LAW OF UNIVERSAL

GRAVITATION

Gravity Sucks!

If I have seen further it is by standing on ye sholders of Giants. This is the statement written in a letter to

Robert Hooke. It also symbolizes that the discoveries of Newton including gravity.

So we’re going to start with a brief history are in the bronze age.

Why is it so difficult to find out about the state of astronomical knowledge of bronze-age civilizations?

1. Written documents from that time are in a language that we don’t understand.

2. There are no written documents documents from that time.

3. Different written documents about their astronomical knowledge often contradict each other.

4. Due to the Earth’s precession, they had a completely different view of the sky than we have today.

5. They didn’t have any astronomical knowledge at all.

Ancient Greek Astronomers Models were based on unproven “first principles”,

believed to be “obvious” and were not questioned:

1. Geocentric “Universe”: The Earth is at the Center of the “Universe”.

2. “Perfect Heavens”: The motions of all celestial bodies can be described by motions involving objects of “perfect” shape, i.e., spheres or circles.

Ptolemy

He was Greco-Roman writer who lived in Alexandria.

Lived between 90 -168 AD

One of the few surviving text about ancient astronomy

• Ptolemy: Geocentric model, including epicycles

1. Imperfect, changeable Earth,

2. Perfect Heavens (described by spheres)

Central guiding principles:

What were the epicycles in Ptolemy’s model supposed to explain?1. The fact that planets are moving against the

background of the stars. 2. The fact that the sun is moving against the background

of the stars.3. The fact that planets are moving eastward for a short

amount of time, while they are usually moving westward.

4. The fact that planets are moving westward for a short amount of time, while they are usually moving eastward.

5. The fact that planets seem to remain stationary for substantial amounts of time.

Epicycles

The ptolemaic system was considered the “standard model” of the Universe

until the Copernican Revolution.

Introduced to explain retrograde (westward) motion of

planets

The Copernican Revolution

Nicolaus Copernicus (1473 – 1543):

Heliocentric Universe (Sun in the Center)

Johannes Kepler (1571 – 1630)• Used the precise

observational tables of Tycho Brahe (1546 – 1601) to study

planetary motion mathematically.

1.Circular motion and

• Planets move around the sun on elliptical paths, with non-uniform velocities.

• Found a consistent description by abandoning both

2.Uniform motion.

New (and correct) explanation for retrograde motion of the planets:

This made Ptolemy’s epicycles

unnecessary.

Retrograde (westward) motion of a

planet occurs when the Earth passes the planet

.

Described in Copernicus’ famous book “De Revolutionibus Orbium Coelestium” (“About the revolutions of celestial objects”)

Galileo Galilei (1564-1642 A.D.)

- Founder of Modern Mechanics and Astronomical use of the Telescope

Proved Aristotle Wrong

1. Many more stars too faint to be seen with eye

2. Moon has mountains and craters like Earth…. Earth and Space made of same material

3. Discovered imperfection in Sun (SUNSPOTS)…Sun is not perfect

Provided more evidence for a Heliocentric Solar System (Venus exhibits a full cycle of phases which is only possible in a Heliocentric system + Jupiter appears as a mini solar system which means that Kepler’s Laws apply for all planets)

Galileo’s Sketch of the Moon Using a Telescope

Isaac Newton (1643 - 1727)

Major achievements:

1. Invented Calculus as a necessary tool to solve mathematical problems related to motion

• Adding physics interpretations to the mathematical descriptions of astronomy by Copernicus, Galileo and Kepler

2. Discovered the three laws of motion

3. Discovered the universal law of mutual gravitation

The “Discovery” of Gravity We’ve all heard the story…

an apple fell on Newton’s head and he discovered gravity.

Most scholars believe that Newton did see an apple fall and it got him wondering about the rules for falling objects.

He wondered if the force that pulled the apple down also affected the Moon.

Remember Newton’s 1st Law He had already explained that straight-line

motion was perfectly natural and moving in a circle required a force

Johannes Kepler had shown that planets and moons moved in an ellipse.

Newton wanted to understand what made them move that way.

His breakthrough was to explain how the same rules apply to little things like apples and big things like the moon.

