UNIVERSAL GRAVITATION

History of Gravity Ptolemy c. AD 90 – c. 168

All objects go to their natural place Geocentric universe Main text for 1600 years

Copernicus 1473 – 1543 Heliocentric universe

Brahe & Keppler (Keppler - (1571 –1630) Keppler used Brahe’s observations to formulate 3 laws of

planetary motion Galileo 1564 –1642

Tested with gravitational affects on Earth, acceleration of gravity

First to come up w/ idea for inertia Contemporary of Keppler

Newton 1643-1727 Law of Universal Gravitation

Einstein 1879–1955) Theory of General Relativity Punked out Isaac Newton

Isaac Newton First credited with the idea that Gravity

extends through the universe Newton knew the concept of inertia,

and that if something changed its speed or direction a force must be responsible

Apple falling from tree… apple changed its motion, so a force must be responsible-- Gravity

He also noticed the moon, and that it was traveling in circular motion Circular motion means changing direction

and this also required a force He reasoned the same force was

responsible for the Moon as was the apple

The Falling Moon- Satellite Motion

Newton realized the moon must be falling or else it would travel away in a straight line

So if it is falling, why is it never getting closer??

hypothesized that moon is just a projectile falling under the influence of gravity.

And it never falls to Earth because of its speed….

Launched projectiles will follow a curved path (see below) until they hit the ground

But the ground also curves (Earth is not flat) Most projectiles have a curve that are much

steeper than the curve of the Earth BUT, if you throw a projectile fast enough, then

curved path of the projectile will match the curve of the Earth

Then, the projectile will never reach the surface and forever be in an orbit

Projectile must also be above the Earth’s atmosphere, so that air resistance does not slow it down

Projectile being thrown a different speeds, from the top of a mountain above the atmosphere

If you could throw something as fast as you wanted and there was no air resistance could you make something orbit?/

Newton’s Law of Universal Gravitation

Force of gravity acting between 2 objects is directly proportional to their mass and inversely proportional to the square of the distance between them

m1 and m2 represents the masses of the two objects involved

d - distance between the center of the two objects G- Universal Gravitation Constant, value of G = 6.67 x 10-11 Nm2/kg2

F - represents the Force of Gravity between any 2 objects Remember that Force of Gravity is the same thing as Weight

What this law means Whatever factor the mass is increased by,

the force of gravity is increased by the same amount Ex. Triple the mass, force of gravity also is

tripled Whatever factor the distance between 2

objects is increased by, the force of gravity is multiplied by the inverse of the square of that number Ex. Triple the distance, force of gravity is 1/9th

as much Inverse square proportion

Examples The force of gravity acting on you while on

the surface of the Earth is 100 N. What would be the force of gravity acting on

you if the Earth’s mass was doubled? Force Directly proportional to mass, So 200 N

What would it be if your mass was doubled and the Earths stayed the same?

Force Directly proportional to mass, So 200 N What would the force of gravity acting on you

be if you were twice as far from the center of the Earth as you are now? Force is Inversely proportional to the square of the distance, so

distance is increased by 2, inverse square of 2 is 1/22 or 1/4th ,

so force of gravity would be 1/4th as much or 25 N.

Why this happens?? Inverse Square Law… Surface Area of a sphere

More examples….

A spaceship with a weight of 10,000 N on the surface of the Earth moves to a distance that is 3 times further away from the center of the Earth, what is the force of gravity on it now?

Distance increased by 3 x , so Fg decreases by the inverse square of this number so Fg goes down by 1/9th. 10,000/9 = 1111.1 N

R

Space ship weighs

1111.1N here

Space ship weighs

10,000N here

Fg = GMm/r2 and Fg =mg

So …. Fg =GMm/d2 = mg

After cancelling out ‘m’ we get g = GM/r2

This gives us the acceleration of gravity, g, (also known as the Gravitational Field Strength) at a distance, r, away from the center of the Earth. so acc. of gravity also follows inverse

square rule

M – mass of central object (ie Earth or Sun)

m-mass of small object (person or satellite)

r- radius.. or distance between objects

Object in orbit around Earth requires a Fc to keep in circular path. This Fc is the force of gravity from the Earth. So Fc = Fg. So if we want to figure out the speed, vt, that an object must go in

order to travel in orbit at a distance, d, away from the center of the Earth… we can solve as follows…

“d” and “r” are representative of the same thing, so just turn ‘d’ into ‘r’ to simplify math

After we cross multiplied, divided and cancelled out like terms we get…

And this tells us the speed, vt,

that we would need to travel in an orbit with radius, r, around an object with mass, mE.

