Download Gravity: the Laws of Motions

Today’s Lectures:
Important Physical Quantities,
The Laws of Gravity
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Speed, velocity, acceleration, force
Mass, momentum and inertia
Newton’s three laws of motion
The Force of Gravity
Assigned Reading
• Chapter 5.1, up to page 92, not including
orbital velocity.
Before Newton
• Kepler understood the phenomenology of planetary
motions, I.e. the three laws, but never understood
*why* planets move the way they do
– He thought of magnetic force from the Sun
– He even thought (and dismissed) that Angels pull planets
along their orbits
• Galileo understood motion and its causes, like
inertia, force and acceleration very well…
• …but fell short of either understand or explain the
causal relationship between force and motion, i.e.
acceleration. He also did not propose a universal
law for the force of gravity, i.e. he did not
recognize gravitation as a universal interaction.
• For that, humanity had to wait for Newton
Describing Motion
• Motion occurs when the position of an object
relative to some reference frame changes with
time
• If position does not change, the object is at rest
• To describe motions we need a reference frame
and we need to monitor position and time
• The rate at which an objects covers a given amount
of space in a given amount of time is called speed
–
v = s/t
Vectors,
or things with a sense of
direction
• There are physical quantities in nature for
which only one number, their intensity,
tells the whole story: e.g. temperature,
mass, luminosity, color.
– We call them SCALARS
• Other quantities need both an intensity
and a direction to be fully described: e.g.
speed, force, acceleration.
• These quantities with a “direction” are
called VECTORS
Velocity and Acceleration
Acceleration (a) is the change of a
body’s velocity (v) with time (t):
a
a = Dv/Dt
Velocity and acceleration are directed
quantities (vectors)!
v
Different cases of acceleration:
1. Acceleration in the conventional
sense (i.e. increasing speed)
2. Deceleration (i.e. decreasing speed)
3. Change of the direction of motion
(e.g., in circular motion)
Vocabulary Words
• Speed is the distance traveled in each
second – it’s measured in meters per
second (m/s).
• Velocity is the speed in a particular
direction – it’s measured in meters per
second (m/s).
• Acceleration is the change in velocity in
each second – it’s measured in meters per
second per second (or m/s2)
More On Acceleration
• An acceleration is a
change in velocity.
• Acceleration occurs when
either the magnitude or
direction of the velocity
(or both) are altered.
• Uniform Circular Motion
is Accelerated Motion
If an object does not accelerate, that means that the total
sum of forces acting on it is zero. It does not mean that
there are no forces on it. Just the they cancel out
Which of the following does NOT
describe an acceleration:
• a car traveling with constant speed
around a bend
• a car decreasing speed on a straight
road
• a car traveling with constant speed on
a straight road
• a planet traveling around a Sun
Momentum
• But Velocity is not the whole story, two objects of different
mass, e.g. a train and bike, might move with the same velcity, but
they carry different “quantity of motion”.
• Momentum is a quantitative way to describe an object’s “quantity
of motion”.
• It also describes the tendency to continue to move in the same
way it was moving previously (“… keep your momentum…”).
• An object’s momentum is simply its
mass times its velocity.
P = m·v
Two objects of different mass subject to the same acceleration
acquire the same speed, but different momentum.
Don’t forget that velocity is a vector, so momentum is a vector: the
direction of motion is very important!
Galileo’s Law of Inertia
• momentum
P = m·v
• An object maintains its “quantity of
motion” (momentum), unless a force is
acted upon it
• Only a force can change the “quantity of
motion” (momentum) of an object, either
by changing its speed, or by changing its
direction of motion, or both
Galileo and the Concept of Inertia
Aristotle held that objects at rest remained
at rest unless a force acted on them, but
that objects in motion did not remain in
motion unless a force acted constantly on
them: F = v
WRONG.
Galileo concluded that an
object in a state of motion
possesses an ``inertia'' that
causes it to remain in that
state of motion unless an
external force acts on it
RIGHT.
Vectors: when the force is not parallel to
velocity (momentum)
• Earth and moon attract each other through gravitation, i.e. there is
a force between them.
