Learning Objectives
By the end of this section, you will be able to do the following:
- Explain Earth's gravitational force
- Describe the gravitational effect of the Moon on Earth
- Discuss weightlessness in space
- Understand the Cavendish experiment
The information presented in this section supports the following AP® learning objectives and science practices:
- 2.B.2.1 The student is able to apply to calculate the gravitational field due to an object with mass M, where the field is a vector directed toward the center of the object of mass M. (S.P. 2.2)
- 2.B.2.2 The student is able to approximate a numerical value of the gravitational field (g) near the surface of an object from its radius and mass relative to those of Earth or other reference objects. (S.P. 2.2)
- 3.A.3.4. The student is able to make claims about the force on an object due to the presence of other objects with the same property: mass and electric charge. (S.P. 6.1, 6.4)
What do aching feet, a falling apple, and the orbit of the Moon have in common? Each is caused by the gravitational force. Our feet are strained by supporting our weight—the force of Earth's gravity on us. An apple falls from a tree because of the same force acting a few meters above Earth's surface. And the Moon orbits Earth because gravity is able to supply the necessary centripetal force at a distance of hundreds of millions of meters. In fact, the same force causes planets to orbit the Sun, stars to orbit the center of the galaxy, and galaxies to cluster together. Gravity is another example of underlying simplicity in nature. It is the weakest of the four basic forces found in nature, and in some ways the least understood. It is a force that acts at a distance, without physical contact, and is expressed by a formula that is valid everywhere in the universe, for masses and distances that vary from the tiny to the immense.
Sir Isaac Newton was the first scientist to precisely define the gravitational force, and to show that it could explain both falling bodies and astronomical motions (see Figure 6.20). But Newton was not the first to suspect that the same force caused both our weight and the motion of planets. His forerunner, Galileo Galilei, had contended that falling bodies and planetary motions had the same cause. Some of Newton's contemporaries, such as Robert Hooke, Christopher Wren, and Edmund Halley, had also made some progress toward understanding gravitation. But Newton was the first to propose an exact mathematical form and to use that form to show that the motion of heavenly bodies should be conic sections—circles, ellipses, parabolas, and hyperbolas. This theoretical prediction was a major triumph. It had been known for some time that moons, planets, and comets follow such paths, but no one had been able to propose a mechanism that caused them to follow these paths and not others. This was one of the earliest examples of a theory derived from empirical evidence doing more than merely describing those empirical results; It made claims about the fundamental workings of the universe.
The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Stated in modern language, Newton's universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Misconception Alert
The magnitude of the force on each object (one has larger mass than the other) is the same, consistent with Newton's third law.
The bodies we are dealing with tend to be large. To simplify the situation we assume that the body acts as if its entire mass is concentrated at one specific point called the center of mass (CM), which will be further explored in Linear Momentum and Collisions. For two bodies having masses and with a distance between their centers of mass, the equation for Newton's universal law of gravitation is
where is the magnitude of the gravitational force and is a proportionality factor called the gravitational constant. is a universal gravitational constant—that is, it is thought to be the same everywhere in the universe. It has been measured experimentally to be
in SI units. Note that the units of are such that a force in Newton's is obtained from , when considering masses in kilograms and distance in meters. For example, two 1.000 kg masses separated by 1.000 m will experience a gravitational attraction of . This is an extraordinarily small force. The small magnitude of the gravitational force is consistent with everyday experience. We are unaware that even large objects like mountains exert gravitational forces on us. In fact, our body weight is the force of attraction of the entire Earth on us with a mass of .
The experiment to measure G was first performed by Cavendish, and is explained in more detail later. The fundamental concept it is based on is having a known mass on a spring with a known force, or spring, constant. Then, a second known mass is placed at multiple known distances from the first, and the amount of stretch in the spring resulting from the gravitational attraction of the two masses is measured.
Recall that the acceleration due to gravity is about on Earth. We can now determine why this is so. The weight of an object mg is the gravitational force between it and Earth. Substituting mg for in Newton's universal law of gravitation gives
where is the mass of the object, is the mass of Earth, and is the distance to the center of Earth, the latter of which is the distance between the centers of mass of the object and Earth. See Figure 6.22. The mass of the object cancels, leaving an equation for
Substituting known values for Earth's mass and radius to three significant figures,
and we obtain a value for the acceleration of a falling body
This is the expected value and is independent of the body's mass. Newton's law of gravitation takes Galileo's observation that all masses fall with the same acceleration a step further, explaining the observation in terms of a force that causes objects to fall—in fact, in terms of a universally existing force of attraction between masses.
