16.1 Hooke’s Law: Stress and Strain Revisited

Learning Objectives

By the end of this section, you will be able to:

Explain Newton’s third law of motion with respect to stress and deformation.
Describe the restoration of force and displacement.
Calculate the energy in Hooke’s Law of deformation, and the stored energy in a spring.

In this figure a hand holding a ruler tightly at the bottom is shown. The other hand pulls the top of the ruler and then releases it. Then the ruler starts vibrating, and oscillates around the equilibrium position. A vertical line is shown to mark the equilibrium position. A curved double-headed arrow shows the span of the oscillation. — **Figure 16.2** When displaced from its vertical equilibrium position, this plastic ruler oscillates back and forth because of the restoring force opposing displacement. When the ruler is on the left, there is a force to the right, and vice versa.

Newton’s first law implies that an object oscillating back and forth is experiencing forces. Without force, the object would move in a straight line at a constant speed rather than oscillate. Consider, for example, plucking a plastic ruler to the left as shown in Figure 16.2. The deformation of the ruler creates a force in the opposite direction, known as a restoring force. Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces dampen the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.

The simplest oscillations occur when the restoring force is directly proportional to displacement. When stress and strain were covered in Newton’s Third Law of Motion, the name was given to this relationship between force and displacement was Hooke’s law:

$F = - kx.$

Here, $F$ is the restoring force, $x$ is the displacement from equilibrium or deformation, and $k$ is a constant related to the difficulty in deforming the system. The minus sign indicates the restoring force is in the direction opposite to the displacement.

A series of illustrations of vibrating plastic rulers is shown demonstrating Hooke’s law. — **Figure 16.3** (a) The plastic ruler has been released, and the restoring force is returning the ruler to its equilibrium position. (b) The net force is zero at the equilibrium position, but the ruler has momentum and continues to move to the right. (c) The restoring force is in the opposite direction. It stops the ruler and moves it back toward equilibrium again. (d) Now the ruler has momentum to the left. (e) In the absence of damping (caused by frictional forces), the ruler reaches its original position. From there, the motion will repeat itself.

The force constant $k$ is related to the rigidity (or stiffness) of a system—the larger the force constant, the greater the restoring force, and the stiffer the system. The units of $k$ are newtons per meter (N/m). For example, $k$ is directly related to Young’s modulus when we stretch a string. Figure 16.4 shows a graph of the absolute value of the restoring force versus the displacement for a system that can be described by Hooke’s law—a simple spring in this case. The slope of the graph equals the force constant $k$ in newtons per meter. A common physics laboratory exercise is to measure restoring forces created by springs, determine if they follow Hooke’s law, and calculate their force constants if they do.

The given figure a is the graph of restoring force versus displacement. The displacement is given by x in meters along x axis, with scales from zero to point zero five zero, then to point one zero, then forward. The restoring force is given by F in unit newton along y axis, with scales from zero to two point zero to four point zero to forward. The graph line starts from zero and goes to upward to point where x is greater than point one zero and F is greater than four point zero with intersection dots at equal distances on the slope line. The slope is depicted by K which is given by rise along y-axis upon run along x axis . The values of mass in kilogram, weight in newtons, and displacement in meters are given along with the graph in a tabular format. In the figure b a horizontal weight bar is shown with three weight measuring springs tied to its lower part, hanging in the downward vertical direction. The first bar has no mass hanging through it, showing zero displacement, as x is equal to zero. It is the least stretched spring downward. The second spring has mass m one tied to it which exerts a force w one, on the spring, which causes displacement in the spring shown here to be x one. Similarly, the third spring is most stretched downward with a mass m two hanging through it with force w two and displacement x two. The values of mass in kg, weight in newtons and displacement in meters are given with the graph in a tabular format. — **Figure 16.4** (a) A graph of absolute value of the restoring force versus displacement is displayed. The fact that the graph is a straight line means that the system obeys Hooke’s law. The slope of the graph is the force constant $k$ . (b) The data in the graph were generated by measuring the displacement of a spring from equilibrium while supporting various weights. The restoring force equals the weight supported, if the mass is stationary.

Example 1 How Stiff Are Car Springs?

Figure 16.5 The mass of a car increases due to the introduction of a passenger. This affects the displacement of the car on its suspension system.The mass of a car increases due to the introduction of a passenger. This affects the displacement of the car on its suspension system. (credit: exfordy on Flickr)

What is the force constant for the suspension system of a car that settles 1.20 cm when an 80.0-kg person gets in?

