Relationship Between Mass And Gravity

Newton'southward Constabulary of Universal Gravitation

As discussed earlier in Lesson 3, Isaac Newton compared the dispatch of the moon to the acceleration of objects on earth. Assertive that gravitational forces were responsible for each, Newton was able to draw an of import conclusion almost the dependence of gravity upon distance. This comparison led him to conclude that the forcefulness of gravitational attraction between the World and other objects is inversely proportional to the distance separating the globe's heart from the object's centre. But distance is non the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equation

F_internet = grand • a

Newton knew that the force that caused the apple's dispatch (gravity) must be dependent upon the mass of the apple. And since the force acting to crusade the apple's downward acceleration too causes the earth'south upward dispatch (Newton's third police force), that strength must besides depend upon the mass of the world. Then for Newton, the force of gravity acting betwixt the earth and whatever other object is directly proportional to the mass of the globe, directly proportional to the mass of the object, and inversely proportional to the square of the distance that separates the centers of the earth and the object.

The UNIVERSAL Gravitation Equation

Only Newton'southward police of universal gravitation extends gravity beyond earth. Newton's law of universal gravitation is about the universality of gravity. Newton's place in the Gravity Hall of Fame is non 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 strength of gravitational attraction is direct dependent upon the masses of both objects and inversely proportional to the foursquare of the distance that separates their centers. Newton'south conclusion about the magnitude of gravitational forces is summarized symbolically every bit

Since the gravitational strength is straight proportional to the mass of both interacting objects, more than massive objects will attract each other with a greater gravitational strength. So every bit the mass of either object increases, the force of gravitational attraction betwixt them also increases. If the mass of one of the objects is doubled, and then the force of gravity between them is doubled. If the mass of ane of the objects is tripled, then the strength of gravity between them is tripled. If the mass of both of the objects is doubled, then the force of gravity between them is quadrupled; and and so on.

Since gravitational force is inversely proportional to the foursquare of the separation altitude between the two interacting objects, more separation altitude volition issue in weaker gravitational forces. And so as two objects are separated from each other, the strength of gravitational attraction betwixt them also decreases. If the separation altitude between two objects is doubled (increased past a factor of two), so the forcefulness of gravitational attraction is decreased by a factor of four (2 raised to the second ability). If the separation distance between any two objects is tripled (increased by a factor of 3), then the force of gravitational attraction is decreased by a factor of 9 (3 raised to the second power).

Thinking Proportionally Most Newton's Equation

The proportionalities expressed by Newton'south universal police of gravitation are represented graphically past the post-obit illustration. Discover how the forcefulness of gravity is directly proportional to the product of the two masses and inversely proportional to the foursquare of the distance of separation.

Some other means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown below.

The abiding of proportionality (M) in the in a higher place equation is known as the universal gravitation constant . The precise value of G was determined experimentally by Henry Cavendish in the century afterward Newton'southward death. (This experiment volition be discussed later in Lesson 3.) The value of 1000 is found to be

M = six.673 x 10^-11 N m^two/kg²

The units on 1000 may seem rather odd; nonetheless they are sensible. When the units on Thou are substituted into the equation to a higher place and multiplied by m₁• g_two units and divided by d² units, the result will exist Newtons - the unit of force.

Using Newton's Gravitation Equation to Solve Problems

Knowing the value of M allows united states to calculate the forcefulness of gravitational allure between any ii objects of known mass and known separation distance. As a get-go example, consider the following trouble.

Sample Problem #1

Determine the force of gravitational allure betwixt the earth (yard = 5.98 ten 10²⁴ kg) and a 70-kg physics student if the student is standing at sea level, a distance of six.38 x x^vi m from world's center.

The solution of the trouble involves substituting known values of Chiliad (6.673 x 10^-11 N m²/kg² ), g₁ (5.98 ten 10²⁴ kg), m₂ (70 kg) and d (half dozen.38 x ten⁶ m) into the universal gravitation equation and solving for F_grav. The solution is as follows:

Sample Problem #two

Decide the force of gravitational attraction between the globe (thou = 5.98 x ten²⁴ kg) and a lxx-kg physics student if the student is in an airplane at 40000 feet to a higher place earth's surface. This would place the student a distance of half dozen.39 ten 10⁶ m from earth's eye.

The solution of the problem involves substituting known values of G (6.673 x x^-11 N m²/kgⁱⁱ ), m₁ (5.98 ten 10²⁴ kg), k_two (70 kg) and d (6.39 x 10⁶ m) into the universal gravitation equation and solving for F_grav. The solution is equally follows:

Two general conceptual comments can exist made about the results of the 2 sample calculations higher up. First, observe that the force of gravity acting upon the student (a.thou.a. the pupil's weight) is less on an airplane at forty 000 feet than at body of water level. This illustrates the changed human relationship between separation altitude and the force of gravity (or in this case, the weight of the educatee). The student weighs less at the higher altitude. Still, a mere change of 40 000 feet farther from the center of the Earth is virtually negligible. This altitude change altered the pupil's weight inverse by 2 Northward that is much less than 1% of the original weight. A altitude of 40 000 feet (from the earth's surface to a loftier distance airplane) is non very far when compared to a distance of 6.38 x ten⁶ m (equivalent to near twenty 000 000 anxiety from the center of the earth to the surface of the world). This alteration of altitude is like a drop in a bucket when compared to the large radius of the Earth. As shown in the diagram beneath, altitude of separation becomes much more influential when a significant variation is made.

