Newton’s theory of

Forces Known to PhysicsThere are four fundamental forces known to physics: • Gravitational Force today • Electromagnetic Force later in Physics 1 and 2 • Weak Nuclear Force • Strong Nucl

Physics I Class 17

Newton’s Theory of

Gravitation

Forces Known to Physics

There are four fundamental forces known to physics:

• Gravitational Force (today)

• Electromagnetic Force (later in Physics 1 and 2)

• Weak Nuclear Force

• Strong Nuclear Force

(All forces we observe are comprised of these fundamental

forces Most forces observable in everyday experience are

electromagnetic on a microscopic level.)

Forces in Physics 1

(so far)

We have encountered the following forces in Physics 1:

• Gravity

• Ideal Springs (Hooke’s Law)

• Pushes and Pulls

• Friction

What makes gravity different from the other three?

(Hint: The ideal spring force is also conservative,

so that isn’t the answer.)

Newton’s Theory of Gravitation

Isaac Newton, 1642-1727

In 1666, our old friend, Isaac Newton, was musing

on the motions of heavenly bodies while sitting in agarden in Lincolnshire England, where he had gone

to escape the plague then ravaging London

What if the force of gravity, the same force that causes an apple to

fall to the ground in this garden, extends much further than usually

thought? What if the force of gravity extends all the way to the

moon? Newton began to calculate the consequences of his

assumption…

Newton’s Law of Universal Gravitation

rˆ r

m

m

G

F  = 1 2 2

The meaning of each term:

F : Gravitational force on object 1 from object 2

G: Universal gravitational constant = 6.673 x 10–11 N m2/kg2

r : Center distance from object 1 to object 2, squared

rˆ: Unit vector from object 1 to object 2

Properties of Gravity

Object 1

Object 2

Gravitational Force on 1 from 2

• Every object with mass is attracted by every other object with mass

• Gravity is a force at a distance (through occupied or empty space)

• Gravity is a “central” force (center-to-center for spherical bodies)

• Gravity varies as the inverse square of the center distance

• Gravity varies as the product of the masses

If Gravity Varies As 1/r2, Where Does g = 9.8 m/s2 Fit In?

Consider the force on an object near the surface of the earth

(Assume the earth is a sphere and ignore rotation effects.)

R = radius of the earth

M = mass of the earth

m = mass of the object

g m

rˆ R

M

G m

rˆ R

M

m

G

F  = 2 = 2 =  (What is the direction?)

g = 9.8 m/s2 only seems constant because we don’t go very far

from the surface of the earth

Gravity is a Conservative Force

Both the mathematical form of Newton’s Law of Universal

Gravitation and experimental evidence show that gravity is a

conservative force Therefore, we can find a gravitational

potential energy for an object with mass m being attracted by

another object with mass M

The gravitational potential energy is defined (for convenience)

to be zero at infinity We can calculate it by finding the

positive work from any point to infinity – you can find the

details in the book in section 13-6

r

M m

G r

d )

r (

M m

G r

d F )

We Have Two Formulas for

Gravitational Potential Energy!

Old: Ug( y ) = m g ( y − y0)

New:

r

M m G )

r

How could these be the same?

Consider a location near the surface of the earth, y0 = R, y = R+h.

The only thing that matters is ∆ U, not U itself.

−

=

∆

h R

1 R

1 M m G R

M m

G h

R

M m G

h M

G

m h

R

1 R

1 M G

(h << R)

h g m

h R

M G m R

h M G

≈

Class #17 Take-Away Concepts

1 Four fundamental forces known to physics:

• Gravitational Force

• Electromagnetic Force

• Weak Nuclear Force

• Strong Nuclear Force

2 Newton’s Law of Universal Gravitation

rˆ r

m

m G

F  = 1 2 2

3 Gravitational Potential Energy (long-range form)

r

M m

G )

r (

Class #17 Problems of the Day

_1 To measure the mass of a planet, with the same radius as

Earth, an astronaut drops an object from rest (relative to the

surface of the planet) from a height h above the surface of the

planet (h is small compared to the radius.) The object

arrives at the surface with a speed that is four (4) times what

it would be if dropped from the same distance above Earth’s

surface If M is the mass of Earth, the mass of the planet is:

C 8 M

Class #17 Problems of the Day

2 Calculate the acceleration due to gravity at the surface of the

planet Mars Assume Mars is a perfect sphere and neglect effects

due to rotation Use M = 6 4 × 10+23kg and R = 3 4 × 10+6m

Activity #17 Gravitation

(Pencil and Paper Activity)

Objective of the Activity:

1 Think about Newton’s Law of Universal Gravitation

2 Consider the implications of Newton’s formula

3 Practice calculating gravitational force vectors

Class #17 Optional Material

Part A - Kepler’s Laws of Orbits

Material on Kepler’s Laws

thanks to

Professor Dan Sperber

Kepler’s Three Laws

of Planetary Motion

1 The Law of Orbits: All planets move in elliptical

orbits having the Sun at one focus.

2 The Law of Areas: A line joining any planet to the

Sun sweeps out equal areas in equal times.

3 The Law of Periods: The square of the period of

any planet about the Sun is proportional to the cube

of the semi-major axis of its orbit.

Newton showed through geometrical reasoning (without calculus)

that his Law of Universal Gravitation explained Kepler’s Laws

Kepler’s Three Laws

of Planetary Motion

Try this link to see an animation:

http://home.cvc.org/science/kepler.htm

The Law of Areas

1 2

2 2

2

( θ )

ω ω

constant

The Law of Periods

ω

π

( )

2

2

Class #17 Optional Material

Part B - General Relativity

Material on General Relativity

thanks to

Albert Einstein

Where Did Newton Go Wrong?

(Again!)Albert Einstein (1879–1955)

(Check back to the optional material for classes 3 and 6 first…)

Einstein realized that something must be wrong with Newton’s

theory of gravity, because it implied that the force of gravity is

transmitted instantaneously to all points in the universe This

contradicts the fundamental limitation in the Theory of Special

Relativity that the fastest speed information or energy of any type

can travel is the speed of light

To overcome this problem Einstein postulated a third principle, the

Principle of Equivalence, to go with his two principles of Special

The Principle of Equivalence

In broad terms, the Principle of Equivalence states that there is no

experiment that one can perform to distinguish a frame of reference

in a gravitational force field from one that is accelerating with a

corresponding magnitude and direction

This is sometimes called the “Elevator Postulate” because we can

imagine a physicist in a closed elevator cab trying to determine

whether he is at rest on earth, or accelerating at 9.8 m/s2 far from

any planet, or perhaps on a planet where gravity is half that of earth

and the elevator is accelerating upward at 4.9 m/s2 According to

Einstein, there is no experiment that could detect a difference

The Principle of Equivalence

General Theory of Relativity

By 1915, Einstein had worked through all the math (with some help)

to show that his postulates led to a new theory of gravity based on

the effect of mass and energy to curve the structure of space and

time His theory has some startling implications, one being the

existence of “black holes” – regions of space where the gravity field

is so high that even light cannot escape The predictions of General

Relativity, including the existence of black holes, have been

confirmed by all experiments to date

Black Holes

Black holes are detected by the characteristic

x-rays given off by matter falling into them

If Newton’s Gravity isn’t true,

why do we still use it?

It’s a good approximation for most engineering purposes.

Massive Black Holes

In Galaxies NGC 3377, NGC 3379 And NGC 4486B