The Foundations of Science
Nature everywhere obeys the
same simple laws.
Sir Isaac Newton (1642-1727)
Inventor of Physics
& Calculus
First to derive
physical laws which
explain and predict
the behavior of
nature.
Newton at Cambridge
Newton enrolled as a law student.
At Cambridge, he realized his
fascination with nature.
He read the standard texts of
time, like Aristotle, and enjoyed
Galileo. But he realized that an
understanding of nature would
necessitate that he learn math.
His work in math cumulated, in
these first 2 years of school, in
the invention of calculus.
Newton’s retreat from the Plague
At 23 yrs old, Newton embarked on his investigations.
Newton and Gravity
Cambridge was closed because
of the Plague. As the story goes,
Newton was sitting under the
apple tree outside his farmhouse
(shown right) and while watching
the apples fall he realized that the
force that made the apples fall
also made the planets orbit the
sun. Using his newly invented
Calculus, Newton was able to
show that Kepler’s 3 laws of
planetary motion followed directly
from this hypothesis.
Link for animation
Newton’s cannonball
From Principia
Newton’s Laws of Motion
1. An object remains at rest or in constant
velocity unless acted on by a force.
2. Force equals mass times acceleration
3. Each action has an equal and opposite
reaction.
Velocity & Acceleration Vectors
Change in speed & direction
• Velocity: a vector whose
magnitude is the
V (t=2)
instantaneous change of
distance with time, and whose
V (t=1)
direction is the in the direction
of motion
• Acceleration: a vector whose
magnitude is the
V (t=2)
instantaneous change in
velocity with time, and whose
direction is in the direction of V (t=1)
the of the change in velocity.
a (t=1-2) = dv/dt
dv
a (t=1-2)
Change in speed
dv
Acceleration:
Any change in
speed and/or
direction of
motion.
For circular motion:
a =
2
v /r
Link to centripetal acceleration
1. An object remains at rest or in
constant velocity unless acted on
by a force
2. Force equals mass times acceleration
F = ma
What’s a force? A push or a pull. It can be caused by many
things including collisions, springs, baseball bats, people,
magnets, and most interestingly, gravity.
What’s Mass? A quantity intrinsic to an object that is a
measure of the object’s resistance to change in state of
motion. On Earth, objects that weigh more have move
mass, but objects have mass even in outer space, where
they weigh nothing. Physicist call the tendency to stay in
the same state inertia and m is called the inertial mass.
3. Each
action
has an
equal
and
opposite
reaction.
Einstein’s Law of Gravitation
One explanation for the motion in
the heavens and on Earth
Newton’s Law of Gravity
• All bodies exert a gravitational force on each other.
• The force is proportional to the product of their
masses and inversely proportional to the square of
their separation.
F = GmM/r2
where m is mass of one object, M is the mass of
the other, and r is their separation.
• G is known as the constant of universal gravitation.
Gravity
F = m1·a
A planet in circular orbit
Cancel out m1: v2/r = Gm2/r2
Multiply both sides by r:
v2 = Gm2/r
The speed of the planet depends on the mass of the
Sun and the planet-Sun distance in a precise way.
m1 = planet’s mass, m2 = Sun’s mass, r = planet-Sun distance, v = speed of planet
Derivation of Kepler’s 3rd Law
We assume a circular orbit:
Velocity of planet: v = 2πr/P
Velocity of an object in circular motion about a
mass, M, a distance r away:
v2 = GM/r
(2πr/P) 2 = GM/r
r3 / P2
= GM/4π2
Which is a constant.
It equals 1 for our Sun in units of year & AU.
Thus:
And:
Newton Explains Galileo
Newton’s 2nd Law:
F = ma
Newton’s law of gravity: F = GMm/R2
Set them equal
ma = GMm/R2
Cancel m on both sides
a = GM/R2
of the equation
The acceleration does not depend on m!
Bodies fall at the same rate regardless of
mass.
Gravity
F = m a = G m ME / r2
a = G M E / r2
where: G = 6.67x10-11 m3kg-1s-2
ME = 5.97x1024 kg
a
•
On Earth’s surface:
r = RE = 6371 km
Thus:
a = G ME / RE2 = 9.82 ms-2
The time of fall of a body is independent of mass, as shown
empirically by Galileo.
The Gravity of all things
The acceleration that we feel just sitting in the class room:
From the Earth:
a = G ME / RE2 = 9.82 ms-2
From the Moon:
a = G MM / DM2 = ? ms-2
From the person sitting next to you:
a = G MP / DP2 = ? ms-2
This is a homework problem for next week!
Constant Acceleration
•
•
The distance that an object travels under
constant acceleration, a, in time, t, starting at
rest:
d = ½ at2
The velocity of an object, starting at rest, after
traveling a time t under constant acceleration:
v = at
These equations were derived by Newton, using calculus.
Motion of a falling object
An object falls a distance, d, starting at
rest. What is its impact velocity?
d = ½ at2
v = at
We want to know the velocity, and we
know the distance traveled and the
acceleration.
v=at
We know t from a and d:
t = (2 d/a)
Then:
v= 2da
d
Jumping Niagara Falls
v = (2da) ½
Where:
a = 9.8 m/s
d = 55 m
Then:
v = 33 m/s
= 118 km/hr
Drop of 180 feet (55 meters)
Pretty silly thought: plunging off the Niagara falls.
Some daredevils
1901, first to survive the plunge
1930, did not survive
Einstein’s Question:
Note the acceleration of an object in a gravitational field:
F = ma = GmM/r2
We cancel the “m” from both sides:
a = GmM/r2
Why should these “m”s cancel?
“m” from “ma” measures the body’s resistance to motion.
“m” from “GmM/r2” measures its gravitational attraction.
Is this a coincidence? Einstein says No!
Einstein’s answer leads to the theory that mass distorts
space (and time) and objects move in natural motion
according to a distortion of space and independent of mass.
If I have seen further than others, it was because
I have been standing on the shoulders of Giants.
- Isaac Newton