The Gravitational Force by David Halliday and Robert Resnick

A gravitational force Fg on a body is a certain type of pull that is directed toward a second body. In this article we do not discuss the nature of this force and usually consider situations in which the second body is Earth. Thus, when we speak of the gravitational force Fg on a body, we usually mean a force that pulls on it directly toward the center of Earth - that is, directly down toward the ground.We shall assume that the ground is an inertial frame.

Free Fall. Suppose a body of mass m is in free fall with the free-fall acceleration of magnitude g. Then, if we neglect the effects of the air, the only force acting on the body is the gravitational force Fg. We can relate this downward force and downward acceleration with Newton's second law . We place a vertical y axis along the body’s path, with the positive direction upward. For this axis, Newton's second law can be written in the form Fnet,y = may, which, in our situation, becomes

-Fg = m(-g)

or Fg = mg. (5-8)

At Rest. This same gravitational force, with the same magnitude, still acts on the body even when the body is not in free fall but is, say, at rest on a pool table or moving across the table. (For the gravitational force to disappear, Earth would have to disappear.)

We can write Newton's second law for the gravitational force in these vector forms:

Fg = -Fg j^ = -mg j^ = mg (5-9)

where j^ is the unit vector that points upward along a y axis, directly away from the ground, and g is the free-fall acceleration (written as a vector), directed downward.


The weight W of a body is the magnitude of the net force required to prevent the body from falling freely, as measured by someone on the ground. For example, to keep a ball at rest in your hand while you stand on the ground, you must provide an upward force to balance the gravitational force on the ball from Earth. Suppose the magnitude of the gravitational force is 2.0 N. Then the magnitude of your upward force must be 2.0 N, and thus the weight W of the ball is 2.0 N. We also say that the ball weighs 2.0 N and speak about the ball weighing 2.0 N.

A ball with a weight of 3.0 N would require a greater force from you - namely, a 3.0 N force - to keep it at rest. The reason is that the gravitational force you must balance has a greater maggnitude - namely, 3.0 N. We say that this second ball is heavier than the first ball.

Now let us generalize the situation. Consider a body that has an acceleration a of zero relative to the ground, which we again assume to be an inertial frame. Two forces act on the body: a downward gravitational force Fnet and a balancing upward force of magnitude W. We can write Newton's second law for a vertical y axis, with the positive direction upward, as

Fnet,y = may.

In our situation, this becomes

W - Fg = m(0) (5-10)

or W = Fg (weight, with ground as inertial frame) (5-11)

This equation tells us (assuming the ground is an inertial frame) that

The weight W of a body is equal to the magnitude Fg of the gravitational force on the body.

Substituting mg for Fg from Eq. 5-8, we find

W = mg (weight), (5-12)

which relates a body’s weight to its mass.

Figure 5-5 An equal-arm balance. When the device is in balance, the gravitational force FgL on the body being weighed (on the left pan) and the total gravitational force FgR on the reference bodies (on the right pan) are equal. Thus, the mass mLR of the reference bodies.

Weighing. To weigh a body means to measure its weight. One way to do this is to place the body on one of the pans of an equal-arm balance (Fig. 5-5) and then place reference bodies (whose masses are known) on the other pan until we strike a balance (so that the gravitational forces on the two sides match). The masses on the pans then match, and we know the mass of the body. If we know the value of g for the location of the balance, we can also find the weight of the body with Eq. 5-12.

Figure 5-6 A spring scale. The reading is proportional to the weight of the object on the pan, and the scale gives that weight if marked in weight units. If, instead, it is marked in mass units, the reading is the object's weight only if the value of g at the location where the scale is being used is the same as the value of g at the location where the scale was calibrated.

We can also weigh a body with a spring scale (Fig. 5-6). The body stretches a spring, moving a pointer along a scale that has been calibrated and marked in either mass or weight units. (Most bathroom scales in the United States work this way and are marked in the force unit pounds.) If the scale is marked in mass units, it is accurate only where the value of g is the same as where the scale was calibrated.

The weight of a body must be measured when the body is not accelerating vertically relative to the ground. For example, you can measure your weight on a scale in your bathroom or on a fast train. However, if you repeat the measurement with the scale in an accelerating elevator, the reading differs from your weight because of the acceleration. Such a measurement is called an apparent weight.

Caution: A body's weight is not its mass. Weight is the magnitude of a force and is related to mass by Eq. 5-12. If you move a body to a point where the value of g is different, the body's mass (an intrinsic property) is not different but the weight is. For example, the weight of a bowling ball having a mass of 7.2 kg is 71 N on Earth but only 12 N on the Moon. The mass is the same on Earth and Moon, but the free-fall acceleration on the Moon is only 1.6 m/s2.

About the Authors

David Halliday was an American physicist known for his physics textbooks, Physics and Fundamentals of Physics, which he wrote with Robert Resnick. Both textbooks have been in continuous use since 1960 and are available in more than 47 languages.

Robert Resnick was a physics educator and author of physics textbooks. He was born in Baltimore, Maryland on January 11, 1923 and graduated from the Baltimore City College high school in 1939. He received his B.A. in 1943 and his Ph.D. in 1949, both in physics from Johns Hopkins University.

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