How To Find Gravitational Force Between Two Objects

Newton's Law of Universal Gravitation

As discussed earlier in Lesson 3, Isaac Newton compared the acceleration of the moon to the acceleration of objects on globe. Believing that gravitational forces were responsible for each, Newton was able to draw an of import conclusion nearly the dependence of gravity upon distance. This comparison led him to conclude that the forcefulness of gravitational attraction betwixt the Earth and other objects is inversely proportional to the altitude separating the earth'due south middle from the object'southward center. But distance is non the simply variable affecting the magnitude of a gravitational force. Consider Newton'southward famous equation

F_internet = m • a

Newton knew that the forcefulness that caused the apple'due south acceleration (gravity) must exist dependent upon the mass of the apple. And since the force acting to cause the apple tree's downwards dispatch as well causes the earth's upward dispatch (Newton's tertiary law), that forcefulness must also depend upon the mass of the earth. And then for Newton, the force of gravity interim between the world and any other object is direct proportional to the mass of the globe, straight proportional to the mass of the object, and inversely proportional to the foursquare of the distance that separates the centers of the earth and the object.

The UNIVERSAL Gravitation Equation

But Newton'south law of universal gravitation extends gravity beyond earth. Newton'due south law of universal gravitation is about the universality of gravity. Newton'south place in the Gravity Hall of Fame is not due to his discovery of gravity, merely rather due to his discovery that gravitation is universal. ALL objects attract each other with a strength of gravitational attraction. Gravity is universal. This force of gravitational allure is directly dependent upon the masses of both objects and inversely proportional to the foursquare of the distance that separates their centers. Newton's decision about the magnitude of gravitational forces is summarized symbolically equally

Since the gravitational force is straight proportional to the mass of both interacting objects, more than massive objects will concenter each other with a greater gravitational force. And so as the mass of either object increases, the forcefulness of gravitational allure between them also increases. If the mass of one of the objects is doubled, so the strength of gravity between them is doubled. If the mass of i of the objects is tripled, and then the force of gravity between them is tripled. If the mass of both of the objects is doubled, and then the force of gravity between them is quadrupled; and then on.

Since gravitational force is inversely proportional to the square of the separation distance between the 2 interacting objects, more separation distance volition issue in weaker gravitational forces. So as two objects are separated from each other, the forcefulness of gravitational allure between them too decreases. If the separation distance betwixt 2 objects is doubled (increased by a factor of 2), then the force of gravitational attraction is decreased by a factor of iv (2 raised to the second power). If the separation distance between any ii objects is tripled (increased by a cistron of 3), then the force of gravitational attraction is decreased by a gene of nine (3 raised to the second power).

Thinking Proportionally Almost Newton's Equation

The proportionalities expressed by Newton's universal law of gravitation are represented graphically by the following illustration. Discover 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.

Another ways of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown beneath.

The constant of proportionality (M) in the above equation is known as the universal gravitation constant . The precise value of Chiliad was determined experimentally by Henry Cavendish in the century after Newton's death. (This experiment will be discussed later in Lesson 3.) The value of K is institute to be

G = 6.673 x 10^-11 Northward k²/kgⁱⁱ

The units on G may seem rather odd; even so they are sensible. When the units on G are substituted into the equation above and multiplied by m₁• yard₂ units and divided by d² units, the result will exist Newtons - the unit of force.

Using Newton'south Gravitation Equation to Solve Problems

Knowing the value of Thousand allows us to summate the strength of gravitational attraction between whatsoever two objects of known mass and known separation distance. As a outset instance, consider the following trouble.

Sample Trouble #one

Make up one's mind the force of gravitational attraction betwixt the earth (m = v.98 ten x²⁴ kg) and a 70-kg physics pupil if the student is standing at sea level, a distance of 6.38 10 x^{half dozen} k from globe'due south center.

The solution of the problem involves substituting known values of G (6.673 ten x^-xi N m²/kgⁱⁱ ), m₁ (5.98 x 10²⁴ kg), yard₂ (70 kg) and d (half-dozen.38 x 10⁶ m) into the universal gravitation equation and solving for F_grav. The solution is as follows:

Sample Problem #2

Determine the force of gravitational attraction between the earth (m = 5.98 10 10²⁴ kg) and a 70-kg physics student if the pupil is in an aeroplane at 40000 feet to a higher place world's surface. This would place the student a distance of 6.39 x 10^vi m from earth's middle.

The solution of the problem involves substituting known values of G (6.673 x x^-11 N thousand²/kg² ), m₁ (five.98 x 10²⁴ kg), 1000₂ (seventy kg) and d (6.39 ten 10⁶ m) into the universal gravitation equation and solving for F_grav. The solution is equally follows:

Two general conceptual comments can be fabricated most the results of the two sample calculations higher up. First, find that the force of gravity acting upon the educatee (a.k.a. the educatee's weight) is less on an plane at xl 000 feet than at sea level. This illustrates the inverse relationship between separation distance and the force of gravity (or in this case, the weight of the student). The student weighs less at the higher distance. Nonetheless, a mere modify of 40 000 anxiety further from the centre of the Earth is virtually negligible. This altitude change altered the student'due south weight changed by 2 Due north that is much less than one% of the original weight. A distance of forty 000 feet (from the earth's surface to a high altitude airplane) is not very far when compared to a distance of six.38 ten 10⁶ m (equivalent to about 20 000 000 feet from the heart of the earth to the surface of the earth). This amending of altitude is like a drop in a saucepan when compared to the big radius of the Earth. Every bit shown in the diagram below, distance of separation becomes much more influential when a meaning variation is made.

