Transcript Ch 10 - Genovese

Chapter 10 - Gravity and Motion
Newton’s First Law of Motion
• 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.
• The natural behavior of objects is to continue to
move however they have been moving (inertia).
• Any time a body changes how it is moving, there
is always some force that caused that change.
Orbital Motion and Gravity
• A force is any kind of push or pull exerted by one
object on another.
• Besides contact, friction, electric, magnetic,
elastic, pressure, etc. forces, Newton said that
objects also exert a gravitational force on each
other.
• The force of gravity causes all bodies to attract
all other bodies.
• Gravity, coupled with laws of motion, enabled
Newton to explain exactly how orbits work.
The Moon and the Earth
• We just learned that the Earth
will exert a gravitational
forces on the Moon pulling
the Moon towards the Earth.
• So, what holds the Moon up?
• Why doesn’t it fall down like
if you drop a rock?
Moon
Gravitational
Force
Earth
The Moon and the Earth (2)
• If the Moon was just sitting up
there, it would fall straight down
onto the Earth.
• But the Moon is moving,
“sideways” at a pretty high speed.
• The Moon does fall down, but it is
moving sideways at the same time.
• Just like if I throw a baseball, it
moves across the room while
falling downwards.
Moon’s
velocity
Moon
Gravitational
Force
Earth
The Moon and the Earth (3)
• Without gravity, the Moon would
move in a straight line, flying away
from the Earth.
• The orbit is a balance between the
natural straight-line motion and the
attractive pull towards the Earth.
• The Moon is always falling towards
the Earth but it is also always
shooting away from the Earth.
Moon’s
velocity
Moon
Gravitational
Force
Earth
Path followed by the Moon
The Sun and Planets
• Orbits of planets around the Sun work just like the
Moon’s orbit around the Earth.
• If the gravitational force and orbital speed are
exactly balanced, a planet will orbit in a perfect
circle.
• If the planet’s speed is a little faster or slower, a
non-circular orbit results.
• If the planet’s speed is much too fast or slow it
may escape the Sun altogether or fall into the Sun.
Unfinished Details
• More details about: exactly how gravity works, the
shapes of orbits, and escaping from the gravity of
a planet will all be discussed later in the chapter.
The Earth, a Pencil, and Gravity
• The Earth exerts a force of gravity on a pencil
causing it to fall (accelerate) to the floor, but
clearly the pencil does not exert an equal force on
the Earth! Right?
• The pencil moves but the Earth just sits there.
• The forces are equal! That does not mean the
accelerations are equal.
a = F/m
• The Earth has a mass 1027 times more than the
pencil, so for the same force it has 1027 times less
acceleration, immeasurably small.
The Law of Gravity
• The three laws of motion would’ve been enough
to make Newton famous, but that’s just a small
fraction of his accomplishments.
• 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 to astronomical objects like the Moon,
Sun, 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 gravitational attraction is given by:
F = G M m / r2
where
F is the gravitational force (in newtons)
M, m are the masses of the attracting bodies (in kilograms)
r is the distance between the (centers) of the bodies
G is a proportionality constant that depends on units
Gravitational Constant G
• In the usual metric units,
G = 6.67 x 10-11 N m2 / kg2
• Fear not, you will not have to memorize this
number or these units, the value and units will be
supplied every time it is needed - which will be
quite a bit.
I know, all these dry equations and laws are boring.
So here’s a change of pace, we will resume the usual
boring lecture momentarily.
Intermission
Law of Gravity Example
Example: Calculate the force of gravity exerted by
the Earth on a 7-kg bowling ball.
Solution: F = GMm/r2
G = 6.67 x 10-11 Nm2/kg2
M = MEarth = 6 x 1024 kg
m = Mball = 7 kg
r = distance between centers = radius of Earth =
6,378,000 m
F = (6.67 x 10-11)(6 x 1024)(7)/(6,378,000)2 N
= 68.9 N (kg and m units all cancelled out)
Second Law Example
Example: If the bowling ball from the previous
example is dropped, how fast will it accelerate?
Solution: a = F/m
F = 68.9 N
m = 7 kg
a = (68.9)/(7) = 9.8 m/s2
The Earth also feels the force of 68.9 N but
accelerates much less (essentially zero
acceleration) because of its far larger mass.
Acceleration of Falling Objects
• To calculate the force of gravity acting on the
bowling ball, we had to multiply by 7 kg (the m in
F = GMm/r2).
• Then to get the acceleration due to this
gravitational force we divided by 7 kg (the m in
a = F/m).
• So the 7 kilograms didn’t matter. We would get
the same acceleration for any object with any mass
at the surface of the Earth, 9.8 m/s2.
Galileo’s Experiment
• Aristotle had claimed that
heavier objects fell faster
than lighter ones. Twice the
weight would fall twice as
fast he said.
• Galileo did experiments that
easily proved this was not
true (although that he did the
experiments/demonstration
at the Leaning Tower of Pisa
is believed to be a myth).