Making sense of one force and two very different results

We already learned that the apple will accelerate at about 9.8 m/s2 downward

Newton also knew that the moon accelerates toward the Earth at 0.00272 m/s2

Distance is only part of the story Distance has a large effect on the force of gravity

(we’ll explore this more in a minute) Remember, though F = m • a

The force of gravity is also affected by the massor more correctly, both masses

Gravitational Force Gravitational Force is the mutual force of

attraction between particles of matter This force always exists between two masses,

regardless of the medium that separates them It is not just between large masses, like the

sun and the Earth. The chair you are sitting on is attracted to the

person next to you. However, the force of friction between the

chair and the carpet is so great that you don’t move.

Newton’s Law of Universal Gravitation is an example of an inverse-square law

This is because the force decreases the further the two objects get from each other.

The distance is measured from the center of each mass.

221

r

mmGFg

Remember, the Force of gravity (Fgrav) that acts on an object is the same as that object’s weight (in Newtons)

Inverse Square Law

4

1/49

1/9161/16

Inverse Square Law

At 2d apple weighs

1/4 N

At 3d apple weighs

1/9 N

At 4d apple weighs

1/16 N

At 5d apple weighs

1/25 N

Connecting the two formulas

Example #1Find the force of gravity between a 30 kg girl and her 10 kg cat if they are 2 meters apart.

m1 = 30 kg

m2 = 10 kg

G = 6.67 x 10-11 N·m2/kg2

r = 2m

NxF

m

kgkgxF

r

mmGF

g

g

g

9

211

221

100025.5

)2(

10301067.6

Example #2Find the distance between a 0.300 kg billiard ball and a 0.400 kg billiard ball if the magnitude of the gravitational force is 8.92 x 10-11 N.

m1 = 0.3 kg

m2 = 0.4 kg

Fg = 8.92 x 10-11 N

r = ?

cmaboutmr

x

x

F

mmGr

F

mmGr

mmGFr

r

mmGF

g

g

g

g

302996.0

1092.8

400.0300.01067.611

1121

212

212

221

Determine the magnitude of the gravitational force between a baseball player with a mass of 100 kg and Earth (5.98 X 1024 kg), if they are separated by a distance of 5.38 X 106 m.

A. [Option 1]

B. [Option 2]

C. [Option 3]

D. [Option 4]

If a large meteor hits the moon, causing it to get closer to the earth. If the moon’s orbits the earth at half of its original radius, would its force be?

A. Double the original force

B. Half the original force

C. Four times the original force

D. One fourth the original force

The acceleration due to gravity on the International space station is 8.7 m/s2. If an 50 kg astronaut stood on a scale what would it read?

A. 435 N

B. 5.7 N

C. .17 N

D. 0 N

Tides Newton also used the inverse square

law to explain the tides. People had known for centuries that the

moon affects the tides. No one until Newton knew how it did

this.

dd-R

d+R

Which of the two forces: moon on left mass (m) or moon on right mass (m) is stronger and why? Fd-R

Tidal Bulges

Ocean tides are the alternate rising and falling of the surface of the ocean that usually occurs in two intervals everyday, between the hours of 7a.m. to 7p.m.

It is caused by the gravitational attraction of the moon occurring unequally on different parts of the earth.

KEPLER: the laws of planetary motion

KEPLER’S FIRSTLAW

KEPLER’S SECOND

LAW

KEPLER’STHIRDLAW

INTERESTINGAPPLETS

Johannes Kepler

Born on December 27, 1571 in Germany

Studied the planetary motion of Mars Used observational data of Brahe

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Instruments

Tyco Brahe only compass and sextant No telescope – naked eye

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Kepler’s FIRST Law

“The orbit of each planet is an ellipse and the Sun is at one focus”

Kepler proved Copernicus wrong – planets didn’t move in circles

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Focus

Focus – one of two special points on the major axis of an ellipse

Foci – plural of focusA+B is always

the same on any point on the ellipse

KEPLER’S FIRSTLAW

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Kepler’s SECOND Law

“The line joining the planet to the sun sweeps out equal areas in equal intervals of time”

