GPS Module 3 - Part 2

Ch 3: Newton’s Laws

Chapter 3 - Newton’s Laws

Weight

W = m g

The pull of gravity at the earth’s surface was given before by:

. . . but this is not very complicated. What happens if you go into earth orbit or to the moon? We need something more complex to handle other possibilities.


Newton’s Law of Universal Gravitation:

The classic cartoon image of this subject is Newton sitting under a tree; an apple falls; hits him on the head and ‘aha gravity’. But the pull of gravity is obvious to all. The Greek theory was that objects naturally seek the earth as their source (specifically ‘earth’ element objects are attracted to the earth – flames are attracted to the sun, water to the ocean, etc.).

In Newton’s time, they were just beginning to see the moon as a big ‘rock’ in the sky and they were trying to figure out solar system structure and motion. An apple falls off the tree, but why doesn’t the moon fall to the ground. What would happen if you went to the moon? Would you still be hanging off dangling toward the earth? These were the big questions in Newton’s time.

In the mid-1600’s a radical new idea was being bounced around (by Newton and perhaps others).



The two members of every pair of objects in the universe exert a gravitational force of attraction on each other. Gravity is not just an ‘earth’ thing, but is a universal property of all matter.

This figure is supposed to illustrate this idea: there is a pull of gravity between you and your book, you and a tree, your book and the moon, etc., etc.



The force of gravity between two masses is given by:

This is a strong statement and not immediately obvious. A fair question is ‘why haven’t I noticed this before?’. It is one thing to make a qualitative statement, but another to be able to back it up with mathematical predictions. That is the real core contribution of Newton; the invention of the physics formula and the birth of the modern approach to the physical sciences.

Looks messy, but we’ll go through this piece by piece.

F = r2

m1G m2



F = r2

m1G m2

‘m1’ and ‘m2’ are the two masses involved.

The force of gravity between two masses is given by:



F = r2

m1G m2The force of gravity between two masses is given by:

‘r’ is the separation distance between the masses. If the masses are not just tiny points, then which distance; there are many distances between the objects. Newton had to invent Calculus to fully handle this issue – but for our purposes the ‘center to center’ distance is usually an excellent approximation. In fact for spheres, the center is the perfect reference point.



F = r2

m1G m2The force of gravity between two masses is given by:

G is just a number. G = ( 6.67 x 10 -112

2kgn m )

G is a very tiny number, with very messy units! That’s why we just give it a one letter abbreviation. It is what is known as a universal physical constant. A basic measurement of the universe, such as speed of light or mass of an electron. It is a measure of the strength of the gravitational force.


What is the force of gravitational attraction between two 100 kg masses, 1m apart?

Example 3.3:



Example 3.3:

F = r2

m1G m2



Example 3.3:

F = 2( 1 m )

( 6.67 x 10 -11 ( 100 kg )2

( 100 kg )2kgn m )

F = r2

m1G m2Everything on the right is known, so just plug it in.



Example 3.3:

F = 2( 1 m )

( 6.67 x 10 -11 ( 100 kg )2

( 100 kg )2kgn m )

F = 2m

( 6.67 x 10 -7kg

2kg2kg

n m

)

The number part and the units part can be handled separately. Gathering all the numbers together yields this.



Example 3.3:

F = 2m

( 6.67 x 10 -7kg

2kg2kg

n m

)

The units are a huge mess, but turns out that most everything cancels. And you don’t have to do it all in one step either. Note that all the kg cancel.



Example 3.3:

F = 2m

( 6.67 x 10 -72n m

)

With kg out of the way, it becomes more obvious that m2 cancels too.

So only ‘newton’ is left – which is good because the ‘newton’ is a force unit and this is supposed to be a force we are finding.



Example 3.3:

F = ( 6.67 x 10 -7 n )Microscopically small! Less than one millionth of a pound. No wonder you haven’t noticed being drawn in by gravity towards your car or a tree.

or 0.00000015 lb


Some Features of Gravity:

The calculations in the previous example are messy. It is less important that you be able to follow every single step, then it is to understand some basic consequences of Newton’s Universal Law of Gravity. Even without doing lots of messy examples, some basic behavior of the Law can be understood. These important features are discussed in the following list.



1.) Since G is so small, the gravitational attraction between ordinary objects at typical separation distances is microscopic.

If you ask why the force in Example 3.3 is so small, it all goes back to the fact that G is so tiny. For any mass or separation distance for everyday objects (people, books, trees, cars,…) the force of gravity is always millionths to billionths of a pound. These were not even measureable in Newton’s time, but the one billionth of a pound force between two lead weights is easily measured with modern techniques.