Everyone thought…

All satellites travel in perfectly circular orbits

NOT TRUE Planets, moons, etc. Were not where they

were supposed to be! Planets did not follow these predicted paths So something must be wrong Then…… along came Johannes Kepler

Kepler’s 3 Laws of planetary motion 1) The paths of the planets are ellipses,

with the sun at one focus (the other focus is just a point in space)

http://www.astro.utoronto.ca/~zhu/ast210/kepler.html

Keppler’s 2nd LawAn Imaginary line from the sun to a planet sweeps out equal areas in equal time intervals.

This means planets move faster when they are closer to the sun and slower when they are further awayhttp://surendranath.tripod.com/Applet

s/Dynamics/Kepler/Kepler1Applet.html

Keppler’s Third Law The square of the ratio of the periods

of any two planets revolving about the sun is equal to the cube of the ratio of their average distances from the sun. Thus, if Ta and Tb are the planets periods, and ra and rb are their average distances from the sun we get

(Ta/Tb)2 =(ra/rb)3

Elliptical paths & Kepler

Kepler came up with Three Laws of Satellite Motion

These laws explained that planets & other satellites travel in ellipses (ovals), not circles

When tested Kepler’s laws almost perfectly predicted paths of planets and moons

This means planets are closer to the Sun at some points in their orbit, and farther from the Sun at other points

Orbits Earth’s orbitVery

nearly circular, but elliptical like all other planets

Elliptical path of Earth does not affect weather at all

Other planets, like Mercury, have more exaggerated elliptical orbits

Conservation of Energy and Satellite motion

Based on the fact that KE + PE always stays constant…..

Planets travel faster when they are closer to the Sun, and slower when they are further away

Slow

Fast

Escape Speed Speed that something needs to travel in

order to permanently escape an objects gravitational pull without getting pulled back

Escape speed for the Earth is roughly 7 miles per sec.

,

M – mass of central objectm- mass of projectile

Escape velocity equation is derived from conservation of Energy Equation- KEi + PEI = KEF + PEF

- Basically the initial kinetic energy has to be greater than or larger than the initial potential energy relative to the center of the object that is being escaped from

- So KEi + PEi = 0 to find minimum escape speed

- Plugging in equations for KE and PE we get…½ mve

2 +-mgri = 0

- and substituting in g= GM/r2 and cancelling out ‘r’ ‘s

- we get ½ mve2 + - GMm/r = 0

- And solving for ve, we get

Big Bang vs. Big Crunch• If KE from big bang is larger than PE before big

bang then universe will expand forever• However if KE from big bang is smaller than PE

before, then universe will eventually collapse back onto itself, thus causing a BIG CRUNCH

• In other words…. If matter had the escape speed of the universe when the big bang began… then it will expand forever, if not big crunch

How do we know universe is still expanding? • Red Shift of galaxies in all directions

– All galaxies are moving away from us• Color of galaxies is shifted to red end (long wavelengths) of visible light• This is caused by Doppler Effect

– Squishing together of wavelengths as object approaches (blue shift)– Spreading apart of wavelengths as object leaves (red shift)– Can see this with sound waves as an ambulance approaches– If we saw a blue shift in galaxies…. That would mean big crunch is happening

and everything is moving towards us

GRAVITATIONAL INTERACTIONS

Gravitational Fields Areas of a certain distance away from a

gravitational source experience the same ‘field strength’

Like ripples in a pond after stone dropped in The field strength on the surface of the Earth is

about 9.8 m/s2 or about 10 m/s2

‘g’ also changes by the inverse square ruleIf you are twice as far away from the center of the Earth ‘g’ = (9.8)/4 m/s2 =2.45 m/s2

g= Gm d2

Hypothetically Going to the center of the Earth According to Newton, the closer you get to

an object the stronger gravity is, but if you were travelling to the center of the Earth, all of the mass above you would start to cancel the gravity below you, so as you travel towards the center of earth gravity will become less and less strong

Weight & Weightlessness How you feel weight, is different than your actual

weight As long as you are near the surface of the Earth you

will always have the same weight but you may “feel” like you have a different weight

This can happen if you are accelerating up or down Imagine an elevator……..

