Having learned to identify forces, we ask the next question: How does an object move when a force is exerted on it? The only way to answer this question is to do experiments. Let’s conduct a “virtual experiment,” one you can easily visualize. Imagine using your fingers to stretch a rubber band to a certain length—say 10 centimeters—that you can measure with a ruler, as shown in FIGURE 5.14. You know that a stretched rubber band exerts a force—a spring force—because your fingers feel the pull. Furthermore, this is a reproducible force; the rubber band exerts the same force every time you stretch it to this length. We’ll call this the standard force F. Not surprisingly, two identical rubber bands exert twice the pull of one rubber band, and $N$ side-by-side rubber bands exert $N$ times the standard force: $F_{\text{net}}=NF$.

FIGURE 5.14 A reproducible force.

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Now attach one rubber band to a 1 kg block and stretch it to the standard length. The object experiences the same force $F$ as did your finger. The rubber band gives us a way of applying a known and reproducible force to an object. Then imagine using the rubber band to pull the block across a horizontal, frictionless table. (We can imagine a frictionless table since this is a virtual experiment, but in practice you could nearly eliminate friction by supporting the object on a cushion of air.)

If you stretch the rubber band and then release the object, the object moves toward your hand. But as it does so, the rubber band gets shorter and the pulling force decreases. To keep the pulling force constant, you must move your hand at just the right speed to keep the length of the rubber band from changing! FIGURE 5.15 shows the experiment being carried out. Once the motion is complete, you can use motion diagrams and kinematics to analyze the object’s motion.

FIGURE 5.15 Measuring the motion of a 1 kg block that is pulled with a constant force.

The first important finding of this experiment is that an object pulled with a constant force moves with a constant acceleration. That is, the answer to the question What does a force do? is: A force causes an object to accelerate, and a constant force produces a constant acceleration.

What happens if you increase the force by using several rubber bands? To find out, use two rubber bands, then three rubber bands, then four, and so on. With $N$ rubber bands, the force on the block is NF. FIGURE 5.16 shows the results of this experiment. You can see that doubling the force causes twice the acceleration, tripling the force causes three times the acceleration, and so on. The graph reveals our second important finding: The acceleration is directly proportional to the force. This result can be written as

where $c$, called the proportionality constant, is the slope of the graph.

FIGURE 5.16 Experiments find that an object’s acceleration is directly proportional to the force acting on it.

MATHEMATICAL ASIDE
Proportionality and proportional reasoning

The concept of proportionality arises frequently in physics. A quantity symbolized by $u$ is proportional to another quantity symbolized by $v$ if

$$u=cv$$

where $c$ (which might have units) is called the proportionality constant. This relationship between $u$ and $v$ is often written

$$u\propto v$$

where the symbol $\propto$ means “is proportional to.”

If $v$ is doubled to $2v$, then $u$ doubles to $c\left(2v\right)=2\left(cv\right)=2u$. In general, if $v$ is changed by any factor $f$, then $u$ changes by the same factor. This is the essence of what we mean by proportionality.

A graph of $u$ versus $v$ is a straight line passing through the origin (i.e., the vertical intercept is zero) with $\text{slope}=c$. Notice that proportionality is a much more specific relationship between $u$ and $v$ than mere linearity. The linear equation $u=cv+b$ has a straight-line graph, but it doesn’t pass through the origin (unless $b$ happens to be zero) and doubling $v$ does not double $u$.

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If $u\propto v$, then $u_1=c{v}_1$ and $u_2=c{v}_2$. Dividing the second equation by the first, we find

$$\frac{u_2}{u_1}=\frac{v_2}{v_1}$$

By working with ratios, we can deduce information about $u$ without needing to know the value of $c$. (This would not be true if the relationship were merely linear.) This is called proportional reasoning.

Proportionality is not limited to being linearly proportional. The graph on the left shows that $u$ is clearly not proportional to $w$. But a graph of $u$ versus $1\text{/}{w}^2$ is a straight line passing through the origin; thus, in this case, $u$ is proportional to $1\text{/}{w}^2$, or $u\propto 1\text{/}{w}^2$. We would say that “$u$ is proportional to the inverse square of $w$.”

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EXAMPLE $u$ is proportional to the inverse square of $w$. By what factor does $u$ change if $w$ is tripled?

SOLUTION This is an opportunity for proportional reasoning; we don’t need to know the proportionality constant. If $u$ is proportional to $1\text{/}{w}^2$, then

$$\frac{u_2}{u_1}=\frac{1\text{/}{w}_2{}^2}{1\text{/}{w}_1{}^2}=\frac{w_1{}^2}{w_2{}^2}={\left(\frac{w_1}{w_2}\right)}^2$$

Tripling $w$, with $w_2\text{/}{w}_1=3$, and thus $w_1\text{/}{w}_2=\frac{1}{3}$, changes $u$ to

$$u_2={\left(\frac{w_1}{w_2}\right)}^2{u}_1={\left(\frac{1}{3}\right)}^2{u}_1=\frac{1}{9}{u}_1$$

Tripling $w$ causes $u$ to become $\frac{1}{9}$ of its original value.

Many Student Workbook and end-of-chapter homework questions will require proportional reasoning. It’s an important skill to learn.

The final question for our virtual experiment is: How does the acceleration depend on the mass of the object being pulled? To find out, apply the same force—for example, the standard force of one rubber band—to a 2 kg block, then a 3 kg block, and so on, and for each measure the acceleration. Doing so gives you the results shown in FIGURE 5.17. An object with twice the mass of the original block has only half the acceleration when both are subjected to the same force.

FIGURE 5.17 An object’s acceleration in response to a force is inversely proportional to its mass.

Mathematically, the graph of FIGURE 5.17 is one of inverse proportionality. That is, the acceleration is inversely proportional to the object’s mass. We can combine these results—that the acceleration is directly proportional to the force applied and inversely proportional to the object’s mass—into the single statement

if we define the basic unit of force as the force that causes a 1 kg mass to accelerate at $1\text{m}/{\text{s}}^2$. That is,

This basic unit of force is called a newton:

One newton is the force that causes a 1 kg mass to accelerate at $1\text{m}\text{/}{\text{s}}^2$. The abbreviation for newton is N. Mathematically, $1\text{N}=1\text{kg m}\text{/}{\text{s}}^2$.

Table 5.1 lists some typical forces. As you can see, “typical” forces on “typical” objects are likely to be in the range 0.01–10,000 N.