2.) If one or both masses is large enough, that can overcome the tiny ‘G’ value and then the force of gravity may be substantial.

F = r2

m1G m2

Tiny GHuge mass (whole moon or planet)

Gravity is actually very weak compared to say the magnetic or electric force on a particle for particle basis. It is only when a whole moon or planet sized mass is involved that gravity is felt.



150 lb = r2

m1G m2

Mass of you Mass of earth

Distance from you to center of earth

Your weight can actually be calculated this way from astronomical values. The weight of a 150 lb person comes from all the pieces of the earth’s mass contributing a small pull.




The larger the masses involved, the larger the force and vice versa. The same person (and hence same mass) will weigh less on the moon, because the moon has much less mass to pull on them.


Same person weighs about 1/6 as much on the moon as on earth. This was handy for the astronauts needing to carry heavy gear. Would also allow people to jump 6 times as high or hit a golf ball 6 times farther, etc.

Force between the same person and …



On Jupiter, this same person weighs about 2.5 times more than on earth. Most people would have trouble even carrying around their own weight.




On smaller solar system objects, such as small moons and asteroids, a person may weigh only a few ounces. So little that they could jump off and not come back down.




The sun would be a hostile destination due to the heat, but perhaps just as bad would be the gravity, which would probably be lethal. This same person would weigh a couple of tons.




3.) The force gets weaker (gradually) as the separation distance between the objects increases.

A person weighing 150 lbs at sea level…


...weighs slightly less on a mountain-top.



The same person would weigh even less up in orbit (although note that the weight is still a long way from being zero).



The force gets very small at large distances (although technically it never goes exactly to zero). Even on the moon, this same person is still pulled to earth by a few hundredths of a pound. Don’t confuse this with the pull of the moon on that person, which is a lot bigger (25 lb).



This was all a pretty amazing leap forward. It is the 1680’s, hundreds of years before space travel could truly test many of these ideas, but the theoretical framework was already there to figure out the pull of gravity on different planets and how quickly gravity would decrease as one left the earth.

Recall that in Newton’s time, in particular, they were interested in the motion of the moon and the planets. This lead to the idea of satellite motion…



1.) Since G is so small, the gravitational attraction between ordinary objects at typical separation distances is microscopic.


Earth

Satellite Motion

An object taken several hundred miles above the earth (well beyond the earth’s atmosphere) will weigh less than it did on the ground, but not that much less.

v = 0hr

miAn object released from rest at typical orbital height would plummet to the ground.

Earth

Satellite Motion

v = 0hr

mi500

hr

mi

Suppose instead an initial ‘sideways’ push is given. The object still falls to the ground, but in an arc that will cover some horizontal distance first.

Earth

Satellite Motion

An object released from rest at typical orbital height would plummet to the ground.

v = 0hr

mi

5000hr

mi

500hr

mi

Earth

The larger the sideways velocity, the longer of an arc the falling object will make.

Satellite Motion

An object released from rest at typical orbital height would plummet to the ground.

Suppose instead an initial ‘sideways’ push is given. The object still falls to the ground, but in an arc that will cover some horizontal distance first.

v = 0hr

mi

16000hr

mi

500hr

mi

5000hr

mi

But the earth’s surface is curved too. So if the object is pushed sideways with enough speed, the arc can be so gentle that the object doesn’t get any closer to the ground. The object is constantly ‘falling’, but is moving so fast horizontally that instead of hitting the earth, it just returns to its starting point and continues to orbit around.

Earth

Satellite Motion


If the pull of gravity is still quite large in low earth orbit, then why do objects/astronauts on the space station or space shuttle appear completely weightless?

On earth, the ground must push up on the bottom of your feet to keep you from falling. Your sense of weight comes from these support forces and not from ‘feeling’ gravity itself.


Drop something and that object, no longer supported by anything, will fall away from you.

On earth, the ground must push up on the bottom of your feet to keep you from falling. Your sense of weight comes from these support forces and not from ‘feeling’ gravity itself.


Drop something and that object, no longer supported by anything, will fall away from you.

Suppose instead you are in a freely falling box. If the box (and hence all the contents) are already falling with the full acceleration of gravity, there is no need for the floor to push on your feet. The support forces and your sense of having any weight will vanish.

Drop something and it will just hover in front of you. It is falling, but it was already falling before you let go (and so are you).


