects with larger masses will change their motion less for a given force.


Section 2.6 - Newton's Third Law
(we've temporarily skipped section 2.5)
3. When two bodies interact, they create equal and opposite forces on each other.
This is a very subtle law which is very often misused.
You may have heard this law before as "for every action there is an equal but opposite reaction".

Every force affects two objects.
If A exerts a force on B, then B will simultaneously be exerting a force on A.
The same type of force, same magnitude, but opposite direction.
If I push on the wall, then the wall is pushing on me!
If in outer space and not connected to anything, my force would cause the wall to go one way, and its force on me would cause me to go the opposite way.

Earth pulls on this eraser with gravity.
By Newton's third law, the eraser is pulling on Earth with gravity.
Each is pulling equally hard on the other.
But if I let go, the eraser flies toward the Earth while Earth just sits there, how can that be if the forces are equal?

The forces are equal, that does not mean the accelerations are equal.
a = F/m
For the same force, the object with the larger mass will accelerate less.
The Earth has a mass about 10^26 times more than the eraser, a huge difference.
So Earth's motion is unmeasurably small.


Section 2.5 - The Law of Gravity
Newton was not finished with his three laws of motion.
His range of accomplishments is astounding.
He also derived a law to explain gravity.
Called the "Law of Universal Gravitation" because he applied the law to both objects on Earth and planets.

Law of Gravitation:
Every mass exerts a force of attraction on every other mass. Further:
The strength of the force is directly proportional to the product of the masses divided by the square of their separation.

The force of attraction is given by
F = G M m / (r^2) where
F = gravitational force
G = proportionality constant = gravitational constant
= 6.67 x 10^-11 N m^2/kg^2 = 6.67 x 10^-11 m^3/(kg s^2)
(just a constant, always has this same value in these units).
M, m = masses of the attracting bodies
r = distance between the masses (centers of masses).

Example: Calculate the force of gravity exerted by the Earth on a 7 kg bowling ball.
Solution: F = G M m / r^2
G = 6.67 x 10^-11 N m^2/kg^2
M = Mearth = 6 x 10^24 kg
m = Mball = 7 kg
r = Rearth = 6378 km (1000 m/km) = 6,378,000 m
F = put in all the numbers = 69 N

Question: If dropped, how fast will the bowling ball accelerate?
F = 69 N, m = 7 kg
a = F/m = 69 / 7 = 9.8 m/s^2

The Earth will also feel a force of 69 N.
But because of its huge mass, will have essentially zero acceleration.

All objects near the surface of Earth fall downward (accelerate) at 9.8 m/s^2, at least they would all fall at this rate if there was no air-resistance.
The same rate for all objects because the masses cancel out as we saw in the previous calculation.
First determined by Galileo.
Aristotle believed larger objects fell faster than smaller ones.
Galileo proved that idea wrong with a very simple experiment.
[Do quick class experiment to illustrate.]
[Drop two objects of different weight simultaneously and observe for yourself!]
[This is also why the pendulum period is independent of mass (weight).]


Newton's laws of motion and law of gravity work for us on Earth's surface.
They also work to explain the motions of planets.
In fact, Newton derived his law of gravity to explain why Kepler's laws worked.
Only this equation for gravity gave elliptical orbits.
In doing so, he found some slight inaccuracies.

By determining the laws underlying planetary motion,
Newton was actually able to improve each of Kepler's Laws.
Let's see how Newton modified Kepler's first law:

K1 Orbits are ellipses with the Sun at one focus.
Newton changed this in two ways.

Sun
The Sun is not located at a focus.
The focus is actually in-between the Sun and planet, at a point called the "center of gravity".
The planets orbit this point, the Sun also orbits this same point.

The Sun moves!
Why not? It a real object, subject to gravity. (Newton's third law.)
It doesn't move much because it's much more massive than any of the planets.
This follows from Newton's 2nd law.

Newton has his own answer to the geocentric/heliocentric debate.
Neither the Earth nor the Sun is at the center of the universe.
Newton believed the universe to be infinite and centerless.
They both move (in fact the whole solar system moves in an orbit around the rest of the galaxy).

Perturbations
The other part of Kepler's first law was also modified by Newton.
Planets do not move in perfect ellipses.
Planets "perturb" (disturb or nudge or deflect) each other with their gravity.
Because Newton explained gravity, these perturbations are entirely predictable.

Perturbations are usually pretty small, undetectable with the naked eye.
But astronomers can easily detect these deviations with telescopes.

Neptune
[In the text, this discovery is discussed in section 1.5, at the start of section 9.4, and in one of the computer labs.]
Astronomers found the planet Uranus to be deviating from its expected orbit, even when perturbations of the other planets were included.
It was guessed that an 8th, unknown, planet might be the source of these perturbations.
Astronomers deduced where the 8th planet would have to be to cause these perturbations and, sure enough, there was Neptune.
Discovered in 1846, sometimes referred to as "the first planet discovered by mathematics".
Quite a theory Newton's got, it allows us to find new planets.

