# 4.3 Newton’s second law of motion: concept of a system  (Page 3/14)

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## Units of force

${\mathbf{\text{F}}}_{\text{net}}=m\mathbf{\text{a}}$ is used to define the units of force in terms of the three basic units for mass, length, and time. The SI unit of force is called the newton (abbreviated N) and is the force needed to accelerate a 1-kg system at the rate of $1{\text{m/s}}^{2}$ . That is, since ${\mathbf{\text{F}}}_{\text{net}}=m\mathbf{\text{a}}$ ,

$\text{1 N}=\text{1 kg}\cdot {\text{m/s}}^{2}.$

While 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.

## 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.

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