Ocean Tides From the Moon Side of the Earth closest to the moon, has a high tide As well as the side of earth furthest from the moon This happens because…

Water flows

There is a large difference in gravitational strength from the

moon on either side of the Earth… because of difference in distance from the moon

Gravity from the moon tugs on Earth just a bit Gravity from moon is of course stronger when moon is close And weaker when far away

Earth

Earth

The Sun also causes Tides The Sun, like the Moon pulls on Earth, so the

side of Earth closest to Sun, receives a high tide from Sun

Tides from the Sun are weaker than from the moon

Why is this?

Sun actually has a lot stronger pull on Earth than the moon does, so why aren’t solar tides larger??

Not that big of a difference in how far the sun is from sunny side compared to dark side So difference in gravity strength is small

Tidal Effects… spring tides, Neap Tides

Earth-Moon Orbit Earth and moon orbit a common point in

space Point is center of gravity of Earth Moon

system

Perturbations Small irregularities in the orbit of a

planet caused by a close proximity to another planet

Gravity of one planet slightly tugging on another

This is how Neptune was discovered http://csep10.phys.utk.edu/astr161/l

ect/history/perturbations.html

Earth Tides and Atmosphere Tides

Tides don’t just happen with water Occur w/ the ground

Earth is really just a thin solid crust on top of molten (liquid) rock

Ground rises and lowers small amount depending on moon’s location

and with the atmosphere Atmosphere has much larger tides than

oceans do

Gravity differences on Earth??

http://www.sciencefriday.com/news/051407/news0514071.html

-Universal Law of Gravitation tells us if we are closer or further from Center of the Earth gravity will change

-gravity slightly weaker at the top of a tall mountain… ie you weigh less

-other inconsistencies…..

- Big Earthquakes change mass distribution of Earth…- 2004 Indonesia Earthquake was so big the rate at which the Earth

orbits slightly changed http://www.uwgb.edu/dutchs/platetec/rotationqk2004.htm

Determine the escape speed in km/s of Pluto, Earth, and Jupiter using below data.

1) Determine the escape velocity from a position of 150 km above the Earth.

• 2) Use Kepler’s 3rd Law to determine Mercury’s orbital period in days?

• mass of the Moon = 7.36 × 1022 kilograms• radius of the moon = 1 737.4 kilometers

• Question… how fast would an astronaut have to throw a moon rock from the surface of the moon for it to orbit around the moon without falling to the surface?

What is the acceleration of the moon as it “falls” toward Earth?

• Distance from Earth to moon = 384,403 kilometers

Determine the speed at which an object would have to be travelling at in order to orbit Earth at an altitude of 300 km above

Earth’s surface.

• Mass of the Earth --- 5.97 x 1024 kg• Radius of the Earth – 6400 km

Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

Radius (Earth=1) 0.382 0.949 1 0.532 11.209 9.44 4.007 3.883

Radius (km) 2,438 6,052 6,378 3,397 71,400 60,000 25,559 24,764mass (Earth=1=5.97 x 1024 kg) 0.055 0.815 1 0.107 318 95 15 17

mean distance from Sun (AU) 0.39 0.72 1 1.52 5.20 9.54 19.18 30.06

orbital period (Earth years) 0.24 0.62 1 1.88 11.86 29.46 84.01 164.8

orbital eccentricity 0.2056 0.0068 0.0167 0.0934 0.0483 0.0560 0.0461 0.0097

mean orbital velocity (km/sec) 47.89 35.03 29.79 24.13 13.06 9.64 6.81 5.43

rotation period (in Earth days) 58.65 -243* 1 1.03 0.41 0.44 -0.72* 0.72

mean temperature at surface (C) -180 to 430 465 -89 to 58 -82 to 0 -150 -170 -200 -210

gravity at equator (Earth=1) 0.38 0.9 1 0.38 2.64 0.93 0.89 1.12

escape velocity (km/sec) 4.25 10.36 11.18 5.02 59.54 35.49 21.29 23.71

mean density (water=1) 5.43 5.25 5.52 3.93 1.33 0.71 1.24 1.67

number of moons 0 0 1 2 63 62 27 13

rings? no no no no yes yes yes yes

Mass of sun = 1.98 x 1030 kg

Radius of the Sun = 695,500 km