Strategy

Consider the car to be in its equilibrium position $x equals 0$ before the person gets in. The car then settles down 1.20 cm, which means it is displaced to a position $x = - 1 . 20 \times 10^{- 2} m$ . At that point, the springs supply a restoring force $F$ equal to the person’s weight $w = mg = (80.0 kg) (9.80 {m/s}^{2}) = 784 N$ . We take this force to be $F$ in Hooke’s law. Knowing $F$ and $x$ , we can then solve the force constant $k$ .

Solution

Solve Hooke’s law, $F = - kx$ , for $k$ :
$k = - \frac{F}{x} .$
Substitute known values and solve $k$ :
$\begin{array}{l} k & = & - \frac{784 N}{- 1 . 20 \times 10^{- 2} m} \\ = & 6 . 53 \times 10^{4} N/m . \end{array}$

Discussion

Note that $F$ and $x$ have opposite signs because they are in opposite directions—the restoring force is up, and the displacement is down. Also, note that the car would oscillate up and down when the person got in if it were not for damping (due to frictional forces) provided by shock absorbers. Bouncing cars are a sure sign of bad shock absorbers.

Energy in Hooke’s Law of Deformation

In order to produce a deformation, work must be done. That is, a force must be exerted through a distance, whether you pluck a guitar string or compress a car spring. If the only result is deformation, and no work goes into thermal, sound, or kinetic energy, then all the work is initially stored in the deformed object as some form of potential energy. The potential energy stored in a spring is ${PE}_{s} = \frac{1}{2} {kx}^{2}$ . Here, we generalize the idea to elastic potential energy for a deformation of any system that can be described by Hooke’s law. Hence,

${PE}_{el} = \frac{1}{2} {kx}^{2},$

where ${PE}_{el}$ is the elastic potential energy stored in any deformed system that obeys Hooke’s law and has a displacement $x$ from equilibrium and a force constant $k$ .

It is possible to find the work done in deforming a system in order to find the energy stored. This work is performed by an applied force $F_{app}$ . The applied force is exactly opposite to the restoring force (action-reaction), and so $F_{app} = kx$ . Figure 16.6 shows a graph of the applied force versus deformation $x$ for a system that can be described by Hooke’s law. Work done on the system is force multiplied by distance, which equals the area under the curve or $open parenthesis 1 / 2 close parenthesis kx squared$ (Method A in the figure). Another way to determine the work is to note that the force increases linearly from 0 to $kx$ , so that the average force is $open parenthesis 1 / 2 close parenthesis kx$ , the distance moved is $x$ , and thus $W = F_{app} d = [(1 / 2) kx] (x) = (1 / 2) {kx}^{2}$ (Method B in the figure).

The graph here represents applied force, given along y-axis, versus deformation or displacement, given along x axis. The slope is linear slanting and the slope area is covered between x axis and the slope, given by F is equal to k multiplied by x, where k is constant and x is displacement. The force applied along y-axis is given by half of k multiplied by x. Along with the graph, two methods are provided to calculate weight, W. The first method gives the solution by multiplying half of b multiplied by h, whereas in the second we can get the solution by multiplying f with x. — **Figure 16.6** A graph of applied force versus distance for the deformation of a system that can be described by Hooke’s law is displayed. The work done on the system equals the area under the graph or the area of the triangle, which is half its base multiplied by its height, or $W equals open parenthesis 1 / 2 close parenthesis kx squared$ .

Example 2 Calculating Stored Energy: A Toy Gun Spring

We can use a toy gun’s spring mechanism to ask and answer two simple questions: (a) How much energy is stored in the spring of a toy gun that has a force constant of 50.0 N/m and is compressed 0.150 m? (b) If you neglect friction and the mass of the spring, at what speed will a 2.00-g projectile be ejected from the gun?

Figure 16.7 (a) In this image of the gun, the spring is uncompressed before being cocked. (b) The spring has been compressed a distance

x

, and the projectile is in place. (c) When released, the spring converts elastic potential energy

{PE}_{el}

into kinetic energy.

Strategy for a

(a): The energy stored in the spring can be found directly from elastic potential energy equation, because $k$ and $x$ are given.

Solution for a

Entering the given values for $k$ and $x$ yields

$\begin{array}{l} {PE}_{el} & = & \frac{1}{2} {kx}^{2} = \frac{1}{2} (50.0 N/m) {(0.150 m)}^{2} = 0 . 563 N \cdot m \\ = & 0.563 J \end{array}$

Strategy for b

Because there is no friction, the potential energy is converted entirely into kinetic energy. The expression for kinetic energy can be solved for the projectile’s speed.

