If you stretch a rubber band, a force tries to pull the rubber band back to its equilibrium, or unstretched, length. A force that restores a system to an equilibrium position is called a restoring force. Objects that exert restoring forces are called elastic. The most basic examples of elasticity are things like springs and rubber bands, but other examples of elasticity and restoring forces abound. For example, the steel beams flex slightly as you drive your car over a bridge, but they are restored to equilibrium after your car passes by. Nearly everything that stretches, compresses, flexes, bends, or twists exhibits a restoring force and can be called elastic.

We didn’t introduce restoring forces in Part I of this textbook because we didn’t have the mathematical tools to deal with them. But now—using work and energy—we do. We’re going to use a simple spring as our model of elasticity. Suppose you have a spring whose equilibrium length is $L_0$. This is the length of the spring when it is neither pushing nor pulling. If you stretch (or compress) the spring, how hard does it pull (or push) back? Measurements show that

• The force is opposite the displacement. This is what we mean by a restoring force.

• If you don’t stretch or compress the spring too much, the force is proportional to the displacement from equilibrium. The farther you push or pull, the larger the force.

FIGURE 9.17 shows a spring along a generic s-axis exerting force $\vec{F}{}_{\text{Sp}}$. Notice that $s_{\text{eq}}$ is the position, or coordinate, of the free end of the spring, not the spring’s equilibrium length $L_0$. When the spring is stretched, the spring displacement $\text{Δ}s=s-s_{\text{eq}}$ is positive while $(F_{\text{Sp}}){}_s$, the s-component of the restoring force, is negative. Similarly, compressing the spring makes $\text{Δ}s<0$ and $(F_{\text{Sp}}){}_s>0$. The graph of force versus displacement is a straight line with negative slope, showing that the spring force is proportional to but opposite the displacement.

FIGURE 9.17 Properties of a spring.

The equation of the straight-line graph passing through the origin is

The minus sign is the mathematical indication of a restoring force, and the constant k—the absolute value of the slope of the line—is called the spring constant of the spring. The units of the spring constant are N/m. This relationship between the force and displacement of a spring was discovered by Robert Hooke, a contemporary (and sometimes bitter rival) of Newton. Hooke’s law is not a true “law of nature,” in the sense that Newton’s laws are, but is actually just a model of a restoring force. It works well for small displacements from equilibrium, but Hooke’s law will fail for any real spring that is compressed or stretched too far. A hypothetical massless spring for which Hooke’s law is true at all displacements is called an ideal spring.

The spring constant k is a property that characterizes a spring, just as mass m characterizes a particle. For a given spring, k is a constant—it does not change as the spring is stretched or compressed. If k is large, it takes a large pull to cause a significant stretch, and we call the spring a “stiff” spring. A spring with small k can be stretched with very little force, and we call it a “soft” spring.

EXAMPLE 9.8 Pull until it slips

FIGURE 9.18 shows a spring attached to a 2.0 kg block. The other end of the spring is pulled by a motorized toy train that moves forward at 5.0 cm/s. The spring constant is 50 N/m, and the coefficient of static friction between the block and the surface is 0.60. The spring is at its equilibrium length at $t=0\text{s}$ when the train starts to move. When does the block slip?

FIGURE 9.18 A toy train stretches the spring until the block slips.

MODEL Model the block as a particle and the spring as an ideal spring obeying Hooke’s law.

VISUALIZE FIGURE 9.19 is a free-body diagram for the block.

FIGURE 9.19 The free-body diagram.

FIGURE 9.19 Full Alternative Text

SOLVE Recall that the tension in a massless string pulls equally at both ends of the string. The same is true for the spring force: It pulls (or pushes) equally at both ends. This is the key to solving the problem. As the right end of the spring moves, stretching the spring, the spring pulls backward on the train and forward on the block with equal strength. As the spring stretches, the static friction force on the block increases in magnitude to keep the block at rest. The block is in static equilibrium, so

$$(F_{\text{net}}){}_x=(F_{\text{Sp}}){}_x+(f_{\text{s}}){}_x=F_{\text{Sp}}-f_{\text{s}}=0$$

where $F_{\text{Sp}}$ is the magnitude of the spring force. The magnitude is $F_{\text{Sp}}=k\text{Δ}x$, where $\text{Δ}x=v_{x}t$ is the distance the train has moved. Thus

$$f_{\text{s}}=F_{\text{Sp}}=k\text{Δ}x$$

The block slips when the static friction force reaches its maximum value $f_{\text{s max}}={\mu }_{s}n={\mu }_{\text{s}}mg$. This occurs when the train has moved

$$\begin{array}{ll}\text{Δ}x\hfill & ={\displaystyle \frac{f_{\text{s}\text{max}}}{k}}={\displaystyle \frac{{\mu }_{\text{s}}mg}{k}}={\displaystyle \frac{\left(0.60\right)\left(2.0\text{kg}\right)(9.80{\text{m/s}}^{\text{2}})}{50\text{N/m}}}\hfill \\ \hfill & =0.235\text{m}=\text{23}\text{.5}\text{cm}\hfill \end{array}$$

The time at which the block slips is

$$t={\displaystyle \frac{\text{Δ}x}{v_x}}={\displaystyle \frac{23.5\text{cm}}{5.0\text{cm}/\text{s}}}=4.7\text{s}$$

This example illustrates a class of motion called stick-slip motion. Once the block slips, it will shoot forward some distance, then stop and stick again. As the train continues, there will be a recurring sequence of stick, slip, stick, slip, stick. . . .

Earthquakes are an important example of stick-slip motion. The large tectonic plates making up the earth’s crust are attempting to slide past each other, but friction causes the edges of the plates to stick together. You may think of rocks as rigid and brittle, but large masses of rock are somewhat elastic and can be “stretched.” Eventually the elastic force of the deformed rocks exceeds the friction force between the plates. An earthquake occurs as the plates slip and lurch forward. Once the tension is released, the plates stick together again and the process starts all over.