Universal Laws
• How did Newton decide that this was the right law
of gravity? (F = GMm/r2)
• Because only this particular equation explained
both objects moving on Earth and the motions of
planets (elliptical orbits, equal areas in equal
times).
• Newton derived the correct law of gravity from
Kepler’s laws.
• In so doing, he then discovered that Kepler’s laws
had some slight inaccuracies.
Newton’s Model
• Newton had his own answer to the geocentric/
heliocentric debate.
• He said neither the Earth nor the Sun is at the
center of the universe.
• Newton believed the universe to be infinite and
center-less with everything moving around.
• The infinite universe remains a viable picture of
the universe today.
Perturbations
• The other part of Kepler’s first
Jupiter
law was also modified, planets
do not move along perfect
ellipses.
• Planets “perturb” (deflect or
Perturbed orbit
Mars
nudge) each other with their
due to Jupiter
gravity, causing tiny wiggles in
the paths the planets follow (only
measurable using telescopes).
Elliptical Orbit
• Newton’s laws allow exact
Sun
predictions of these
perturbations.
Measuring Mass Using Orbital Motion
• We’ve learned that a lot of things are connected.
– The masses of bodies and their separation determines
how much gravitational force they exert on each other.
– The force on a body and its mass determines what
acceleration it will experience.
– Acceleration is the rate of change of velocity, so
knowing how big an orbit a planet is following and how
long it takes to orbit would allow you to calculate what
acceleration it is undergoing.
Gravitational Force
• The gravitational force between objects is directly
proportional to the mass of the particles and inversely
proportional to the square of the distance between them.
• F= mxM / d2
• F=Force, m=mass of object, M=mass of earth/mass of
second object, d=distance apart
• The quantity of the universal constant of gravitation, G
must be inserted
• G = 6.67x10-11 Nm2/kg2
Gravitational Force
• Which gives us,
– Force = G x m1 x M2 / d2
– Gravitational force is measured in newtons, like all
other forces
• Examples
– The gravitational force of attraction between the Earth
and the sun is 1.6x1023. What would be the force if
Earth were twice as big?
• F=Gm1M2 / d2
• = 2(1.6x1023N)
• = 3.2x1023N
Gravitational Force Example
• Ben, whose mass is 85kg, sits 2.0m apart from
Nick, whose mass is 100kg. What is the force of
attraction between Ben and Nick? Why don’t Ben
and Nick drift toward one another?
– Force = G x m1 x m2 / d2
–
= (6.67x10-11 Nm2/kg2)(85kg)(100kg) / 2m2
–
= 1.41x10-7 N
Gravitational Force Example
• When Sammy was 10 years old, she had a mass of
22kg. By the time she was 16 she then weighed
45kg. How much larer is the gravitational force
on earth at age 16 compared to age 10?
– Force = G x m1 x m2 / d2
–
= (6.67x10-11 Nm2/kg2)(22)
–
= 1.46x10-9
– Force = G x m1 x m2 / d2
–
= (6.67x10-11 Nm2/kg2)(45)
–
= 3.00x10-9
Gravitational Acceleration
• You can use the law of universal gravitation to
find the gravitational acceleration, g, of any body
if you know that body’s mass and radius.
• g = GM2 / d2
• d here is equal to the radius of the celestial body
Gravitational Acceleration
• Andy Ford is standing in the lunch line at
6.38x106m from the center of the earth. The
earth’s mass is 5.98x1024kg. What is his
acceleration due to gravity?
• g = GM2 / d2
• = (6.67x10-11 Nm2/kg2)(5.98x1024kg) / 6.38x106m
• = 9.8m/s2
Gravitational Acceleration
• What is the acceleration due to gravity on the sun
if the radius of the sun is
• g = GM2 / d2
Escape Velocity
• Throw an object upwards: it goes up, stops, falls
back down.
• Throw the object up with a faster speed and it will
go higher before falling back.
• Throw an object fast enough (called the escape
velocity) and Earth’s gravity is not enough to stop
it and bring it back. It escapes into space.
• Guess what? You can use Newton’s laws to derive
a formula for escape velocity! You want to know
what the formula is? Okay…
Escape Velocity Formula
• Escape velocity vesc2 = 2 G m / r
G = gravitational constant
M = mass of planet
R = radius of planet
where
• Example:
•
Calculate the escape velocity from Earth given
that the mass of Earth is 6 x 1024 kilograms and its
radius 6 x 106 meters.
Example: Earth’s Escape Velocity
• Solution:
• Escape velocity vesc2 = 2 G M / R
G = 7 x 10-11 m3/(kg s2)
m = 6 x 1024 kg
r = 6 x 106 m
vesc2 = 2 (7e-11) (6e24) / (6e6) m2/s2
= 11,832 m/s
= 11,832 m/s (1 km/1000 m) = 11.8 km/s
End of Lecture Review
• What should you have learned?
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–
–
–
–
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How gravity and inertia cause orbits
Newton’s law of gravity
The Sun moves and planets perturb each other
That orbital properties can be used to find masses
What “surface gravity” and “escape velocity” are
How to calculate gravitational force, gravitational
acceleration, and escape velocity