Pluto
Neptune also showed unexpected perturbations.
Calculations and searching led to the discovery of Pluto in 1930.
Pluto is very small and was not causing significant perturbations.
This was a lucky accident.
There were some errors for planet masses.
Recent measurements indicate there are no current unexplained perturbations.
No more large planets seem likely.


Section 2.7 - Measuring Mass Using Orbital Motion
Imagine a planet orbiting the Sun in a circular (for simplicity) orbit of radius r.
This distance r would be the semimajor axis length.
The planet follows a curved path, not a straight line, because of the gravitational pull of the Sun (because of a force).

Knowing the size of the orbit (r) and the period of time for the planet to orbit (P), you can figure out the acceleration being experienced by the planet and from that (using F = ma) you can figure out the force.
Knowing the force (F), the planet's mass (m), and the distance between Sun and planet (r), you can use Newton's law of gravity (F = G M m / r^2) to solve for the mass of the Sun (M).

P,r => a => F => M,m

The textbook goes through this derivation in detail, but that's not crucial.
[Extending Our Reach section on page 84.]
The result in the text is M = (4 pi^2 r^3) / (G P^2).
The text then plugs in numbers using the Earth's orbit around the Sun and correctly determines the mass of the Sun, M = 2 x 10^30 kg.

Motions and masses are related through gravity and Newton's laws of motion.
So the size and speed of orbits can be used to determine masses.

Using his laws, Newton was able to derive this result:
P^2 = (4 (pi)^2/(G (Msun + Mplanet)) a^3
This is Kepler's third law (P^2 = k a^3).
Essentially, Newton figured out what determines the constant k (the Sun's mass).

Rearranging gives: M + m = (4 (pi)^2/G) (a^3/P^2)
The (4 (pi)^2/G) is just a constant, =1 if we chose the right units.
M + m = a^3/P^2
M and m are the masses of objects orbiting each other in solar masses.
a = average distance between the objects in AU.
P = period of the orbit in years.

This is an incredibly useful result.
Whenever we see things orbiting each other in space, we can measure the average distance between them and the period of time for them to orbit each other.
Plug into this formula and out pops the total mass of the two objects.

Astronomers determined the exact mass of Earth by carefully measuring the orbit of a satellite.
Watching Mercury go around the Sun can be used to calculate the mass of the Sun.
Watching a moon going around Jupiter can be used to calculate the mass of Jupiter.
Watching two stars orbit each other (a "binary" star system) can be used to calculate the masses of the stars.
Galaxies orbit each other, from that we can determine the masses of entire galaxies.

Again the basic idea, gravitational forces bend objects into orbits, the masses of the objects determine the gravitational force, so from the orbit, masses can be calculated.
Basically, whenever we talk about masses of objects out in space, the mass was calculated using Newton's version of Kepler's third law.

< Usual end of lecture. >


Section 2.8 - Surface Gravity
The acceleration of falling objects at the surface of Earth is a = g = 9.8 m/s^2.
[This is also called the Surface Gravity of Earth.]

This value depends on the mass of Earth and radius of Earth, specifically
g = G M / R^2
where G = gravitational constant, M = mass of planet, R = radius of planet.
The formula can be used to determine the surface gravity on other planets

Section 2.9 - Escape Velocity
Throw an object upwards, it goes up, stops, and falls back down.
Throw the object with a faster speed, it will go up higher before falling back.
Throw an object fast enough to start (called escape velocity) and Earth's gravity is not enough to stop it and bring it back, it escapes to deep space.

Using Newton's laws, the formula for escape velocity can be derived, the result is
Vesc = square root ( 2 G M / R)
where G = gravitational constant, M = mass of planet, and R = radius of planet.


Example: (Chapter 2 Problem 2)
Calculate the escape velocity from Earth, given that the mass of Earth is 6 x 10^24 kilograms and its radius 6 x 10^6 meters. In this problem, round off G to 7 x 10^-11 meters^3/(kg s^2).

Solution: Everything is in the same units (for instance all the lengths are in meters) so we don't need to do any unit conversions.
This is just a plug-in-the-numbers problem.

Vesc = sq root (2GM/R) = sq root [2 (7 x 10^-11 m^3/kg/s^2) (6 x 10^24 kg) / (6 x 10^6 m)] = sq rt (140,000,000 m^2/s^2) = 11,832 m/s

The textbook normally changes escape velocities into units of km/s so I'll do the same,
Vesc = 11,832 m/s (1 km / 1000 m) = 11.8 km/s.


End of chapter 2.

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