Newton’s Universal

Law of Gravitation

The Gravitational Force

Newton’s Universal Law of Gravitation states that every particle in the universe exerts an attractive force on every other

particle.

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Where “G” is the “universal gravitational constant” G = 6.67 x 10-11

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What happens to the Force if one of the masses is doubled?

It is also doubled.

What happens to the Force if both of the masses were doubled?

It will be four times larger.

What happens to the Force if one of the masses is doubled and then other one is halved?

The Force will remain the same.

This is an “inverse square” law, since the

Force is proportional to the inverse of the distance squared.

Example:

At twice the distance, the gravitational force between two objects would be less. How much less?

?2

1

d

1F

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Two objects are separated by some distance, d. How would the

gravitational force differ if the distance was tripled?

1/9 the original force

What if the distance was 4d?

1/16 the original force

5d?

10d?

½ d?

4 times the original force

Example: Two masses of 8 kg and 12 kg are separated by 1.5 m. What is the gravitational force they exert on each other?

How do you enter all those numbers in your calculator? Use your exponent button for “G”!!

6.67E-11*8*12÷1.52 =F = 2.85 x 10-9 N

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G = 6.67 x 10-11

What is the gravitational force between a 600 kg mass and a 850 kg mass if they are 0.4 meters apart?

F = 2.126 x 10-4 N

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G = 6.67 x 10-11

Example: Two masses of 3 x 103 kg and 1.8 x 1015 kg are separated by

d = 1.4 x 1021m. What is the gravitational force they exert on each other?

How do you enter all those numbers in your calculator? Use your exponent button!!

6.67E-11*3E3*1.8E15÷1.4E21 2 =F = 1.84 x 10-34 N

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G = 6.67 x 10-11

If the gravitational force between a 75 kg mass and a 120 kg mass is 4.2 x 10-4N, how far apart are they?

What’s the shortcut to get d2 out of the denominator?

Trade places with F!!

d = 0.0378 m

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G = 6.67 x 10-11

NET Gravitational Force

Two masses pull on the central mass.

How would you get the NET gravitational force?

Subtract the two forces.

NET Gravitational Force

Two masses pull on the left mass.

How would you get the NET gravitational force?

Add the two forces. (Be careful about your distances!)

NET Gravitational ForceTwo masses pull on

the mass at the origin.

How would you get the NET gravitational force?

Pythagorize the two forces.

2nd tan for angle.

Cavendish and “G”, the gravitational constantHenry Cavendish, a British scientist,

first devised an experiment to determine G in 1797.

He suspended two small known masses from a “torsion wire” of which he knew the strength. These two small masses were gravitationally attracted to two large known masses, which caused the wire to twist until the torsion force was balanced by the gravitational force. Because he knew the strength of the torsion force, he also knew the strength of the gravitational force.

With known masses, known Force, and known distance, the only “unknown” left was G!

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Finding “g”Weight is the gravitational force a planet

exerts. Weight = Gravitational Force

mg = G

“g”, the acceleration due to gravity can be found by canceling an “m”.

The distance, d, is measured from the center of the planet to the location of interest. (often, the radius)

The acceleration due to gravity, “g”, is also called the “gravitational field strength”.

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How large is “g” on the planet Venus, which has a mass of 4.87 x 1024 kg and has a radius of 6,050,000 meters?

6.67E -11x 4.87 E24 ÷ 6,050,0002 =

g = 8.87 m/s2

2planet

d

mGg

Example: An asteroid of radius 500 m has a mass of 6.5 x 1013 kg. What is the gravitational field strength at its surface?

6.67E -11 x 6.5 E13 ÷ 5002 =

g = 0.0173 m/s2

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mGg

AristotleGeocentric universe384 BC

“geocentric” – Earth centered universe…… WRONG!

Ptolemy, 83 ADPtolemy (also geocentric universe) presented his astronomical models in convenient tables, which could be used to compute the future or past position of the planets, the Sun, and Moon, the rising and setting of the stars, and eclipses of the Sun and Moon. His model showed the planets turning in small circles as they orbited the Earth!

The tables actually produced fairly good predictions, but his model and his geocentric universe was….. WRONG!

Ptolemy was also the first to use latitude and longitude lines.

Copernicus 1473heliocentric universe

Although others before him had

proposed that the planets orbit the sun rather than the Earth, Copernicus was the

first to publish mathematical

evidence

“sun-centered” universe

• Tycho Brahe

• 1546

• Built “The Castle of the Stars”

• Had an accident in a duel

• Died an unusual death…

Johannes Kepler 1571

A mathematician hired as Brahe’s

assistant

• Wrote Three Laws of Planetary Motion

1. The Law of Orbits: All planets have elliptical orbits with the sun at one focus.

Comets have highly elliptical orbits

Planets’ orbits are only slightly elliptical

2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times.

(Planets move faster when they are closer to the sun.)

3. The Law of Periods: The square of the period of a planet is proportional to the cube of its average orbital radius.

(earth years) T2 = r3 (AU)

An AU is an “astronomical unit” and is the distance from

the Sun to the Earth.

T2 = r3

Venus is located 0.72 AU’s from the Sun. How many years does it take Venus to orbit the sun?

