**Vida’s Wiki Page: Newton’s Universal Gravitational Law**

**Table of Contents:**
Sequence in Wiki Page Title

## How did Newton come up with is Universal Gravitational Law?Edit

Before we begin, here is a picture of Isaac Newton himself:

In 1666, Isaac Newton was living at home in England because schools were closed due to the black plague – so when you are sitting in class bored out of your mind wondering who in the world thought up these seemingly meaningless equations, thinking that the reason was solely based upon making your life miserable, be thankful you are in school because Newton thought this stuff up while school was closed! Imagine what he would have come up with if school were open…oh jeez… It had already been shown that if the path of a planet were an ellipse, then the magnitude of force F on the planet resulting from the sun must vary inversely with the square of the distance between the center of the planet and the center of the sun (in agreement with Kepler’s first law of planetary motion: (1)paths of planets are ellipses, with the sun at one focus; (2)an imaginary line from sun to a planet streams out equal areas in equal time intervals, so planets move faster when they are closer to the sun and slower when they are farther away from the sun; (3)the square of the ration periods of any two planets revolving around the sun is equal to the cube of their ratio of their average distances from the sun):

F (is proportional to) 1 / d²
where (sign) means is proportional to and d is the distance between the sun and the centers of two bodies
Newton showed that the force acted in the direction of the line connecting the two centers of the two bodies, however, what still remained a mystery was if the force acting between the planet and the sun the same force that caused objects to fall to Earth.
Newton saw an apple fall from a tree and realized that it fell straight down because Earth attracted it. He did not know if this same force that attracted the apple to fall to the Earth was the same force that attracted the planets to the sun. It seemed like a far stretch that the same force that could pull an apple down could pull planets together, but it also seemed slightly plausible. He thought that the force pulling an apple down must be proportional to its mass somehow. According to his third law of motion, the apple must also attract the Earth (all forces come in pairs, and the two forces in a pair that act on different objects and are equal in magnitude are opposite in direction: F a on b = -F b on a), so the force of attraction must also be proportional to the mass of the Earth. This attractive force that exists between all objects is today known as **GRAVITATIONAL FORCE**
Newton was confident that the laws governing motion on Earth would work anywhere else in the universe that he assumed that the same force of attraction would act between any two masses, mA and mB. He then proposed his LAW OF UNIVERSAL GRAVITATION, which can be shown by the equation:

(Reference 2) where d is the distance between two centers of the masses, and G is a universal constant – meaning that it is the same everywhere.

Here is another visual representation of the equation:

(Reference 3) (Reference 4) The proportionalities expressed by Newton's universal law of gravitation is represented graphically by the following illustration. Observe how the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance of separation:

## Wait a Minute! How did he know that his equation was right?Edit

Newton made his laws so that they ally to the motion of planets around the sun, which agreed with Kepler’s third law of planetary motion, see above, which was all the proof Newton needed to know that his laws agreed with the observations and beliefs of the day, because no one actually knows if his equation is right. Science is all theories with substantial amounts of proof behind those theories, and they are accepted as true until proven wrong.

## How on earth do you find the weight of the Earth? Is there some huge scale that the Earth was just plunked on?Edit

First of all, the weight of the Earth is represented by a letter, G, and the amount that G represents can be found on page one of the reference tables. G is equal to:

6.67 x 10 ˆ11 N•m²/kg²

It is very difficult to measure the force of attraction between two objects, and took a very long time before an apparatus that was sensitive enough to do this was developed. In 1798, Henry Cavendish was the first person to experimentally determine the value of G to within one percent of its accepted value today. Here’s how he did it:

He used a torsion (twisting) balance in his “Cavendish Experiment” to conduct his measurement. It functions with a horizontal rod with small test weights on each end that are hung at its center by a thin metal wire, called the torsion fiber. When an external mass, such as an apple, for example, is brought up close to one of the tiny weights, the gravitational attraction causes the fiber to twist slightly. The mass then falls into the external weight, such as the apple, rather than the other way around, as when Newton observed an apple falling into the earth. The Cavendish torsion balance experiment can be used to determine the weight of the earth by comparing the attraction of the tiny test mass first to the Earth and then to the external mass, like an apple. The result is that the Earth weighs in at six thousand six hundred billion billion tons (6,000,000,000,000,000,000,000,000 kilograms). Wow! That’s twenty-four zeros! Surprisingly enough, the earth weighs about twice what it should if it were made of rock, which lead some scientists to believe that maybe the Earth was not made of rock. Today, the accepted reason for this discrepancy is that the planet has a heavy nickel-iron core.

