# Newton's Law of Gravitation and Masses

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Learning Goal: To understand Newton's law of gravitation and the distinction between inertial and gravitational masses.

In this problem, you will practice using Newton's law of gravitation. According to that law, the magnitude of the gravitational force between two small particles of masses and , separated by a distance , is given by

,

where is the universal gravitational constant, whose numerical value (in SI units) is .

This formula applies not only to small particles, but also to spherical objects. In fact, the gravitational force between two uniform spheres is the same as if we concentrated all the mass of each sphere at its center. Thus, by modeling the Earth and the Moon as uniform spheres, you can use the particle approximation when calculating the force of gravity between them.

Be careful in using Newton's law to choose the correct value for . To calculate the force of gravitational attraction between two uniform spheres, the distance in the equation for Newton's law of gravitation is the distance between the centers of the spheres. For instance, if a small object such as an elephant is located on the surface of the Earth, the radius of the Earth would be used in the equation. Note that the force of gravity acting on an object located near the surface of a planet is often called weight.

Also note that in situations involving satellites, you are often given the altitude of the satellite, that is, the distance from the satellite to the surface of the planet; this is not the distance to be used in the formula for the law of gravitation.

There is a potentially confusing issue involving mass. Mass is defined as a measure of an object's inertia, that is, its ability to resist acceleration. Newton's second law demonstrates the relationship between mass, acceleration, and the net force acting on an object: . We can now refer to this measure of inertia more precisely as the inertial mass.

On the other hand, the masses of the particles that appear in the expression for the law of gravity seem to have nothing to do with inertia: Rather, they serve as a measure of the strength of gravitational interactions. It would be reasonable to call such a property gravitational mass.

Does this mean that every object has two different masses? Generally speaking, yes. However, the good news is that according to the latest, highly precise, measurements, the inertial and the gravitational mass of an object are, in fact, equal to each other; it is an established consensus among physicists that there is only one mass after all, which is a measure of both the object's inertia and its ability to engage in gravitational interactions. Note that this consensus, like everything else in science, is open to possible amendments in the future.

In this problem, you will answer several questions that require the use of Newton's law of gravitation.

A) Two particles are separated by a certain distance. The force of gravitational interaction between them is . Now the separation between the particles is tripled. Find the new force of gravitational interaction .

Express your answer in terms of .

=

B) A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitational interaction between the satellite and the planet is . Then the satellite moves to a different orbit, so that its altitude is tripled. Find the new force of gravitational interaction .

Express your answer in terms of .

=

C) A satellite revolves around a planet at an altitude equal to the radius of the planet. The force of gravitational interaction between the satellite and the planet is . Then the satellite is brought back to the surface of the planet. Find the new force of gravitational interaction .

Express your answer in terms of .

=

D) Two satellites revolve around the Earth. Satellite A has mass and has an orbit of radius . Satellite B has mass and an orbit of unknown radius . The forces of gravitational attraction between each satellite and the Earth is the same. Find .

Express your answer in terms of .

=

E) An adult elephant has a mass of about 5.0 tons. An adult elephant shrew has a mass of about 50 grams. How far from the center of the Earth should an elephant be placed so that its weight equals that of the elephant shrew on the surface of the Earth? The radius of the Earth is 6400 . ( .)

Express your answer in kilometers.

=

F) The table below gives the masses of the Earth, the Moon, and the Sun.

Name Mass (kg)

Earth

Moon

Sun

The average distance between the Earth and the Moon is . The average distance between the Earth and the Sun is . Use this information to answer the following questions.

Find the net gravitational force acting on the Earth in the Sun-Earth-Moon system during the new moon (when the moon is located directly between the Earth and the Sun).

Express your answer in newtons to three significant figures.

=

G) Find the net gravitational force acting on the Earth in the Sun-Earth-Moon system during the full moon (when the Earth is located directly between the moon and the sun).

Express your answer in newtons to three significant figures.

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