# 5.4 Mass and weight

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By the end of the section, you will be able to:
• Explain the difference between mass and weight
• Explain why falling objects on Earth are never truly in free fall
• Describe the concept of weightlessness

Mass and weight are often used interchangeably in everyday conversation. For example, our medical records often show our weight in kilograms but never in the correct units of newtons. In physics, however, there is an important distinction. Weight is the pull of Earth on an object. It depends on the distance from the center of Earth. Unlike weight, mass does not vary with location. The mass of an object is the same on Earth, in orbit, or on the surface of the Moon.

## Units of force

The equation ${F}_{\text{net}}=ma$ is used to define net force in terms of mass, length, and time. As explained earlier, the SI unit of force is the newton. Since ${F}_{\text{net}}=ma,$

$1\phantom{\rule{0.2em}{0ex}}\text{N}=1\phantom{\rule{0.2em}{0ex}}\text{kg}·{\text{m/s}}^{2}.$

Although almost the entire world uses the newton for the unit of force, in the United States, the most familiar unit of force is the pound (lb), where 1 N = 0.225 lb. Thus, a 225-lb person weighs 1000 N.

## Weight and gravitational force

When an object is dropped, it accelerates toward the center of Earth. Newton’s second law says 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     $\stackrel{\to }{w}$ , or its force due to gravity acting on an object of mass m . Weight can be denoted as a vector 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 toward Earth. It experiences only the downward force of gravity, which is the weight $\stackrel{\to }{w}$ . Newton’s second law says that the magnitude of the net external force on an object is ${\stackrel{\to }{F}}_{\text{net}}=m\stackrel{\to }{a}.$ We know that the acceleration of an object due to gravity is $\stackrel{\to }{g},$ or $\stackrel{\to }{a}=\stackrel{\to }{g}$ . Substituting these into Newton’s second law gives us the following equations.

## Weight

The gravitational force on a mass is its weight. We can write this in vector form, where $\stackrel{\to }{w}$ is weight and m is mass, as

$\stackrel{\to }{w}=m\stackrel{\to }{g}.$

In scalar form, we can write

$w=mg.$

Since $g=9.80\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$ on Earth, the weight of a 1.00-kg object on Earth is 9.80 N:

$w=mg=\left(1.00\phantom{\rule{0.2em}{0ex}}\text{kg}\right)\left({9.80\phantom{\rule{0.2em}{0ex}}\text{m/s}}^{2}\right)=9.80\phantom{\rule{0.2em}{0ex}}\text{N}.$

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 gravity. However, when objects on Earth fall downward, they are never truly in free fall because there is always some upward resistance force from the air acting on the object.

Acceleration due to gravity g varies slightly over the surface of Earth, so the weight of an object depends on its location and is not an intrinsic property of the object. Weight varies dramatically if we leave Earth’s surface. On the Moon, for example, acceleration due to gravity is only ${1.67\phantom{\rule{0.2em}{0ex}}\text{m/s}}^{2}$ . A 1.0-kg mass thus has a weight of 9.8 N on Earth and only about 1.7 N on the Moon.

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