The Foundations of Science

Nature everywhere obeys the same simple laws.

Sir Isaac Newton (1642-1727)

First to derive physical laws which explain and predict the behavior of nature.

Inventor of Physics & Calculus

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

Link for animation

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.

Newton’s cannonball

From Principia

Newton’s Laws of MotionNewton’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 Velocity & Acceleration Vectors

• Velocity: a vector whose magnitude is the instantaneous change of distance with time, and whose direction is the in the direction of motion

• Acceleration: a vector whose magnitude is the instantaneous change in velocity with time, and whose direction is in the direction of the of the change in velocity.

V (t=2)

V (t=1)

a (t=1-2)

V (t=1)

V (t=2)

a (t=1-2) = dv/dt

Change in speed

Change in speed & direction

dv

dv

Acceleration:

Any change in speed and/or direction of motion.

For circular motion:

a = v2/rLink 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 GravityNewton’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.

GravityGravity

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 LawWe 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

Thus: (2πr/P) 2 = GM/rAnd: r3 / P2 = GM/4π2

Which is a constant. It equals 1 for our Sun in units of year & AU.

Newton Explains Galileo

The acceleration does not depend on m!Bodies fall at the same rate regardless of

mass.

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 of the equation

a = GM/R2

GravityGravity F = m a = G m ME / r2

a = G ME / r2

where: G = 6.67x10-11 m3kg-1s-2

ME = 5.97x1024 kg

On Earth’s surface:

r = RE = 6371 km Thus:

a = G ME / RE2 = 9.82 ms-2

•

a

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 objectd = ½ at2

v = atAn object falls a distance, d, starting at rest. What is its impact velocity?

We want to know the velocity, and we know the distance traveled and the acceleration.

v = a t

We know t from a and d:

t = (2 d/a)

Then: v = 2 d ad

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