Understanding Newton’s Law of Universal Gravitation

Learning Goal: To understand Newton’s law of universal gravitation and be able to apply it in two-object situations and (collinear) three-object situations; to distinguish between the use of

G and g.

In the late 1600s, Isaac Newton proposed a rule to quantify the attractive force known as gravity between objects that have mass, such as those shown in the figure. Newton’s law of universal gravitation describes the magnitude of the attractive gravitational force  between two objects with masses m₁ and m₂ as

where r is the distance between the centers of the two objects and G is the gravitational constant.

The gravitational force is attractive, so in the figure it pulls to the right on m₁ (toward m₂) and toward the left on m₂ (toward m₁). The gravitational force acting on m₁ is equal in size to, but exactly opposite in direction from, the gravitational force acting on m₂, as required by Newton’s third law. The magnitude of both forces is calculated with the equation given above.

The gravitational constant G has the value

and should not be confused with the magnitude of the gravitational free-fall acceleration constant, denoted by g, which equals 9.80 ms/² near the surface of the earth. The size of G in SI units is tiny. This means that gravitational forces are sizeable only in the vicinity of very massive objects, such as the earth. You are in fact gravitationally attracted toward all the objects around you, such as the computer you are using, but the size of that force is too small to be noticed without extremely sensitive equipment.

Consider the earth following its nearly circular orbit (dashed curve) about the sun. (Intro 2 figure) The earth has mass m_earth = 5.98 x 10²⁴ kg and the sun has mass m_sun = 1.99 x 10³⁰ kg. They are seperated, center to center, r= 93 million miles = 150 million km.

A) What is the size of the gravitational force acting on the earth due to the sun?

=> 3.53×10²²

B) At the moment shown in the figure of the earth and sun,  what is the direction of the gravitational force acting on the earth? The possible directions are displayed in this figure.

=> A

C) What is the size of the gravitational force acting on the sun due to the earth?

=> The earth exerts  of force on the sun, exactly the same amount of force the sun exerts on the earth found in Part A.

D) Which of the following changes to the earth-sun system would reduce the magnitude of the force between them to one-fourth the value found in Part A?

=> Reduce the mass of the earth to one-fourth its normal value.

Reduce the mass of the sun to one-fourth its normal value.

Reduce the mass of the earth to one-half its normal value and the mass of the sun to one-half its normal value.

E) With the sun and the earth back in their regular positions, consider a space probe with mass m_p = 125 kg launched from the earth toward the sun. When the probe is exactly halfway. between the earth and the sun along the line connecting them, what is the direction of the net gravitational force acting on the probe?   Ignore the effects of other massive objects in the solar system, such as the moon and other planets.

=> The force is toward the sun.

F) What is the value of the composite constant , to be multiplied by the mass of the object m₀ in the equation above?

=> 9.8 m/s2