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In Another Words…

The area from one time to another time is equal to another area with the same time interval

All of the areas (in yellow and peach) have equal intervals of time

KEPLER’S SECOND

LAW

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Acceleration of Planets

Planet moves faster when closer to the sun Force acting on the planet increases as

distance decreases and planet accelerates in its orbit

Planet moves slowerwhen farther from the sun

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KEPLER’S SECOND

LAW

Kepler’s THIRD Law

“The square of the period of any planet is proportional to the cube of the semi-major of its axis”

Also referred to as the Harmonic Law

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T² a³

T = orbital period in years a = semi-major axis in

astronomical unit (AU)Can calculate how long it takes

(period) for planets to orbit if semi-major axis is known

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KEPLER’STHIRDLAW

Astronomical Unit

Astronomical unit – AU AU is the mean distance between

Earth and the Sun1 AU ≈ 1.5 x 108 km ≈ 9.3 x 107

miles

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KEPLER’STHIRDLAW

Examples of 3rd Law

Calculating the orbital period of 1AU T² = a³ T² = (1)³ = 1 T = 1 year

Calculating the orbital period of 4AU T² = a³ T² = (4)³ = 64 T = 8 years

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KEPLER’STHIRDLAW

The planet Saturn is located 9.6 AU from the sun. What is the length of a year on Saturn?

A. 885 years

B. 4.5 years

C. 30 years

D. 45 years

CometsAlthough Kepler’s

laws were intended to describe the motion of planets around the sun, the laws also apply to comets

Comets are good examples because they have very elliptical orbits

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So what would happen is fell in hole through the center of the earth?

Well Lets see!!!

So what would happen is fell in hole through the center of the earth?

So lets recapThe deeper you get into the earth, the weaker

the gravity because there is less mass pulling you down because some is pulling you up.

The center of the earth has no gravity because the mass is surrounding you on all sides.

Once you reach the other side of the earth, you’d stop and be pulled back toward the center.

Gravity

Recap:

Newton’s universal law of gravitation:

Acceleration due to gravity:

Acceleration that an object experiences when it a distance r from the center

F = GMm R2

g = GM R2

GPE in a uniform field

When we do vertical work on a book, lifting it onto a shelf, we increase its gravitational potential energy (Ug). If the field is uniform (e.g. Only for very short distances above the surface of the Earth) we can say...

GPE gained (Ug) = Work done = F x d

= Weight x Change in height

so... ΔUg = mg∆h

E.g. In many projectile motion questions we assume the gravitational field strength (g) is constant.

GPE in non-uniform fields

However, as Newton’s universal theory of gravity says, the force between two masses is not constant if their separation changes significantly. Also, the true zero of GPE is arbitrarily taken not as Earth’s surface but at ‘infinity’.

If work must be done to “lift” a small mass from near Earth to zero at infinity then at all points GPE must be negative. (This is not the same as change in GPE which can be + or -)

‘Infinity’Ep = 0

Lots of positive work must be done on the small mass!

Ep = negative

The GPE of any mass will always be due to another mass (after all, what is attracting it from infinity?)

Strictly speaking, the GPE is thus a property of the two masses.

E.g. Calculate the potential energy of a 5kg mass at a point 200km above the surface of Earth. ( G = 6.67 10-11 N m2 kg-2 , mE= 6.0 1024 kg, rE= 6.4 106 m )

The gravitational potential energy of a mass at any point is defined as the work done in moving the mass from infinity to that point.

Ug = - GMm r

E.g. Calculate the potential energy of a 5kg mass at a point 200km above the surface of Earth.

( G = 6.67 10-11 N m2 kg-2 , mE= 6.0 1024 kg, rE= 6.4 106 m )

Escape speed

If a ball is thrown upwards, Earth’s gravitational field does work against it, slowing it down. To fully escape from Earth’s field, the ball must be given enough kinetic energy to enable it to reach infinity.

Loss of KE = Gain in GPE

½ mv2 = GMm (Note this also = Vm) r

So... but… so…

The escape speed is the minimum launch speed needed for a body to escape from the gravitational field of a larger body (i.e. to move to infinity).