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Chapter Review

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In this chapter we study gravity - a fundamental force of nature. Of the four fundamental forces, gravity is the most familiar in everyday life. The other fundamental forces will be discussed later in the text.

12-1 - 12-2 Newton's Law of Universal Gravitation and the Attraction of Spherical Bodies

Gravitation is the phenomenon that between every two objects there is a force of attraction. Newton's law of universal gravitation describes the behavior of this force. Between any two point masses m₁ and m₂, the magnitude of the gravitational force on each mass due to the other is given by

,

where r is the distance between the two masses and G is a constant called the universal gravitational constant. The value of this constant is

.

The force on each mass points directly at the other mass because each mass attracts the other toward it.

For situations involving more than two masses we apply the principle of superposition:

The net gravitational force on any given mass due to two or more other masses is the vector sum of the gravitational forces due to each of the other masses individually.

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Exercise 12.1 Earth and Venus: On average Earth and Venus are separated by about 4.14 x 10¹⁰ m. What is the magnitude of the gravitational force between them at this separation?

Solution:

In this problem we are only given the distance between Eath and Venus. To calculate the gravitational force between them we will need to look up their masses. The masses are as follows:

M_E = 5.97 x 10²⁴ kg, M_V = 4.87 x 10²⁴ kg.

Having all the data we need, the force can now be calculated.

.

Remember, this is the magnitude of the force exerted on both Earth and Venus.

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The above law of universal gravitation is stated for point masses. The detailed calculations for extended bodies can become complicated but Newton figured out that the final result for spherical bodies becomes simple again. Basically, Newton showed that two completely separated, uniform spherical bodies attract each other as if they are point masses with all of their mass located at their respective centers. This fact explains why the point mass formula given above works so well for large objects like the earth and the moon.

Treating the earth as a point mass M_E located at the center of the earth provides further insight into the acceleration due to gravity that we measure near Earth's surface. Objects near the surface are a distance from the center roughly equal to the radius of the earth R_E. Using Newton's law of gravity we can conclude that the acceleration due to gravity near the surface is given by

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With this result you can also see that the higher you go above the surface, the further you are from Earth's center, and therefore, the weaker the effect of gravity resulting in a smaller acceleration.

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Example 12.2 Martian Gravity: What is the acceleration due to gravity on the surface of Mars?

Picture the Problem The picture shows a representation of the planet Mars, with its radius extending from the center to the surface.

Strategy We need to adapt the expression for Earth's surface acceleration of gravity to Mars.

Solution

1. Look up the mass of Mars:	MM = 6.4 x 1032 kg
2. Look up the radius of Mars:	RM = 3.4 x 106 m
3. Use the mass and radius of Mars in place of those for the Earth in the expression for g:

Insight A similar substitution can be made for any of the spherical astronomical bodies.

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Physlet Illustration: Newton's Law of Gravitation
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m = kg
A planet of variable mass (1023 kg < m < 1028 kg) and a Sun (mass 2 x 1030 kg) experience a gravitational attraction. The distance grid is in Astronomical Units (AU). The force between them is shown in Newtons. Can you verify Newton's Law of Universal Gravitation? Start
Hints: How does the force change if you change the mass of the planet while keeping the distance between the Sun and the planet constant? Drag the planet closer to or farther from the Sun. How does the force change? Is the magnitude of the force what you would expect? (Recall that G = 6.67 x 10-11 Nm2/kg2 and that 1 AU = 1.5 x 1011 m.)

Practice Quiz

Two objects are separated by a distance r. If the mass of each object is doubled, how does the gravitational force between them change? It increases to twice as much. It decreases to half as much. It increases to four times as much. It decreases to one-fourth as much. [None of the above.]

Two objects are separated by a distance r. If the distance between them is doubled, how does the gravitational force between them change? It increases to twice as much. It decreases to half as much. It increases to four times as much. It decreases to one-fourth as much. [None of the above.]

What is the acceleration due to gravity at a height of 1000 km above Earth's surface? 9.78 m/s2 4.89 m/s2 3.98 m/s2 7.31 m/s2 [None of the above.]

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