• Earth’s gravitational force constantly
accelerates the moon towards Earth.
Dv
• This acceleration is along the direction
of the force, i.e. along the radius
•The velocity is perpendicular to the
force
•This force is constantly changing the
moon’s velocity.
v
•But it only changes the direction of the
F
velocity vector, not its intensity (speed).
•So, when the force is perpendicular
to the velocity, velocity varies in
direction not in strength
v’
Moon
Earth
Sir Isaac Newton and the
Unification of Physics & Astronomy
• Newton was by many standards the
most important figure in the
development of modern science.
• He demonstrated that the laws
that govern the heavens are the
same laws that govern the motion
on the surface of the Earth: the
universality of gravity, and hence
of physical laws.
• Newton's Three Laws of Motion.
• Theory of Universal Gravitation
(1642-1727)
Newton’s First Law of Motion
(really, Galielo’s Inertia Law)
• In the absence of a net force, an object
moves with constant velocity, or it
conserves its momentum.
• (paraphrased) If nothing acts on an object,
the object will keep moving in a straight line
and at a constant speed.
• The same is true for zero velocity.
Newton’s Laws of Motion (1)
1. A body continues
at rest or in
uniform motion in
a straight line
unless acted upon
by some net force.
An astronaut floating in space
will continue to float forever in
a straight line unless some
external force is accelerating
him/her.
Newton’s Laws of Motion (2)
•
A body subject to a
force, changes its
momentum, i.e.
accelerates, and the
acceleration a of a body
is:
1. inversely
proportional to its
mass m,
2. directly proportional
to the net force F,
3. and in the same
direction as the net
force.
a = F/m  F = m a
Newton’s Laws of Motion (3)
3. To every action,
there is an
equal and
opposite
reaction.
The same force that
is accelerating the
boy forward, is
accelerating the
skateboard backward.
Newton’s First Law can also be stated
as:
• In the absence of a net force, the
acceleration of an object must be zero
• If an object is being accelerated, there
must be a net force exerting on it
Newton’s Second Law Of Motion
• Acceleration is caused by force but also
related to the mass of the object
Force = Mass x Acceleration
F = m·a
Or
a = F/m
Gravity is such that acceleration is constant in any given
planet, not the force. So, should a heavy object fall
faster than a light one, as Aristotle believed?
Smaller force
Bigger force
Why di they fall with the same
speed? The it is not true that
F = v…
The Falling Body Experiment on the
Moon
The crew of the Falcon L.E.M., of
the Apollo 15 mission, repeating
Galileo’s experiment on the Moon
No air on the Moon, so drag by air
not a factor in the experiment
Survey Question
Two identical spacecraft are to be accelerated by
rockets. The first rocket fires with a force 4 times
as great as that of the second rocket. The
acceleration of the first rocket is _____ times as
great as the acceleration of the second rocket.
1)
2)
3)
4)
5)
1/4
1/2
the same
2 times
4 times
Newton’s Third Law Of Motion
• For every ACTION (I.e. application of a
force) there is an equal and opposite reACTION
F1 = -F2
Newton’s Third Law
For any force, there always is an equal and
opposite reaction force.
Box pushes down
on table due to gravity.
Table
pushes
back on
box
Example: The Jet Engine
• What makes a jet-propelled airplane move
forward?
The Third Law demonstrated: without
external force, momentum is conserved
If one isolated system (i.e. skate-board + human) is at rest
(I.e. v=0) is suddenly separated into two parts (i.e. the
human jumps off), one moves in one direction and the
other must move in the opposite direction!
Before separation: P1 + P2 = 0
After separation: P1 = m1·v1
P2 = m2·v2
CONSERVATION: P1 + P2 = m1·v1 + m2·v2 = 0
THUS: m1·v1 = -m2·v2
THUS:
To the force that created the first motion…
there is an opposite-acting force that creates the
second motion
Gravity
• What force is responsible for motions in the
universe?
• What force makes objects fall?
• What keeps us on the rotating Earth?