## For the smarty pants reading this who asks, “Is that all the equation is? Or is there more that can be done with it?” My answer to you is, “Of course there’s more!”Edit

If you rearrange Newton’s Universal Gravitational Law and incorporate some previous knowledge, you can actually come up with the time it takes for a planet to make one revolution around the sun. If mp is the mass of a planet, ms is the mass of the sun, r is the radius of the planet’s orbit Newton’s second law of motion states that force is equal to mass multiplied by acceleration, represented by the equation:

F = ma

Where F is force, m is mass, and a is acceleration. This equation can be re-written as:

F = mpac

Where F is the gravitational force, mp is the mass, and ac is the centripetal acceleration of the planet, assuming it is a circular orbiting path the planet is following. Now you must remember that there is an equation to find centripetal acceleration:

** ac = 4πr²**
____
T²

where T is time in seconds. Thus the equation F = mpac can be expressed as:

** F = mp4πr²**
______
T²

Now, if you make this equation equal to Newton’s Universal Gravitational equation, you get something that looks like this:

** msmp mp4π²r**
G_____ = _______
r² T²

Now for a little cross-multiplication and rearranging that requires some lovely algebra skills, and after that is done, a final equation is born:

** 4π²**
T² = ____ • r³
Gms

This equation beautifully illustrates Kepler’s third law (see above) and was derived from Newton’s Universal Gravitational Law. Amazing the power of one equation, isn’t it?

## Okay, so now I understand WHAT Newton’s Universal Gravitational Law is, and how he came up with it and knew it was right, but how does this help me in my physics class? How do I use these ideas and relationships and this equation and apply it to problems to pass my class and my tests?Edit

To give some example of how to apply and understand Newton’s equation, if the mass of a planet near the sun were doubled, the force of attraction would also be doubled; if the planet were near a star having twice the mass of the sun, the force between the two bodies would be twice as great as well; if the planet were twice the distance from the sun, the gravitational force would only be one quarter as strong; since force depends on 1/d², it is called an inverse square law Many problems about this law just have to do with plugging numbers into the equation and chugging them out on your calculator…unless you are a math buff and prefer to do all of the calculations by hand! Much of the success with correctly solving problems having to do with Newton’s Universal Gravitational Law deals with recognizing which numbers are representative of which value.

## Practice ProblemsEdit

**1.** Suppose that two objects attract each other with a force of 16 units. If the distance between the two objects is doubled, what is the new force of attraction between the two objects?

**2.** Suppose that two objects attract each other with a force of 16 units. If the distance between the two objects is reduced in half, then what is the new force of attraction between the two objects?

**3.** Suppose that two objects attract each other with a force of 16 units. If the mass of both objects was doubled, and if the distance between the objects remained the same, then what would be the new force of attraction between the two objects?

**4.** Suppose that two objects attract each other with a force of 16 units. If the mass of both objects was doubled, and if the distance between the objects was doubled, then what would be the new force of attraction between the two objects?

**5.** Suppose that two objects attract each other with a force of 16 units. If the mass of both objects was tripled, and if the distance between the objects was doubled, then what would be the new force of attraction between the two objects?

**6.** Suppose that two objects attract each other with a force of 16 units. If the mass of object 1 was doubled, and if the distance between the objects was tripled, then what would be the new force of attraction between the two objects?

**7.** What would happen to your weight if the mass of the Earth somehow increased by 10%?

**8.** The planet Jupiter is more than 300 times as massive as Earth, so it might seem that a body on the surface of Jupiter would weigh 300 times as much as on Earth. But it so happens a body would scarcely weigh three times as much on the surface of Jupiter as it would on the surface of the Earth. Explain why this is so.

9.with what force an apple weighing 1N attracts the earth?

## Answers to Practice ProblemsEdit

**1.** If the distance is increased by a factor of 2, then force will be decreased by a factor of 4 (22). The new force is then 1/4 of the original 16 units.
F = (16 N) / 4 = 4 units

**2.** If the distance is decreased by a factor of 2, then force will be increased by a factor of 4 (22). The new force is then 4 times the original 16 units.
F = (16 N) • 4 = 64 units

**3.** If each mass is increased by a factor of 2, then force will be increased by a factor of 4 (2*2). The new force is then t times the original 16 units.
F = (16 N) • 4 = 64 units

**4.** If each mass is increased by a factor of 2, then force will be increased by a factor of 4 (2*2). But this effect is offset by the doubling of the distance. Doubling the distance would cause the force to be decreased by a factor of 4 (22); the result is that there is no net effect on force.
F = (16 N) • 4 / 4 = 16 units

**5.** If each mass is increased by a factor of 3, then force will be increased by a factor of 9 (3*3). But this effect is partly offset by the doubling of the distance. Doubling the distance would cause the force to be decreased by a factor of 4 (22). the net effect on force is that it increased by 9/4.
F = (16 N) * 9 / 4 = 36 units

**6.** If each mass is increased by a factor of 3, then force will be increased by a factor of 9 (3*3). But this effect is partly offset by the doubling of the distance. Doubling the distance would cause the force to be decreased by a factor of 4 (22). The net effect on force is that it increased by 9/4.
F = (16 N) • 9 / 4 = 36 units

**7.** You would weigh 10% MORE since my weight is DIRECTLY PROPORTIONAL to the mass of the Earth.

**8.** The effect of the greater mass of Jupiter is partly different because the RADIUS of Jupiter is larger. An object on Jupiter's surface is 10 TIMES FARTHER from Jupiter’s center than it would be if on the surface of the Earth. Thus, the 300-fold increase in force (caused by the greater mass) must be divided by 100 since the distance is 10 times greater.

## ReferencesEdit

1. Textbook, p.177; 181-4 Zitzewitz, Paul W., Ph.D. NY, Ny. 1999. Glencoe. *Italic text*Physics Problems and Principles*Italic text*.

2. Formulas

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