Solution for b

Identify known quantities:
${KE}_{f} = {PE}_{el} or 1 / 2 {mv}^{2} = (1 / 2) {kx}^{2} = {PE}_{el} = 0 . 563 J$
Solve for $v$ :
$v = {[\frac{2 {PE}_{el}}{m}]}^{1 / 2} = {[\frac{2 (0.563 J)}{0.002 kg}]}^{1 / 2} = 23.7 {(J/kg)}^{1 / 2}$
Convert units: $23.7 m / s$

Discussion

(a) and (b): This projectile speed is impressive for a toy gun (more than 80 km/h). The numbers in this problem seem reasonable. The force needed to compress the spring is small enough for an adult to manage, and the energy imparted to the dart is small enough to limit the damage it might do, especially because the darts in many of these guns are made of soft material with a rubber tip. Yet, the speed of the dart is great enough for it to travel an acceptable distance.

Envision holding the end of a ruler with one hand and deforming it with the other. When you let go, you can see the oscillations of the ruler. In what way could you modify this simple experiment to increase the rigidity of the system?

You could hold the ruler at its midpoint so that the part of the ruler that oscillates is half as long as in the original experiment.

If you apply a deforming force on an object and let it come to equilibrium, what happened to the work you did on the system?

It was stored in the object as potential energy.

Test Prep for AP Courses

Which of the following represents the distance (how much ground the particle covers) moved by a particle in a simple harmonic motion in one time period? (Here, A represents the amplitude of the oscillation.)

0 cm
A cm
2A cm
4A cm

(d)

A spring has a spring constant of 80 N∙m⁻¹. What is the force required to (a) compress the spring by 5 cm and (b) expand the spring by 15 cm?

In the formula $F = - k x$ , what does the minus sign indicate?

It indicates that the restoring force is in the direction of the displacement.
It indicates that the restoring force is in the direction opposite the displacement.
It indicates that mechanical energy in the system decreases when a system undergoes oscillation.
None of the above

(b)

The splashing of a liquid resembles an oscillation. The restoring force in this scenario will be due to which of the following?

Potential energy
Kinetic energy
Gravity
Mechanical energy

Section Summary

An oscillation is a back and forth motion of an object between two points of deformation.
An oscillation may create a wave, which is a disturbance that propagates from where it was created.
The simplest type of oscillations and waves are related to systems that can be described by Hooke’s law:
$F = - kx,$
where $F$ is the restoring force, $x$ is the displacement from equilibrium or deformation, and $k$ is the force constant of the system.
Elastic potential energy ${PE}_{el}$ stored in the deformation of a system that can be described by Hooke’s law is given by
$PE sub el equals open parenthesis 1 / 2 close parenthesis kx squared.$

Conceptual Questions

Describe a system in which elastic potential energy is stored.

Problems & Exercises

Fish are hung on a spring scale to determine their mass (most fishermen feel no obligation to truthfully report the mass).

(a) What is the force constant of the spring in such a scale if it the spring stretches 8.00 cm for a 10.0 kg load?

(b) What is the mass of a fish that stretches the spring 5.50 cm?

(a) $1. 23 times 10 cubed N/m$

(b) $6. 88 kg$

It is weigh-in time for the local under-85-kg rugby team. The bathroom scale used to assess eligibility can be described by Hooke’s law and is depressed 0.75 cm by its maximum load of 120 kg. (a) What is the spring’s effective spring constant? (b) A player stands on the scales and depresses it by 0.48 cm. Is he eligible to play on this under-85 kg team?

One type of BB gun uses a spring-driven plunger to blow the BB from its barrel. (a) Calculate the force constant of its plunger’s spring if you must compress it 0.150 m to drive the 0.0500-kg plunger to a top speed of 20.0 m/s. (b) What force must be exerted to compress the spring?

(a) 889 N/m

(b) 133 N

(a) The springs of a pickup truck act like a single spring with a force constant of $1 . 30 \times 10^{5} N/m$ . By how much will the truck be depressed by its maximum load of 1000 kg?

(b) If the pickup truck has four identical springs, what is the force constant of each?

When an 80.0-kg man stands on a pogo stick, the spring is compressed 0.120 m.

(a) What is the force constant of the spring? (b) Will the spring be compressed more when he hops down the road?

(a) $6. 53 times 10 cubed N/m$

(b) Yes

A spring has a length of 0.200 m when a 0.300-kg mass hangs from it, and a length of 0.750 m when a 1.95-kg mass hangs from it. (a) What is the force constant of the spring? (b) What is the unloaded length of the spring?

Adapted from College Physics 2e by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.