The 2d conceptual comment to exist made near the above sample calculations is that the use of Newton's universal gravitation equation to calculate the strength of gravity (or weight) yields the same result as when calculating it using the equation presented in Unit ii:

F_grav = m•one thousand = (70 kg)•(9.8 thou/s²) = 686 Due north

Both equations reach the same event because (as we will study afterward in Lesson three) the value of g is equivalent to the ratio of (G•Chiliad_earth)/(R_earth)².

The Universality of Gravity

Gravitational interactions practise not just exist between the earth and other objects; and non simply between the dominicus and other planets. Gravitational interactions exist between all objects with an intensity that is directly proportional to the product of their masses. So every bit you 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 thought was that gravity is universal - ALL objects attract in proportion to the product of their masses. Gravity is universal. Of grade, most gravitational forces are so minimal to exist noticed. Gravitational forces are only recognizable every bit the masses of objects become big. To illustrate this, utilize Newton's universal gravitation equation to calculate the force of gravity between the post-obit familiar objects. Click the buttons to check answers.

	Mass of Object one (kg)	Mass of Object ii (kg)	Separation Distance (thousand)	Forcefulness of Gravity (N)
a.	Football Player 100 kg	Earth v.98 x10²⁴ kg	6.38 x 10^{half dozen} m (on surface)
b.	Ballerina 40 kg	Earth 5.98 x10²⁴ kg	half dozen.38 ten 10^six g (on surface)
c.	Physics Student lxx kg	Earth five.98 x10²⁴ kg	half dozen.lx ten x⁶ m (low-tiptop orbit)
d.	Physics Student 70 kg	Physics Student 70 kg	ane one thousand
e.	Physics Student 70 kg	Physics Pupil 70 kg	0.2 grand
f.	Physics Student 70 kg	Physics Book 1 kg	1 chiliad
1000.	Physics Student 70 kg	Moon seven.34 x 10²² kg	1.71 x 10⁶ thousand (on surface)
h.	Physics Educatee 70 kg	Jupiter 1.901 x 10²⁷ kg	6.98 x x⁷ m (on surface)

Today, Newton'due south police force of universal gravitation is a widely accustomed theory. Information technology guides the efforts of scientists in their written report of planetary orbits. Knowing that all objects exert gravitational influences on each other, the small perturbations in a planet'southward elliptical move tin can be hands explained. As the planet Jupiter approaches the planet Saturn in its orbit, it tends to deviate from its otherwise smooth path; this deviation, or perturbation , is easily explained when because the effect of the gravitational pull between Saturn and Jupiter. Newton's comparison of the acceleration of the apple to that of the moon led to a surprisingly simple decision about the nature of gravity that is woven into the entire universe. All objects concenter each other with a force that is directly proportional to the product of their masses and inversely proportional to their distance of separation.

Investigate!

Utilise the Newton'due south Police of Universal Gravitation widget below to investigate the effect of the object masses and separation distance upon the corporeality of gravitational attraction. Enter the masses of the ii objects and their separation distance. Then click the Submit button to view the gravitational force. Experiment with various values of mass and distance.

How did Newton constitute that information technology was the force of gravity between the sun and the planets was the force responsible for keeping the planets in motility forth their elliptical path? Click to encounter.

We Would Like to Suggest ...

Sometimes information technology isn't enough to but read about it. You take to interact with it! And that'south exactly what yous do when you use one of The Physics Classroom's Interactives. We would like to advise that you combine the reading of this page with the utilize of our Gravitation Interactive. You can discover information technology in the Physics Interactives section of our website. The Gravitation Interactive allows a learner to interactively explore the changed square law of gravitation.

Cheque Your Agreement

one. Suppose that 2 objects attract each other with a gravitational force of sixteen units. If the distance between the two objects is doubled, what is the new strength of attraction between the 2 objects?

ii. Suppose that two objects attract each other with a gravitational force of sixteen units. If the altitude between the two objects is reduced in one-half, then what is the new force of attraction between the 2 objects?

3. Suppose that two objects attract each other with a gravitational force of 16 units. If the mass of both objects was doubled, and if the distance between the objects remained the aforementioned, then what would be the new force of attraction betwixt the 2 objects?

4. Suppose that two objects concenter each other with a gravitational forcefulness 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?

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

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

seven. As a star ages, it is believed to undergo a variety of changes. One of the final phases of a star's life is to gravitationally plummet into a black pigsty. What will happen to the orbit of the planets of the solar system if our star (the Sun shrinks into a black hole)? (And of course, this assumes that the planets are unaffected past prior stages of the Sun's evolving stages.)

8. Having recently completed her first Physics course, Dawn Well has devised a new business plan based on her teacher'southward Physics for Meliorate Living theme. Dawn learned that objects weigh dissimilar amounts at dissimilar distances from World's center. Her plan involves buying gilded by the weight at one altitude and then selling information technology at another altitude at the same cost per weight. Should Dawn buy at a loftier distance and sell at a low altitude or vice versa?

ix. Fred is very concerned well-nigh his weight but seldom does anything well-nigh information technology. Afterward learning about Newton's police of universal gravitation in Physics course, he becomes all concerned most the possible consequence of a change in Earth'south mass upon his weight. During a (rare) free moment at the lunch table, he speaks upwardly "How would my weight modify if the mass of the World increased by ten%?" How would you reply Fred?

10. When comparison mass and size data for the planets Earth and Jupiter, it is observed that Jupiter is about 300 times more massive than Earth. Ane might chop-chop conclude that an object on the surface of Jupiter would counterbalance 300 times more than on the surface of the Earth. For instance, one might expect a person who weighs 500 N on Earth would weigh 150000 N on the surface of Jupiter. Yet this is not the case. In fact, a 500-Northward person on Earth weighs about 1500 N on the surface of Jupiter. Explain how this can be.