LAW OF UNIVERSAL GRAVITATION!

Background

As we have studied gravity, we have mentioned Aristotle, Galileo, Newton, and how their theories have shaped what we now know about the world around us. During this section we will discuss gravity in more detail and learn how factors such as mass and distance can affect the “gravity” or gravitational pull on objects throughout the universe.

One day Isaac Newton was sitting under an apple tree, when……. BAM!!! Being that it probably pissed him off, he began to think about it like this: The apple is accelerated, because it was hanging on a tree and then came down. He thought there must be a force that acts on the apple to cause this acceleration. Let's call this force "gravity", and the associated acceleration the "acceleration due to gravity". Then he imagined the apple tree twice as high....Again, we expect the apple to be accelerated toward the ground, so this suggests that this force that we call gravity reaches to the top of the tallest apple tree – and always make the apple come down!

So after many days and nights of measurements, calculations, and apple pies, Newton began to expand his theory. Newton knew that the force that caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth.

Isaac Newton then compared the acceleration of the moon to the acceleration of objects on earth. Believing that gravitational forces were responsible for each, Newton was able to draw an important conclusion about the dependence of gravity upon distance. This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center.

So, to sum it all up, the gravitational force between two objects depends on: the mass of those two objects and the distance2 (squared) between them.

What is the Law of Universal Gravitation?

The law of universal gravitation allows you to calculate the gravitational force between ANY two objects IN THE UNIVERSE from their masses and the distance between them. The Law of Universal gravitation says that every mass exerts an attractive force on every other mass. If the two masses are m1 and m2 and the distance between them is r, the magnitude of the force (F) is given by the equation:

WTH is G = 6.673 x 10-11 N m2/kg2 ???? The constant of proportionality (G) in the above equation is known as the universal gravitation constant. This number was determined after Newton died…. But for some reason it is what it is. The units on G may seem rather odd; nonetheless they are sensible. When the units on G are substituted into the equation and multiplied by m1• m2 units and divided by d2 units, the result will be Newtons - the unit of force.

G = 6.673 x 10-11 N m2/kg2

Newton's law of universal gravitation extends gravity beyond earth. Newton's law of universal gravitation is about the universal application of gravity. Newton's place in the Gravity Hall of Fame is not due to his discovery of gravity, but rather due to his discovery that gravitation is universal. ALL objects attract each other with a force of gravitational attraction. Gravity is universal. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance that separates their centers.

So, we are all attracted to each other?

YES! Even though, you may be ugly as sin, everyone (and everything) is attracted to you!! You and the person sitting next to you are attracted to each other! (you both have mass and a distance between). You are also attracted to the Earth!! That’s why when you jump up….. you come straight back down!!! EARTH LOVES YOU – whether you like it or not!!

So why don’t you and the person next to you “stick” together like magnets – if you are attracted? Well, the reason why is because both of your masses are SMALL compared to the Earth. You are attracted (and you stick) to Earth because of its HUGE mass! But two people wont “stick” together because relative to the Earth – you two are too small to make a difference so the Force between the two of you is VERY small.

Solving for the force between two objects

YOU MUST use parenthesis when solving for F!!!

F = (( G ) (m1) (m2)) ( (r)2 )

Where: F (Force) measured in Newtons M1 (Mass of first object) in kg M2 (Mass of second object) in kg d (distance) this distance between the two objects ß this is squared in equation

SIDE NOTE!!!! DISTANCE – the distance you measure is from the center of gravity of both objects. Example: Consider two spheres, with radius of 0.3 meters, next to each other. What is the distance between them? When you measure from the center of gravity, you basically multiply the radius by two.

The distance between the two spheres is actually 0.6 meters! Ok!! Back to Solving for F!!

This equation allows you to figure the gravitational force between any two masses. What, for example, is the pull between the Sun and the Earth? The sun has a mass of about 1.99 × 1030 kilograms, and the Earth has a mass of about 5.98 × 1024 kilograms. A distance of about 1.50 × 1011 meters separates the two bodies. Plugging the numbers into Newton’s equation gives you

There is NO distance?!? WRONG!!!!

You have to measure from the CENTER OF GRAVITY!

VS

On the land-based end of the spectrum, say that you’re out for your daily physics observations when you notice two people on a park bench, looking at each other and smiling. As time goes on, you notice that they seem to be sitting closer and closer to each other each time you take a glance. In fact, after a while, they’re sitting right next to each other. What could be causing this attraction? If the two lovebirds have masses of about 75 kilograms each, what’s the force of gravity pulling them together, assuming they started out 0.50 meters away? Your calculation looks like this:

Law of Universal Gravitation is an Inverse Square Law!

What is the Inverse Square law???

The Inverse Square Law is any Physical Law stating that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity.

How does the inverse square law apply to law of universal gravitation?

Basically, it means….The Intensity (Pull) you feel from the earth decreases as you move away from the Source (center of the earth)

Example

Lets say you turn on a stove at home and leave it on for a few minutes. You place your hand close to the stove and how does it feel? - HOT!! No DUH! But, if you move your hand away from the stove…. You feel the heat less and less… right?

So how does that apply to gravity?

Well consider the following example.

The Intensity (Pull) you feel from the earth decreases as you move away from the Source (center of the earth)

You actually weigh less at higher altitudes because the gravitational force is a bit weaker. Its sooo small that it does not make a difference.

How does the Inverse Square Law work? Before we can begin, we need to make sure we have a clear understanding of what WEIGHT is. WEIGHT = MASS x GRAVITY Weight is a force, also known as the gravitational force… Law of Universal Gravitation solves for the gravitational force….

WEIGHT is THE SAME as the GRAVITATIONAL FORCE!!!!

As you move away from the Earth, you weigh less and less because the Earth’s pull on you depends on the distance between you and the Earth.

We consider the CENTER of the EARTH as our SOURCE.

If you are on the SURFACE (or 1 radius away from the source) the of the EARTH you have a normal weight : W=mg

If you move 2 Earth Radii away from the source, you will be ¼ of your weight.

Think about it…. 2r… our number is 2… FLIP IT ½ then SQUARE it ( ½ )2 = ( ¼ ) W

So….

3r (or 3 radii away) from the source means…

Our number is 3…. FLIP IT… ( 1/3 ) then SQUARE it (1/3)2 = (1/9) W

That means if you move 3r’s (3 times the Earths radius away) from the Earth, you weigh 1/9th of your normal weight.

Example:

An object weighs 100 Newtons on Earth’s surface. When it is moved to a distance that is one Earth radius above Earth’s surface, it will weigh-

At 1 radius above the Earth, an object will weigh ¼ of its weight on the surface of the Earth.

¼ of 100N = 25N