• Why don’t planets move in straight lines, but
orbit around the Sun instead?
Gravity
• We can summarize the universal law of
gravitation with the following statements:
– Every mass attracts every other mass through the
force of gravity.
– If mass #1 exerts force on mass #2, and mass#2
exxerts force on mass#1, the force must depend o
both masses, namely:
– The force of attraction is directly proportional to
the product of the two masses.
– The force of attraction is inversely proportional to
the square of the distance between the masses.
How did Newton figure out Gravity: if the apple was
initially at rest and then started to fall, there must be a
force acting on the apple. What force?
More importantly, if the force of
gravity reaches to the tree, might
it not reach even further?
In particular, might it not reach
all the way to the orbit of the
Moon?
Newton calculated what the
Earth’s gravity would be at the
position of the Moon. And found
that it is what would be required
to keep the Moon in its orbit
The Universal Law of Gravity
• Any two bodies are attracting each other
through gravitation, with a force proportional
to the product of their masses and inversely
proportional to the square of their distance:
F=-G
Mm
r2
(G is the Universal constant of gravity.)
How did Newton calculate
Earth’s gravity at the
position of the Moon?
F~
2
1/D
To calculate Earths’ gravity
on the Moon, Newton assumed that
Gravity obeys the inverse square law
The Law of Gravity
Near Earth’s
surface
M 1M 2
Fg  G
2
d
Fg  gM 2
G = 6.67x10-11 m3/kg/s2
M1
g G 2
d
2
 9.8m/s
d
M1
M2
Mass and Weight
• Mass is a measure of how much material is
in an object.
• Weight is the force exterted by gravity on
a massive body (body with mass), e.g.
placed on the surface of Earth
• Weight is a measure of the gravitational
force exerted on that material.
• Thus, mass is constant for an object, but
weight depends on the location of the
object.
• Your mass is the same on the moon, but
your weight on the surface of the moon is
smaller
Remember these two very, very important concepts:
1)Your mass decides the total force of gravity acting on you
• Both you and a truck have the same distance to the center of
earth.
• Yet, the force of gravity on the truck is bigger
2)The gravity of Earth decides your acceleration. In other words, when
object fall under gravity, they all have the same acceleration, regardless
of their mass (remember the two bodies falling from Pisa’s leaning
tower?)
The gravitational force on an object near
the surface of Earth is:
Fgrav = m·g
(g = 9.8m/s2)
Question: How do objects accelerate due
to the force of gravity?
M 1M 2
Survey Question Fg  G d 2
Two equal masses, m, separated by a distance,
d, exert a force, F, on each other due to
their gravitational attraction. How large is
the gravitational force between an object of
mass m and an object of mass 2m separated
by the same distance d?
¼F
½F
F
2F
4F
M 1M 2
Survey Question Fg  G d 2
Two equal masses, m, separated by a distance,
d, exert a force, F, on each other due to
their gravitational attraction. How large
would their gravitational attraction be if the
distance between them was doubled?
1) ¼ F
2) ½ F
3) F
4) 2F
5) 4F
M 1M 2
Survey Question Fg  G d 2
If aliens magically turned our Sun into a black hole
of the same mass but 10 times smaller in diameter,
what would change about the Earth’s orbit?
1) it would be 10 times smaller in radius
2) it would spiral into the black hole
3) nothing would change
4) it would spiral away from the black hole
5) it would be 10 times larger in radius
… so why don’t planets just fall
into the sun?
M1
M2
… because they miss (that is, they
have enough tangential velocity to
always miss)
v
Fg
Fg
M1
This is the concept of an orbit.
M2
Why doesn't the earth fall to
the sun?
• It has a velocity
and it has
inertia!
• Force of gravity
causes change in
the direction of
velocity --acceleration.
• The earth is
falling towards
the sun all the
time!
V=8km/s
Why don't they fall?
They are circuling Earth at a speed of 8 km/s!
Quiz


Astronauts inside the space shuttle float
around because ____
they are falling in the same way as the
space shuttle.
If you are in a free-falling elevator, you
are massless. (true or false)
false