Chapter 11Newton

When, in the year of Galileo's death,Newton, the mightiest of the sons of light,Was born to lift the splendor of this torchAnd carry it, as I heard that Tycho saidLong since to Kepler, 'carry it out of sight,Into the great new age I must not know,Into the great new realm I must not tread."

What is force?

Force is the agency to realize a change in the state of motion, i.e. acceleration. Precise definition of Force was vague up to the arrival of Newton

"A body remains in its state of rest or uniform motion in a straight line, unless it is compelled to change that state by an outside force acting on the body."

Newton extended Galileo’s Law of Inertia

Force causes a change in the state of motion: acceleration.When the same force is applied to different objects, the acceleration is different.What makes acceleration different is a unique property of the body, Newton called it mass.

The more the mass, the more the resistance to any change in velocity, i.e., the more the resistance to acceleration.

a = F/m

Put together, the two concepts of force and mass spell: F = ma

Key to the mechanical universe

F = ma unifies all motion

a Fa 1/m

m

Equilibrium

position

m

m

Force generated by fan motor on trolley (demo)Acceleration = Force/mass

Increase massAcceleration decreases as expected from a = F/m

Force is a vector : direction and magnitude

Mass is an intrinsic property of all matterIt unifies the motion properties of all matterMass resists any change in the state of motion

Mass is not weight - Weight is a force - the force with which the earth attracts a body = F = mg

Unit of mass = kgUnit of Force = kg x m/sec2 = newton

What is the force of "attraction of the earth" (gravity) on an object of mass = 1kg.

m = 1kga = g = 9.8 m/sec2

F = ma = 1 x 9.8 kg m/sec2 = 9.8 newtons

The meaning of mass can be understood from its relationship to(a) how much force is needed to stop a moving object(b) how much force is needed to accelerate an object(c) how much force is needed to change the direction of an object moving at constant speed(d) all of the above.

A body of the same mass can have different weightsweight at equator is slightly different from weight at poles, why?Because the acceleration is differentBut mass is the same

Later we will see how Your weight on the moon is less than your weight on earth

OrWeight in orbiting satellite can be zero, but mass is the same

F = mg

F = (g – 0.0338) ---- Weigh Less at the Equator!!!

F = ma - The Key to the Mechanical Universe

Newton used the same key in two waysFirst method: a = F/m

If you know the force(s), you can deduce the motion - acceleration = a

> velocity vs. time > distance vs. time, time taken to travel a particular distance, or the period of oscillation

Vf = Vi + atVf

2 = Vi2t = 2ad

d = vit + (1/2)at2

Example - EquilibriumYou know that F = 0 (Also definition of equilibrium)

=> a = 0=> velocity is constant = v, => d = vt

If you do not know the force, you can deduce the force from the motion.Example: We can deduce the existence of a Reaction Force

Weight = mg

Reaction = mg

a= 0 F net = 0

F1

F 1 = mg

F net = F 1 + R = 0

R = -F 1

Look at gravity force, earth on mass

Increase massAcceleration stays the same! (remember Galileo at Pisa)

What does this tell us about the gravity force?

Earth’s gravity force also increases with mass

Fgrav mass……

different from F= ma

Force generated by fan motor on trolley (demo again)Acceleration = Force/mass

Increase massAcceleration decreases as expected from a = F/m

As if the donkey pulls harder to provide the same accelerationknowing that the farmer doubles the load

Force of gravity on apple proportional) mass of apple

F gravity mF gravity = K m(K = constant)

Apply F = ma (Newton’s Law of Motion)

F = ma = F gravity = Km=> ma = Km= > A = KAcceleration is constant.

Constant acceleration motion revealed a fundamental property of gravity to Newton

F gravity mDifferent from F = ma!!!

Did Newton, dreaming in his orchard thereBeside the dreaming Witham, see the moonBurn like a huge gold apple in the boughsAnd wonder why should moons not fall like fruit?Or did he see as those old tales declare...A ripe fruit fall from some immortal treeOf knowledge, while he wondered at what heightWould this earth-magnet lose its darkling power?Would not the fruit fall earthward, though it grewHigh o'er the hills as yonder brightening cloud?Would not the selfsame power that plucked the fruitDraw the white moon, then, sailing in the blue?Then, in one flash, as light and song are born,And the soul wakes, he saw it-this dark earthHolding the moon that else would fly through spaceTo her sure orbit, as a stone is heldIn a whirled sling; and, by the selfsame power,Her sister planets guiding all their moons;While exquisitely balanced and controlledIn one vast system, moons and planets wheeledAround one sovran majesty, the sun.