T2 = a3

33 72.0aT

0.61 years

Math button or use housetop

X cubed

If it takes an asteroid 5 Earth years to orbit the sun, how far is the asteroid from the sun?

T2 = a3

3 23 2 5Ta

2.92 AU

Math button

cube root

Kepler’s Third Law in fundamental units of seconds and meters

We know that the centripetal force for planets is the gravitational force.

We also know the velocity for any circular motion.

Substituting that v into our centripetal force yields a new equation.

Cancelling and rearranging give us Kepler’s Third Law with the standard units of meters and seconds.

This law will work for orbits around ANY body as long as you know the mass of that body.

vr

T

2

Gm m

dm

v

rm

rTr

planet sun

planet planet2

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TGm

rsun

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“Newton’s Cannon”

If you fire a rocket horizontally from the top of a very high mountain, gravity will pull it towards the center of the Earth and it will go a certain distance before it hits the ground.

“Newton’s Cannon”

The faster the rocket is launched, the further it will go as it falls until it hits the ground.

“Newton’s Cannon”If it is launched fast enough, the pathway of the

rocket as it falls will exactly match the curvature of the Earth. The satellite will continue to fall and fall and fall, but it will never fall to the ground. It goes into orbit about the Earth!

“Newton’s Cannon”The horizontal speed of the orbiting rocket must be

very high and it must maintain that speed or it will fall into the Earth. Satellites in orbit are always FALLING… and FALLING… and FALLING…

“Newton’s Cannon”This is why the astronauts appear to be

weightless. There is in fact plenty of gravity at the elevation of the space shuttle and space station, but since they are always falling, they appear to be in zero gravity- weightless.

Satellite MotionThe gravitational force provides the

centripetal force for an orbiting satellite in circular orbit.

rv

md

mmG

2

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mGv planet

sat

For any object moving in a circle the velocity is given by

v = circumference / time = 2r / T

Where T is the period of the motion – the time to go around once.

Therefore, for a satellite in circular orbit:

= 2r / T d

mGv planet

sat

If a rocket is launched VERTICALLY from the surface of a planet, is it true that what goes up must come down?

If we throw a ball into the air, it reaches some highest point and then gravity pulls it back down.

If we throw it faster, it will go higher before it comes back down.

However, if we throw it fast enough, it can escape the gravitational pull of the Earth and keep moving upward.

The speed at which that will occur is called the ESCAPE SPEED. The escape speed for a planet is given by

ESCAPE SPEED = d

mG2v planet

escape

Geosynchronous SatellitesA geosynchronous satellite is one whose orbital period is

the same as Earth’s rotational period.

So, as Earth rotates once every 24 hours, the satellite orbits the Earth once every 24 hours.

This means that when the orbit lies entirely over the equator, the satellite remains stationary relative to the Earth's surface and the antennae’s do not have to “track” it continually. These satellites are used for communications, and intelligence!

How many satellites are in Earth’s orbit?

One natural satellite

Over 8000 artificial satellites!

There are two main processes constantly going on in the super massive stars:

nuclear fusion (which tends to blow the star's hydrogen outward from the star's center) and

gravitation (which tends to pull all hydrogen back in the direction it had come).

These two processes balance one another until all the star's hydrogen is exhausted, allowing gravitation to take over.

Once gravitation dominates, the star becomes unstable and starts to collapse.

Once a super massive star starts to collapse, it does not stop, and the star (and ultimately its atoms) will cave inward upon itself, resulting in the formation of a black hole (Hewitt 186).

When a star has exhausted its fuel supply, gravitational forces crush the star to one of three

possible outcomes:  1) The star shrinks and

stabilizes into a white dwarf.  2) The star crunches into a

neutron star.  3) The star collapses to a black

hole.

A star less than 1.4 times the mass of the sun will become a white dwarf. A star between 1.4

and 3 times the mass of the sun will become a neutron star.

It's only those stars greater than 3 times the mass of the sun that become black holes

upon collapse.

How do you “see” a black hole when it can’t be seen??

When a star collapses and changes into a black hole, the strength of its gravitational field still remains the same as it had been before the collapse. Therefore the planets in orbit would not be affected. The planets would continue in their orbits as usual and would not be drawn into the black hole. Because black holes do not give off any light, the planets would appear to be orbiting around nothing. There is reason to believe that the planets could just be orbiting about a star that is too faint to be seen, but there is an equal chance that a black hole could be present (Hewitt 187).

Because the gravity of a black hole is so intense, dust particles from nearby stars and dust clouds are pulled into the black hole. As the dust particles speed and heat up, they emit x-rays. Objects that emit x-rays can be detected by x-ray telescopes outside of the Earth's atmosphere (Miller).

Black holes can also be detected through a technique called gravity lensing. Gravity lensing occurs when a massive object, in this case a black hole, passes between a star and the Earth. The black hole acts as a lens when its gravity bends the star's light rays and focuses them on the Earth. From an observer's point of view on the Earth, the star would appears to brighten or to be distorted or to be in a different part of the sky.

The event horizon is the boundary around a black hole where gravity has become so strong that nothing- not even light- can escape. The escape velocity at the event horizon = c. The escape velocity inside the event horizon > c, therefore, escape is impossible. The event horizon is the point of no return.