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Weight and the Gravitational Force

Weight and the Gravitational Force

When an object is dropped, it accelerates toward the center of Earth. Newton’s second law states that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight \(\mathbf{\text{w}}\).

Weight can be denoted as a vector \(\mathbf{\text{w}}\) because it has a direction; down is, by definition, the direction of gravity, and hence weight is a downward force. The magnitude of weight is denoted as \(w\).

Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration \(g\). Using Galileo’s result and Newton’s second law, we can derive an equation for weight.

Consider an object with mass \(m\) falling downward toward Earth. It experiences only the downward force of gravity, which has magnitude \(w\). Newton’s second law states that the magnitude of the net external force on an object is \({F}_{\text{net}}=\text{ma}\).

Since the object experiences only the downward force of gravity, \({F}_{\text{net}}=w\). We know that the acceleration of an object due to gravity is \(g\), or \(a=g\). Substituting these into Newton’s second law gives

Weight

This is the equation for weight—the gravitational force on a mass \(m\):

\(w=\text{mg}.\)

Since \(g=9.80\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\) on Earth, the weight of a 1.0 kg object on Earth is 9.8 N, as we see:

\(w=\text{mg}=\left(1\text{.}\text{0 kg}\right)\left(9.80\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)=9.8\phantom{\rule{0.25em}{0ex}}\text{N}.\)

Recall that \(g\) can take a positive or negative value, depending on the positive direction in the coordinate system. Be sure to take this into consideration when solving problems with weight.

When the net external force on an object is its weight, we say that it is in free-fall. That is, the only force acting on the object is the force of gravity. In the real world, when objects fall downward toward Earth, they are never truly in free-fall because there is always some upward force from the air acting on the object.

The acceleration due to gravity \(g\) varies slightly over the surface of Earth, so that the weight of an object depends on location and is not an intrinsic property of the object. Weight varies dramatically if one leaves Earth’s surface. On the Moon, for example, the acceleration due to gravity is only \(1.67\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\). A 1.0-kg mass thus has a weight of 9.8 N on Earth and only about 1.7 N on the Moon.

The broadest definition of weight in this sense is that the weight of an object is the gravitational force on it from the nearest large body, such as Earth, the Moon, the Sun, and so on. This is the most common and useful definition of weight in physics.

It differs dramatically, however, from the definition of weight used by NASA and the popular media in relation to space travel and exploration. When they speak of “weightlessness” and “microgravity,” they are really referring to the phenomenon we call “free-fall” in physics. We shall use the above definition of weight, and we will make careful distinctions between free-fall and actual weightlessness.

It is important to be aware that weight and mass are very different physical quantities, although they are closely related. Mass is the quantity of matter (how much “stuff”) and does not vary in classical physics, whereas weight is the gravitational force and does vary depending on gravity.

It is tempting to equate the two, since most of our examples take place on Earth, where the weight of an object only varies a little with the location of the object. Furthermore, the terms mass and weight are used interchangeably in everyday language; for example, our medical records often show our “weight” in kilograms, but never in the correct units of newtons.

Common Misconceptions: Mass vs. Weight

Mass and weight are often used interchangeably in everyday language. However, in science, these terms are distinctly different from one another. Mass is a measure of how much matter is in an object. The typical measure of mass is the kilogram (or the “slug” in English units).

Weight, on the other hand, is a measure of the force of gravity acting on an object. Weight is equal to the mass of an object (\(m\)) multiplied by the acceleration due to gravity (\(g\)). Like any other force, weight is measured in terms of newtons (or pounds in English units).

Assuming the mass of an object is kept intact, it will remain the same, regardless of its location. However, because weight depends on the acceleration due to gravity, the weight of an object can change when the object enters into a region with stronger or weaker gravity.

For example, the acceleration due to gravity on the Moon is \(1.67\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\) (which is much less than the acceleration due to gravity on Earth, \(9.80\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\)).

If you measured your weight on Earth and then measured your weight on the Moon, you would find that you “weigh” much less, even though you do not look any skinnier. This is because the force of gravity is weaker on the Moon. In fact, when people say that they are “losing weight,” they really mean that they are losing “mass” (which in turn causes them to weigh less).

Optional Take-Home Experiment: Mass and Weight

What do bathroom scales measure? When you stand on a bathroom scale, what happens to the scale? It depresses slightly. The scale contains springs that compress in proportion to your weight—similar to rubber bands expanding when pulled. The springs provide a measure of your weight (for an object which is not accelerating).

This is a force in newtons (or pounds). In most countries, the measurement is divided by 9.80 to give a reading in mass units of kilograms. The scale measures weight but is calibrated to provide information about mass. While standing on a bathroom scale, push down on a table next to you. What happens to the reading? Why? Would your scale measure the same “mass” on Earth as on the Moon?

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