The earth exerts the force of gravity on a falling apple.Any object on the surface of the earth experiences a downward force of magnitude mg.

How far does this force of gravity extend?Could it possibly stretch all the way up to the moon?

Newton was sitting in his garden at home contemplating such questions when he had a brilliant insight - a pivotal event for physics

If the moon goes around the earth in circular motion, there must be a centripetal force acting on the moon, a force directed toward the center of the earth.

Could that centripetal force be the very same attraction of the earth?Could it be gravity which attracts the apple to the center of earth? Was the centripetal acceleration = g= 9.8 m/sec2 ?

He was eager to try out the numbers.

With this data Newton had the numbers to calculate the centripetal acceleration of the moon

Much smaller than the value of g = 9.8 m/sec2

V = 2 πRT

=2π 38.4x x10 7

2.36 x 10 6

V2

R = 10212

38.4x107 = 0.002715

= 1021 = 1021 m/secm/sec

aamm = = m/secm/sec22

Period of the moon's revolution,

T = 29.32 days = 2.36x106 sec

The distance (R) from the earth to the moon is 38.4x107 m

Earth rotation speed = 464 m/s

At this point a lesser mind would have given up the idea!

But Newton saw an interesting pattern in the numbers..

Recognizing mathematical patterns in nature…again

6.4 million m

V = 1021 m/secMoon

acceleration = V

2

R

=

0.0027 m

sec2

R = 385 million m

Earth-moon distance

Earth radius

≈ 60

Earth

=radius

9.81/0.00271 = 3609 ≈ 3600There is a nice relationship between 60 and 3600 => The moon’s acceleration is 602 times smaller than the apple’s acceleration.a m = g/3600

Why?Because the earth’s attractive force on the moon must be 602 times weaker, since the moon is 60 times farther away from earth-center than the apple is =

F moon = Mmoon g/3600

The force of gravity decreases with the square of the distance

Force of Gravity = F G 1

R

2

If the moon were at the surface of the earth, the earth would attract the moon by a much larger force, larger by a factor of 602

The reason why the moon orbits the earth is because the earth attracts the moon with the force of gravity…but the force is weaker

It is the same reason why the apple from the tree falls to the ground - to the center of the earth.

The moon also falls to the earth!

Why does it not fall all the way to the earth?Newton's answer:As the moon moves horizontally, it falls just enough toward the earth to follow the curvature of the earth. The moon is forever falling, but its horizontal motion makes it move in a curved path, which happens to be a circle (nearly).

6.4 million m

V = 1021 m/secMoon

acceleration = V

2

R

=

0.0027 m

sec2

R = 385 million m

Earth-moon distance

Earth radius

≈ 60

Earth

=radius

In one second the apple will fall(1/2)gt2 = (1/2)9.8 (1) 2 = 4. 9m = 5mIn one second the moon will fall(5/(60) 2) = 5/3600 = 1.4mm

In 3600 seconds the moon will fall 5mGravity force of earth on moon is diluted by 602 = 3600

Calculate how far an apple will fall in one second. Use d = (1/2)gt2 (g = 9.8m/sec2). Then calculate how far the moon will “fall” in one hour, keeping in mind that the acceleration of the moon is much less than g. Compare the two answers. A) The apple falls much farther, since it is close to earth.B) The moon falls much farther because of the longer timeC) They both fall about the same distance.D) There is not enough information given.

No bouts of ecstasy!

Newton’s explanation that the moon is “falling” to the earth in just the same way as the apple falls to the earth rested on an important assumption:

The earth and the moon are far from each other so it is reasonable to treat both the earth and the moon as single points when considering the force that the earth exerts on the moon.

For the apple the earth behaves as if it exerts its attractive force from a single point at the center of the earth.

Spherical Symmetry

Newton’s Two Crucial Insights in Understanding the Motion of the Moon

1) The attractive force of gravity which is responsible for the apple falling to the earth is also responsible for the moon’s orbit around the earth.

The same force applies to motion of earthly bodies as to motion of a heavenly body.

It was not the first time that someone had perceived unity between heaven and earth.

Galileo: the moon has mountains the moon is made of the same stuff as the earth

2) The force of gravity decreases as 1/R2 - A mathematical description for the force of gravity

NowCould the force of gravity be also responsible for the motion of

the planets around the sun?

Newton could prove from Kepler’s 3rd Law :

The inverse square law of gravitational attraction.

Here is his proof for a circular orbit:

If each planet moves in a circular orbit

The acceleration for this circular motion is produced by a force between the sun and the planet.

The force is directed toward the center of the circle, i.e. the sun.

