162. Hooke’s Law: Stress and Strain RevisitedLearning Objectives
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 1. The deformation of the ruler creates a force in the opposite direction, known as a 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: $$F=\text{kx.}$$Here, $F$ is the restoring force, $x$ is the displacement from equilibrium or The Example 1: How Stiff Are Car Springs?What is the force constant for the suspension system of a car that settles 1.20 cm when an 80.0kg person gets in? Strategy Consider the car to be in its equilibrium position $x=0$ before the person gets in. The car then settles down 1.20 cm, which means it is displaced to a position $x=1\text{.}\text{20}\times {\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\text{m}$. At that point, the springs supply a restoring force $F$ equal to the person’s weight $w=\text{mg}=\left(\text{80}\text{.}0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{kg}\right)\left(9\text{.}\text{80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)=\text{784}\phantom{\rule{0.25em}{0ex}}\text{N}$. We take this force to be $F$ in Hooke’s law. Knowing $F$ and $x$, we can then solve the force constant $k$. Solution
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 DeformationIn 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 ${\text{PE}}_{\text{el}}=\frac{1}{2}{\mathrm{kx}}^{2}$. Here, we generalize the idea to elastic potential energy for a deformation of any system that can be described by Hooke’s law. Hence, $${\text{PE}}_{\text{el}}=\frac{1}{2}{\mathrm{kx}}^{2},$$where ${\text{PE}}_{\text{el}}$ is the 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 ${F}_{\text{app}}$. The applied force is exactly opposite to the restoring force (actionreaction), and so ${F}_{\text{app}}=\text{kx}$. Figure 5 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 $(1/2){\mathrm{kx}}^{2}$(Method A in the figure). Another way to determine the work is to note that the force increases linearly from 0 to $\mathrm{kx}$, so that the average force is $(1/2)\mathrm{kx}$, the distance moved is $x$, and thus $W={F}_{\text{app}}d=[(1/2)\text{kx}](x)=(1/2){\mathrm{kx}}^{2}$ (Method B in the figure). Example 2: Calculating Stored Energy: A Tranquilizer Gun SpringWe 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 tranquilizer 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 2.00g projectile be ejected from the gun? 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 $$\begin{array}{lll}{\text{PE}}_{\text{el}}& =& \frac{1}{2}{\mathrm{kx}}^{2}=\frac{1}{2}\left(\text{50}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{N/m}\right){\left(0\text{.}\text{150}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}=0\text{.}\text{563}\phantom{\rule{0.25em}{0ex}}\text{N}\cdot \text{m}\\ & =& 0\text{.}\text{563}\phantom{\rule{0.25em}{0ex}}\text{J}\end{array}$$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
Discussion (a) and (b): This projectile speed is impressive for a tranquilizer 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. Yet, the speed of the dart is great enough for it to travel an acceptable distance. Check your UnderstandingEnvision 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? Show/Hide Solution AnswerYou 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. Check your UnderstandingIf you apply a deforming force on an object and let it come to equilibrium, what happened to the work you did on the system? Show/Hide Solution AnswerIt was stored in the object as potential energy. Section Summary
Conceptual QuestionsExercise 1Describe a system in which elastic potential energy is stored. Problems & ExercisesExercise 1Fish 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? (c) How far apart are the halfkilogram marks on the scale? Show/Hide Solution Solution(a) $1\text{.}\text{23}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{N/m}$ (b) $6\text{.}\text{88 kg}$ (c) $4\text{.00 mm}$ Exercise 2It is weighin time for the local under85kg 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 under85 kg team? Exercise 3One type of BB gun uses a springdriven 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 0.0500kg plunger to a top speed of 20.0 m/s. (b) What force must be exerted to compress the spring? Show/Hide Solution Solution(a) 889 N/m (b) 133 N Exercise 4(a) The springs of a pickup truck act like a single spring with a force constant of $1\text{.}\text{30}\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/m}$. 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? Exercise 5When an 80.0kg 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? Show/Hide Solution Solution(a) $6\text{.}\text{53}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{N/m}$ (b) Yes Exercise 6A spring has a length of 0.200 m when a 0.300kg mass hangs from it, and a length of 0.750 m when a 1.95kg mass hangs from it. (a) What is the force constant of the spring? (b) What is the unloaded length of the spring?