Gravitational Mass and Inertial Mass
Notice that, in Figure 6.21, the mass of the objects under consideration is directly proportional to the gravitational force. More mass means greater forces, and vice versa. However, we have already seen the concept of mass before in a different context.
In Chapter 4, you read that mass is a measure of inertia. However, we normally measure the mass of an object by measuring the force of gravity (F) on it.
How do we know that inertial mass is identical to gravitational mass? Assume that we compare the mass of two objects. The objects have inertial masses m1 and m2. If the objects balance each other on a pan balance, we can conclude that they have the same gravitational mass, that is, that they experience the same force due to gravity, F. Using Newton's second law of motion, F = ma, we can write m1 a1 = m2 a2.
If we can show that the two objects experience the same acceleration due to gravity, we can conclude that m1 = m2, that is, that the objects' inertial masses are equal.
In fact, Galileo and others conducted experiments to show that, when factors such as wind resistance are kept constant, all objects, regardless of their mass, experience the same acceleration due to gravity. Galileo is famously said to have dropped two balls of different masses off the leaning tower of Pisa to demonstrate this. The balls accelerated at the same rate. Since acceleration due to gravity is constant for all objects on Earth, regardless of their mass or composition, that is, a1 = a2, then m1 = m2. Thus, we can conclude that inertial mass is identical to gravitational mass. This allows us to calculate the acceleration of free fall due to gravity, such as in the orbits of planets.
Take-Home Experiment
Take a marble, a ball, and a spoon and drop them from the same height. Do they hit the floor at the same time? If you drop a piece of paper as well, does it behave like the other objects? Explain your observations.
Making Connections: Gravitation, Other Forces, and General Relativity
Attempts are still being made to understand the gravitational force. Modern physics is exploring the connections of gravity to other forces, space, and time. General relativity alters our view of gravitation, leading us to think of gravitation as bending space and time.
Applying the Science Practices: All Objects Have Gravitational Fields
We can use the formula developed above, , to calculate the gravitational fields of other objects.
For example, the Moon has a radius of 1.7 × 106 m and a mass of 7.3 × 1022 kg. The gravitational field on the surface of the Moon can be expressed as
This is about 1/6 of the gravity on Earth, which seems reasonable, since the Moon has a much smaller mass than Earth does.
A person has a mass of 50 kg. The gravitational field 1.0 m from the person's center of mass can be expressed as
This is less than one-millionth of the gravitational field at the surface of Earth.
In the following example, we make a comparison similar to one made by Newton himself. He noted that if the gravitational force caused the Moon to orbit Earth, then the acceleration due to gravity should equal the centripetal acceleration of the Moon in its orbit. Newton found that the two accelerations agreed “pretty nearly.”
Example 6.6 Earth's Gravitational Force is the Centripetal Force Making the Moon Move in a Curved Path
(a) Find the acceleration due to Earth's gravity at the distance of the Moon.
(b) Calculate the centripetal acceleration needed to keep the Moon in its orbit, assuming a circular orbit about a fixed Earth, and compare it with the value of the acceleration due to Earth's gravity that you have just found.
Strategy for (a)
This calculation is the same as the one finding the acceleration due to gravity at Earth's surface, except that is the distance from the center of Earth to the center of the Moon. The radius of the Moon's nearly circular orbit is .
Solution for (a)
Substituting known values into the expression for found above, remembering that is the mass of Earth not the Moon, yields
Strategy for (b)
Centripetal acceleration can be calculated using either form of
We choose to use the second form
where is the angular velocity of the Moon about Earth.
Solution for (b)
Given that the period of time it takes to make one complete rotation of the Moon's orbit is 27.3 days, (d) and using
we see that
The centripetal acceleration is
The direction of the acceleration is toward the center of Earth.
Discussion
The centripetal acceleration of the Moon found in (b) differs by less than 1 percent from the acceleration due to Earth's gravity found in (a). This agreement is approximate because the Moon's orbit is slightly elliptical, and Earth is not stationary; The Earth-Moon system rotates about its center of mass, which is located some 1,700 km below Earth's surface. The clear implication is that Earth's gravitational force causes the Moon to orbit Earth.
Why does Earth not remain stationary as the Moon orbits it? This is because, as expected from Newton's third law, if Earth exerts a force on the Moon, then the Moon should exert an equal and opposite force on Earth (see Figure 6.23). We do not sense the Moon's effect on Earth's motion because the Moon's gravity moves our bodies right along with Earth, but there are other signs on Earth that clearly show the effect of the Moon's gravitational force as discussed in Satellites and Kepler's Laws: An Argument for Simplicity.