R3

T2

= constant

Sun

Earth

Moon

60

23440

3x10^4 m/sec

Ah! so the earth and everything on it does fall toward the center of the sun.

Just as the moon’s circular motion is equivalent to falling into the earth.

The earth is also forever falling into the sun, but its horizontal motion makes it move in a curved path, which happens to be a circle (nearly).

If the sun attracts the planets (like earth) with “a force of gravity”:

Does that force of gravity also decrease as 1/R2

Yes!

Newton could prove this from Kepler’s 3rd Law

and much more.

Again, we see how a new theory embraces what is previously understood, and guides us to new understanding.

Sun

Earth

Moon

60

23440

3x10^4 m/sec

Newtonian Synthesisof Celestial and Terrestrial Motion

The planet speed is

V = distance traveled

time taken

= 2 π R

T

V

2

= 4 π

2

R

2

T

2

F = M planet

(4 π

2

R

2

)

T

2

1

R

= M planet

(4 π

2

R )

T

2

Now we apply Kepler's 3rd Law:

R

3

T

2

= constant = K

which is equivalent to

T

2

= R

3

x K'

F = M planet

(4 π

2

)R

R

3

x K'

F = 4 π

2

K'

M planet

R

2

Kepler's harmonic law [R3/T2 = constant] seems mysterious, but Newton established that it is the same as the inverse square law of gravitational attraction.

The force which the sun exerts on each planet is the same force of gravity that Makes an apple fall to the ground from a tree Makes the moon go round the earth Makes planets go around the sun

Byron:

When Newton saw an apple fall, he foundIn that slight startle from his contemplation

A mode of proving that the earth turn’d round [the sun]In a most natural whirl, called “gravitation”;And this is the sole mortal who could grapple,

Since Adam, with a fall, or with an Apple.

The force of gravity Varies as 1/R2. Varies as mass of the object it acts on (apple, moon, planet)

“And God said let there be a firmament in the midst of the waters, and let it divide the waters from the waters. And God made the firmament, and divided the waters which were under the firmament from the waters which were above the firmament.”

-Genesis

Von Carolsfeld

Artists always separated heaven from earth

Corregio, Heaven and

Earth1526-30

St. Ignazius Church (1770)

Pietro da Cortona. Glorification of the

Reign of Urban VII

If the force of gravity between the sun and an orbiting object varies with distance as 1/R2, then the orbit of the object could be

(a) a circle (b) ellipse (c) parabola (d) any of the above

Grander Synthesis Newton’s Law of Gravitation

R 12

1 2

M1 M2

Fgrav = G M1M2

R122

+

Newton’s Law of Motion: a = F/m (Newton’s Second Law)

Together:Reveal the mechanics of the universe

Deeper Insight…Mutual Attraction

We have discussed the -1st Law of Motion (principle of inertia…Galileo) -And the 2nd Law of Motion: F = m a

Now we come to Newton’s 3rd Law of Motion, the third pillar of mechanics.

-A single particle, or a single body by itself, can neither exert nor experience any force at all.

-Forces emerge only as a result of interactions between two or more entities.

Whenever there is an interaction between bodies (A and B)

The force exerted by body A and on body B is equal and opposite to the force exerted by body B on body A.

Demo

Symmetry Sun attracts planets with gravity force : Planets attract sun

with SAME gravity force, and opposite in direction

Asymmetry

If the earth attracts the apple with a certain force,

then the apple attracts the earth with the same force.

So, why does the earth not accelerate toward the apple?

It does, but the acceleration of earth is much smaller, because the mass of earth >> mass of apple. Demo

If the earth attracts the moon with a certain force then the moon attracts the earth with the same force.... (One of the causes of Tides ! - later)

F grav-on-earth m earth

F grav- on-sun m sun

Symmetry Commands:

F grav-on-earth = - F grav-on-sun

How can that be?? masssun >> massearth

F on-earth m earthm sun

F on-sun m sunm earth

F grav-on-apple m apple

Put it all together

F grav m planet m sun (1/ Rsun-planet2)

Is this true for earth-moon – Yes

Is this true for earth-apple – Yes

Generalize to mass1 (m) and mass2 (M).

Replace proportionality by constant:

F = G mM

R2

The sublime principle of universal gravitation:

“All matter moves as if every particle attracts every other particle with a force proportional to the product of their masses, and

inversely proportional to the distance between them. That force is universal gravitation.”

Poet Francis Thompson captures the universality and reciprocal symmetry of gravity:

“All things by immortal powerNear or farHiddenly

To each other linked areThat thou canst not stir a flowerWithout the troubling of a star.”

This was the first synthesis.