Newton’s Law of Universal Gravitation

There is a popular story that an apple fell on Newton’s head when he was sitting under an apple tree, and he suddenly thought of the “Universal Law of Gravitation”. As in all such aspects, this is almost definitely not true in its details, but now we are going to discuss what actually happened.

**What really happened with Apple?**

Probably the more correct version of the story is that Newton, upon observing an apple fall from a tree, began to think along the following lines: The apple is accelerated, since its velocity changes from zero as it is hanging on the tree and moves toward the ground. Thus, by Newton’s 2^{nd} Law there must be a force that acts on the apple to cause this acceleration. Let’s call this force “gravity”, and the associated acceleration would be the “acceleration due to gravity”. Then imagine the apple tree is 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 the top of the tallest apple tree.

**What was Sir Isaac’s Most Excellent Idea?**

Now, we will discuss Newton’s truly brilliant insight: if the force of gravity reaches the top of the highest tree, might it not reach even further; in particular, might it not reach all the way to the orbit of the Moon. Then, the orbit of the Moon about the Earth could be a consequence of the gravitational force, because the acceleration due to gravity could change the Moon’s velocity in such a way that it followed an orbit around the earth.

It was clear to Newton that the force which caused the apple’s acceleration (gravity) must be dependent upon the apple’s mass. However, the force acting to cause the apple’s downward acceleration also causes the earth’s upward acceleration (Newton’s 3^{rd} law), that force must also depend upon the earth’s mass. Therefore, the force of gravity acting between the earth and any other object is directly proportional to the earth’s mass, also directly proportional to the object’s mass, and inversely proportional to the square of the distance that separates the center of the earth and the center of the object.

**Definition of Law of Universal Gravitation**

Every object in the Universe attracts every other object with a force directed along the line of centers for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects.

**F _{g} = G m_{1}m_{2}/r^{2}**

F_{g} is the Gravitational Force

m_{1} & m_{2 } are the masses of two objects

r is the separation between the objects

G is the Universal Gravitational Constant

The constant of proportionality G is also termed as a “universal constant” because it can be thought to be the same at all times and at all places, and thereby universally characterizes the intrinsic strength of the gravitational force. G has a very small numerical value, which basically represents the reason behind the force of gravity to be the weakest force of nature.

The constant of proportionality G is also termed as a “universal constant” because it can be thought to be the same at all times and at all places, and thereby universally characterizes the intrinsic strength of the gravitational force. G has a very small numerical value, which basically represents the reason behind the force of gravity to be the weakest force of nature.

**
**But Newton’s law of universal gravitation extends gravity beyond earth. Newton’s law of universal gravitation is all about the universality 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 which separates their centers.

**Weight and the Gravitational Force**

We have seen that in the Universal Law of Gravitation the crucial quantity is mass. In popular language mass and weight are often used to mean the same thing; in reality, they are related but quite different things. What we commonly call weight is really just the *gravitational force* exerted on an object of a certain mass. We can illustrate by choosing the Earth as one of the two masses in the previous illustration of the Law of Gravitation:

**Weight = F _{g} = G Mm/r^{2 }= mg**

M is the mass of the Earth

m is the mass of the object

r is the radius of the Earth

g is the acceleration due to gravity at the Earth’s surface

Thus, the weight of an object at the surface of the Earth can be obtained by multiplying the mass *m* with acceleration due to gravity, *g*, at the Earth’s surface. The acceleration due to gravity is approximately the product of the mass of the Earth *M* and the universal gravitational constant *G*, divided by the square of Earth’s radius. The measured gravitational acceleration at the Earth’s surface is found to be about 980 cm/s^{2}.

**Mass and Weight**

Mass is a measure of material in an object, but weight is a measure of the gravitational force exerted on that material in a gravitational field. Therefore, mass and weight are proportional to each other, with the acceleration due to gravity as the constant of proportionality. Thus, it follows that mass is constant for an object, but the weight of the object depends on its location. For example, if we transported the preceding object of mass *m* to the Moon’s surface, the gravitational acceleration would change because both the mass and radius of the Moon differ from those of the Earth. Thus, our object has mass *m* both on the surface of the Earth as well as on the surface of the Moon, but its weight would be much less on the Moon’s surface because of the gravitational acceleration there is a factor of 6 less than at the surface of the Earth.

**Using Equations as a Guide to Thinking**

The inverse square law proposed by Newton depicts that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the object’s centers. Alteration of separation distance (r) results in an alteration in the force of gravity acting between the objects. Since the two quantities are inversely proportional, an increase in one quantity results in a decrease in the value of the other quantity. Thus, an increase in the separation distance causes a decrease in the force of gravity, and a decrease in the separation distance causes an increase in the force of gravity.

Furthermore, the factor by which the force of gravity is changed is the square of the factor by which the separation distance is changed. So if the separation distance is doubled, then the force of gravity is decreased by a factor of four. And if the separation distance (r) is tripled, then the force of gravity is decreased by a factor of nine. Thinking of the force-distance relationship in this way involves using a mathematical relationship as a guide to thinking about how an alteration in one variable affects the other variable. Equations can be more than mere recipes for algebraic problem-solving.

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.

In the above figure, the left-hand side represents the effect of “mass” if the distance “d” between the two objects remains fixed. The right-hand figure indicates the effect of changing the distance while the mass remains constant, and the effect of changing both the distance and the mass is shown in the last part.

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