The second conceptual comment to be fabricated almost the above sample calculations is that the apply of Newton's universal gravitation equation to calculate the force of gravity (or weight) yields the aforementioned effect as when calculating it using the equation presented in Unit of measurement two:

F_grav = m•thou = (70 kg)•(nine.8 g/s^two) = 686 Northward

Both equations accomplish the same event considering (every bit we will study subsequently in Lesson 3) the value of k is equivalent to the ratio of (G•M_earth)/(R_world)².

The Universality of Gravity

Gravitational interactions do not simply exist betwixt the earth and other objects; and not simply between the sun and other planets. Gravitational interactions exist between all objects with an intensity that is direct proportional to the production of their masses. So equally you lot sit in your seat in the physics classroom, you are gravitationally attracted to your lab partner, to the desk you lot are working at, and even to your physics book. Newton's revolutionary idea was that gravity is universal - ALL objects attract in proportion to the product of their masses. Gravity is universal. Of grade, about gravitational forces are then minimal to be noticed. Gravitational forces are but recognizable every bit the masses of objects go large. To illustrate this, use Newton's universal gravitation equation to calculate the forcefulness of gravity between the post-obit familiar objects. Click the buttons to check answers.

	Mass of Object ane (kg)	Mass of Object 2 (kg)	Separation Distance (m)	Force of Gravity (N)
a.	Football game Player 100 kg	Globe 5.98 x10²⁴ kg	6.38 x ten⁶ thousand (on surface)
b.	Ballerina 40 kg	Earth 5.98 x10²⁴ kg	6.38 x 10⁶ thou (on surface)
c.	Physics Pupil 70 kg	Globe 5.98 x10²⁴ kg	six.threescore ten 10^vi m (low-height orbit)
d.	Physics Student 70 kg	Physics Student seventy kg	1 grand
due east.	Physics Student lxx kg	Physics Student 70 kg	0.ii m
f.	Physics Educatee 70 kg	Physics Book ane kg	i k
m.	Physics Student 70 kg	Moon seven.34 x x²² kg	1.71 x 10⁶ 1000 (on surface)
h.	Physics Student lxx kg	Jupiter 1.901 x 10²⁷ kg	6.98 x 10⁷ m (on surface)

Today, Newton'due south police force of universal gravitation is a widely accepted theory. Information technology guides the efforts of scientists in their study of planetary orbits. Knowing that all objects exert gravitational influences on each other, the small perturbations in a planet's elliptical motion tin be easily explained. As the planet Jupiter approaches the planet Saturn in its orbit, it tends to deviate from its otherwise smooth path; this difference, or perturbation , is hands explained when because the consequence of the gravitational pull between Saturn and Jupiter. Newton'south comparing of the acceleration of the apple to that of the moon led to a surprisingly simple conclusion virtually the nature of gravity that is woven into the entire universe. All objects attract each other with a strength that is directly proportional to the product of their masses and inversely proportional to their altitude of separation.

Investigate!

Use the Newton's Law of Universal Gravitation widget below to investigate the effect of the object masses and separation distance upon the amount of gravitational attraction. Enter the masses of the two objects and their separation altitude. And then click the Submit push to view the gravitational forcefulness. Experiment with diverse values of mass and distance.

How did Newton establish that it was the force of gravity betwixt the sun and the planets was the forcefulness responsible for keeping the planets in motility along their elliptical path? Click to run into.

We Would Similar to Suggest ...

Sometimes information technology isn't enough to just read nearly it. Y'all have to interact with it! And that's exactly what you do when you use i of The Physics Classroom's Interactives. We would similar to propose that y'all combine the reading of this page with the use of our Gravitation Interactive. You can observe it in the Physics Interactives section of our website. The Gravitation Interactive allows a learner to interactively explore the inverse square law of gravitation.

Check Your Understanding

1. Suppose that ii objects attract each other with a gravitational force of xvi units. If the altitude between the two objects is doubled, what is the new forcefulness of attraction between the two objects?

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

iii. Suppose that 2 objects attract each other with a gravitational force of sixteen units. If the mass of both objects was doubled, and if the distance betwixt the objects remained the same, then what would be the new force of allure betwixt the ii objects?

4. Suppose that ii objects attract each other with a gravitational strength of 16 units. If the mass of both objects was doubled, and if the distance betwixt the objects was doubled, and then what would exist the new forcefulness of allure between the ii objects?

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

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

7. As a star ages, information technology is believed to undergo a diversity of changes. I of the concluding phases of a star'southward life is to gravitationally collapse into a black hole. What will happen to the orbit of the planets of the solar organization if our star (the Lord's day shrinks into a black hole)? (And of form, this assumes that the planets are unaffected past prior stages of the Sun's evolving stages.)

eight. Having recently completed her first Physics course, Dawn Well has devised a new business concern plan based on her teacher's Physics for Better Living theme. Dawn learned that objects weigh different amounts at different distances from Earth's center. Her plan involves buying golden by the weight at ane altitude and then selling it at another altitude at the aforementioned price per weight. Should Dawn purchase at a high altitude and sell at a low altitude or vice versa?

nine. Fred is very concerned nigh his weight but seldom does anything well-nigh it. After learning almost Newton'south law of universal gravitation in Physics course, he becomes all concerned about the possible effect of a change in World's mass upon his weight. During a (rare) free moment at the luncheon table, he speaks up "How would my weight change if the mass of the Earth increased by x%?" How would you answer Fred?

x. When comparing mass and size information for the planets Earth and Jupiter, it is observed that Jupiter is about 300 times more than massive than World. I might quickly conclude that an object on the surface of Jupiter would weigh 300 times more than on the surface of the Earth. For instance, one might expect a person who weighs 500 N on Earth would counterbalance 150000 N on the surface of Jupiter. Yet this is non the instance. In fact, a 500-Northward person on Globe weighs about 1500 N on the surface of Jupiter. Explain how this can be.