Objects inside a falling box or an orbiting spaceship are all ‘falling’ at the same rate. They appear weightless because nothing needs to rest on top of anything else.

People paying good money to experience brief freefall and the associated sense of weightlessness.


An airplane in a dive* can also mimic weightlessness. Astronauts take these training flights and it is one of the few space tourism things that a private individual can purchase.

Photo is from a training vehicle affectionately known as the ‘vomit comet’.

If you don’t like roller coasters or airplane turbulence, then astronaut training may not be for you.

* A powered dive. Because of air resistance, just cutting the engines isn’t enough. You have to mimic the path of a rock falling out of the sky.




Space walking astronaut is still pulled fairly strongly by the earth, but is ‘falling’ at the same rate as the nearby ship, the tools, etc.


Figure explaining satellite motion from Newton’s main work, Principia Mathematica, circa 1680’s

Satellite Motion

So Newton figured out how to put an artificial satellite around the earth about 300 yrs before technology made it possible. But he also realized that the moon orbiting the earth and the planets orbiting the sun are natural examples of satellite motion.

We’ll finish chapter 3 with some miscellaneous related items.


The lower a satellite orbits, the faster it orbits.

v = 17000hr

mi

Period = 1.5 hour

Low Earth Orbit

Low satellites feel a stronger gravity, so they must move faster sideways to stay ‘up’. The period of an orbit is the time required to complete one orbit.



v = 17000hr

mi

Period = 1.5 hour

Low Earth Orbit

v = 15000hr

mi

Period = 3 hour



v = 17000hr

mi

Period = 1.5 hour

Low Earth Orbit

v = 15000hr

mi

Period = 3 hour

v = 7000hr

mi

Period = 24 hour

Geosynchronous Orbit



v = 17000hr

mi

Period = 1.5 hour

Low Earth Orbit

v = 15000hr

mi

Period = 3 hour

v = 7000hr

mi

Period = 24 hour

Geosynchronous Orbit

v = 2000hr

mi

Period = 28.5 days

Moon’s Orbit


Geosynchronous Orbit occurs at the height where satellites orbit in a period of 24 hrs. At this height an interesting effect occurs.





The satellite has to keep moving, but at this rate it keeps up exactly with the daily turning of the earth.

So it appears to hang motionless in the same place in the sky, as seen by a ground observer.


Having a 24 hr orbital period is not enough to be geosynchronous. The geometry must be correct also (orbit is over the equator and headed ‘east’).


Falling Satellites:

Satellite in a low orbit (400 mile high), drawn to scale.


Virtually all of the earth’s atmosphere is within 100 miles of the ground. But it doesn’t just abruptly end and the microscopic amount left in low earth orbit does provide a tiny air resistance. A satellite can very slowly loose speed and drop lower each orbit. This accelerates toward the end and the satellite comes crashing down.

Falling Satellites:

Satellite in a low orbit (400 mile high), drawn to scale.


Fell 1979

Launched 1973

Satellites in low earth orbit routinely fall out of the sky. The Russians put up the first two artificial satellites, but they both fell fairly quickly. The first US satellite, Explorer 1, was placed in a much higher orbit and remains the oldest artificial satellite around the planet.

Probably the most famous satellite to fall out of the sky was our first space station - Skylab


Watching Satellites:

Satellites can be observed with the un-aided eye. Under dark conditions, usually a few can be seen every night during a couple different time windows.



The first hour or two after sunset (and an hour or two before sunrise) it is dark at ground level, but still ‘sunny’ 400 miles above.



An hour after sunset (and an hour before sunrise) it is dark at ground level, but still ‘sunny’ 400 miles above.Satellites will appear as a tiny dot of white light (like a star) that slowly moves uniformly across the sky.

Ground observer shortly after nightfall, can still see sunlight glinting off of a satellite passing overhead. Satellites have no lights of their own, as they are un-manned. Blinking lights, colored lights, or multiple lights means an airplane.


Artificial Gravity:

Objects don’t change velocity without being pushed. In a box moving in a curve, ‘loose’ objects tend to keep going in a straight line, but this will appear to observer’s in the box as a tendency for them to be thrown to the outside of the curve.


Artificial Gravity:



Artificial Gravity:



Artificial Gravity:



This is known as the centrifugal effect.

Artificial Gravity:


This could be the basis for an artificial gravity in orbit.

Spinning a 100 m wide drum at 3.3 rev/min yields the effect of objects being ‘pressed’ to the outside with equivalent of earth’s gravity.

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GPS Module 3 - Part 2

Education