75. Conservative Forces and Potential EnergyLearning Objectives
Potential Energy and Conservative ForcesWork is done by a force, and some forces, such as weight, have special characteristics. A Potential Energy and Conservative Forces:Potential energy is the energy a system has due to position, shape, or configuration. It is stored energy that is completely recoverable. A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy $(\text{PE})$ for any conservative force. The work done against a conservative force to reach a final configuration depends on the configuration, not the path followed, and is the potential energy added. Potential Energy of a SpringFirst, let us obtain an expression for the potential energy stored in a spring (${\text{PE}}_{\mathrm{s}}$). We calculate the work done to stretch or compress a spring that obeys Hooke’s law. (Hooke’s law was examined in Elasticity: Stress and Strain, and states that the magnitude of force $F$ on the spring and the resulting deformation $\mathrm{\Delta}L$ are proportional, $F=k\mathrm{\Delta}L$.) (See Figure 1.) For our spring, we will replace $\mathrm{\Delta}L$ (the amount of deformation produced by a force $F$) by the distance $x$ that the spring is stretched or compressed along its length. So the force needed to stretch the spring has magnitude $\text{F = kx}$, where $k$ is the spring’s force constant. The force increases linearly from 0 at the start to $\text{kx}$ in the fully stretched position. The average force is $\text{kx}/2$. Thus the work done in stretching or compressing the spring is ${W}_{\mathrm{s}}=\text{Fd}=\left(\frac{\text{kx}}{2}\right)x=\frac{1}{2}{\text{kx}}^{2}$. Alternatively, we noted in Kinetic Energy and the WorkEnergy Theorem that the area under a graph of $F$ vs. $x$ is the work done by the force. In Figure 1(c) we see that this area is also $\frac{1}{2}{\text{kx}}^{2}$. We therefore define the where $k$ is the spring’s force constant and $x$ is the displacement from its undeformed position. The potential energy represents the work done on the spring and the energy stored in it as a result of stretching or compressing it a distance $x$. The potential energy of the spring ${\text{PE}}_{\mathrm{s}}$ does not depend on the path taken; it depends only on the stretch or squeeze $x$ in the final configuration. The equation ${\text{PE}}_{\mathrm{s}}=\frac{1}{2}{\text{kx}}^{2}$ has general validity beyond the special case for which it was derived. Potential energy can be stored in any elastic medium by deforming it. Indeed, the general definition of Conservation of Mechanical EnergyLet us now consider what form the workenergy theorem takes when only conservative forces are involved. This will lead us to the conservation of energy principle. The workenergy theorem states that the net work done by all forces acting on a system equals its change in kinetic energy. In equation form, this is $${W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}\frac{1}{2}{{\text{mv}}_{0}}^{2}=\mathrm{\Delta}\text{KE.}$$If only conservative forces act, then $${W}_{\text{net}}={W}_{\text{c}}\text{,}$$where ${W}_{\mathrm{c}}$ is the total work done by all conservative forces. Thus, $${W}_{\text{c}}=\text{\Delta}\text{KE.}$$Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, ${W}_{\text{c}}=\text{\Delta}\text{PE}$. Therefore, $$\text{\Delta}\text{PE}=\text{\Delta}\text{KE}$$or $$\text{\Delta}\text{KE}+\text{\Delta}\text{PE}=0.$$This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is, $$\begin{array}{cc}& \text{KE}+\text{PE}=\text{constant}\\ \text{or}& \\ & {\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}}={\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}\end{array}\}\text{(conservativeforcesonly),}$$where i and f denote initial and final values. This equation is a form of the workenergy theorem for conservative forces; it is known as the Example 1: Using Conservation of Mechanical Energy to Calculate the Speed of a Toy CarA 0.100kg toy car is propelled by a compressed spring, as shown in Figure 3. The car follows a track that rises 0.180 m above the starting point. The spring is compressed 4.00 cm and has a force constant of 250.0 N/m. Assuming work done by friction to be negligible, find (a) how fast the car is going before it starts up the slope and (b) how fast it is going at the top of the slope. Strategy The spring force and the gravitational force are conservative forces, so conservation of mechanical energy can be used. Thus, $${\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}}={\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}$$or $$\frac{1}{2}{{\text{mv}}_{\mathrm{i}}}^{2}+{\text{mgh}}_{\mathrm{i}}+\frac{1}{2}{{\text{kx}}_{\mathrm{i}}}^{2}=\frac{1}{2}{{\text{mv}}_{\mathrm{f}}}^{2}+{\text{mgh}}_{\mathrm{f}}+\frac{1}{2}{{\text{kx}}_{\mathrm{f}}}^{2},$$where $h$ is the height (vertical position) and $x$ is the compression of the spring. This general statement looks complex but becomes much simpler when we start considering specific situations. First, we must identify the initial and final conditions in a problem; then, we enter them into the last equation to solve for an unknown. Solution for (a) This part of the problem is limited to conditions just before the car is released and just after it leaves the spring. Take the initial height to be zero, so that both ${h}_{\text{i}}$ and ${h}_{\text{f}}$ are zero. Furthermore, the initial speed ${v}_{\text{i}}$ is zero and the final compression of the spring ${x}_{\text{f}}$ is zero, and so several terms in the conservation of mechanical energy equation are zero and it simplifies to $$\frac{1}{2}{{\text{kx}}_{\mathrm{i}}}^{2}=\frac{1}{2}{{\text{mv}}_{\mathrm{f}}}^{2}\text{.}$$In other words, the initial potential energy in the spring is converted completely to kinetic energy in the absence of friction. Solving for the final speed and entering known values yields $$\begin{array}{lll}{v}_{\mathrm{f}}& =& \sqrt{\frac{k}{m}}{x}_{\mathrm{i}}\\ & =& \sqrt{\frac{\text{250}\text{.0 N/m}}{\text{0.100 kg}}}(\text{0.0400 m})\\ & =& \text{2.00 m/s.}\end{array}$$Solution for (b) One method of finding the speed at the top of the slope is to consider conditions just before the car is released and just after it reaches the top of the slope, completely ignoring everything in between. Doing the same type of analysis to find which terms are zero, the conservation of mechanical energy becomes $$\frac{1}{2}{\text{kx}}_{\text{i}}^{2}=\frac{1}{2}{\text{mv}}_{\text{f}}^{\text{2}}+{\text{mgh}}_{\text{f}}\text{.}$$This form of the equation means that the spring’s initial potential energy is converted partly to gravitational potential energy and partly to kinetic energy. The final speed at the top of the slope will be less than at the bottom. Solving for ${v}_{\text{f}}$ and substituting known values gives $$\begin{array}{lll}{v}_{\mathrm{f}}& =& \sqrt{\frac{{{\text{kx}}_{\mathrm{i}}}^{2}}{m}2{\text{gh}}_{\mathrm{f}}}\\ & =& \sqrt{\left(\frac{\text{250.0 N/m}}{\text{0.100 kg}}\right)(\text{0.0400 m}{)}^{2}2(\text{9.80}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2})(\text{0.180 m})}\\ & =& \text{0.687 m/s.}\end{array}$$Discussion Another way to solve this problem is to realize that the car’s kinetic energy before it goes up the slope is converted partly to potential energy—that is, to take the final conditions in part (a) to be the initial conditions in part (b). Note that, for conservative forces, we do not directly calculate the work they do; rather, we consider their effects through their corresponding potential energies, just as we did in Example 1. Note also that we do not consider details of the path taken—only the starting and ending points are important (as long as the path is not impossible). This assumption is usually a tremendous simplification, because the path may be complicated and forces may vary along the way. PhET Explorations: Energy Skate Park:Learn about conservation of energy with a skater dude! Build tracks, ramps and jumps for the skater and view the kinetic energy, potential energy and friction as he moves. You can also take the skater to different planets or even space! Section Summary
where i and f denote initial and final values. This is known as the conservation of mechanical energy. Conceptual QuestionsExercise 1What is a conservative force? Exercise 2The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential energy of a diving board as a swimmer dives from it, starting just before the swimmer steps on the board until just after his feet leave it. Exercise 3Define mechanical energy. What is the relationship of mechanical energy to nonconservative forces? What happens to mechanical energy if only conservative forces act? Exercise 4What is the relationship of potential energy to conservative force? Problems & ExercisesExercise 1A $5\text{.}\text{00}\times {\text{10}}^{5}\text{kg}$ subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a large spring bumper at the end of its track. What is the force constant $k$ of the spring? Show/Hide Solution Solution$$\text{7.81}\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{N/m}$$Exercise 2A pogo stick has a spring with a force constant of $2\text{.}\text{50}\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N/m}$, which can be compressed 12.0 cm. To what maximum height can a child jump on the stick using only the energy in the spring, if the child and stick have a total mass of 40.0 kg? Explicitly show how you follow the steps in the ProblemSolving Strategies for Energy.
