# 5.5: Conservative Forces, Potential Energy, and Conservation of Energy

- Page ID
- 475692

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)##### Learning Objectives

- Define conservative force, potential energy, and mechanical energy.
- Apply conservation of mechanical energy to simple physical situations.
- Explain the law of the conservation of energy.

### Potential Energy and Conservative Forces

Work is done by a force, and some forces, such as weight, have special characteristics. A **conservative force** is one, like the gravitational force, 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. This happens if the force is a function of position alone and not time, velocity, or other parameters. We can define a **potential energy** (**PE**) for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is *conservative*. That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy.

##### 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 a force that is a function of position alone, with the result that the work done by the force depends only on the starting and ending points of a motion and not on the particular path taken.

Change in potential energy comes from the work done against a conservative force. Or flipping this around, potential energy increases as a conservative force does a negative work. That is,

\[\Delta \mathrm{PE}=-W_{c}, \nonumber \]

where \(W_{c}\) is work done by a conservative force.

### Conservation of Mechanical Energy

Consider an object in a system. If it gains any kinetic energy, this is a result of a net work done on the object, according to the work-kinetic energy theorem. If only conservative forces, such as the gravitational force or a spring force, do work in this system, then as kinetic energy increases with the net work done by the conservative forces, the system loses potential energy. That is, \(\Delta \mathrm{KE}=W_{c}=-\Delta \mathrm{PE}\). In other words,

\[\Delta \mathrm{KE}+\Delta \mathrm{PE}=0. \nonumber \]

This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,

\[

\left.\begin{array}{l}

\text { or } \begin{array}{l}

\mathrm{KE}+\mathrm{PE}=\text { constant } \\ \\[4pt]

\mathrm{KE}_{\mathrm{i}}+\mathrm{PE}_{\mathrm{i}}=\mathrm{KE}_{\mathrm{f}}+\mathrm{PE}_{\mathrm{f}}

\end{array}

\end{array}\right\} \text { (conservative forces only), } \nonumber \]

where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the **conservation of mechanical energy** principle. Remember that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its **mechanical energy**, (**KE+PE**). In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between **KE** and the various types of **PE**, with the total energy remaining constant.

##### Example \(\PageIndex{1}\): Using Conservation of Mechanical Energy to Calculate the Speed of a Toy Car

A 0.100-kg toy car is propelled by a compressed spring, as shown in Figure \(\PageIndex{1}\). The car follows a track that rises 0.180 m above the starting point. The compressed spring has a potential energy of 0.200 J. 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,

\[\mathrm{KE}_{\mathrm{i}}+\mathrm{PE}_{\mathrm{i}}=\mathrm{KE}_{\mathrm{f}}+\mathrm{PE}_{\mathrm{f}} \nonumber\]

or

\[\frac{1}{2} m v_{\mathrm{i}}^{2}+m g h_{\mathrm{i}}+\mathrm{PE}_{\mathrm{si}}=\frac{1}{2} m v_{\mathrm{f}}^{2}+m g h_{\mathrm{f}}+\mathrm{PE}_{\mathrm{sf}} \nonumber\]

where \(h\) is the height (vertical position) and \(\mathrm{PE}_{\mathrm{s}}\) is the potential energy 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 \(h_{\mathrm{i}}\) and \(h_{\mathrm{f}}\) are zero. Furthermore, the initial speed \(v_{\mathrm{i}}\) is zero and the final compression of the spring is zero, and so several terms in the conservation of mechanical energy equation are zero and it simplifies to

\[\mathrm{PE}_{\mathrm{si}}=\frac{1}{2} m v_{\mathrm{f}}^{2}. \nonumber\]

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

\[v_{\mathrm{f}}=\sqrt{\frac{2 \mathrm{PE}_{\mathrm{si}}}{m}}=\sqrt{\frac{(2)(0.200 \mathrm{~J})}{0.100 \mathrm{~kg}}}=2.00 \mathrm{~m} / \mathrm{s}. \nonumber\]

###### 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

\[\mathrm{PE}_{\mathrm{si}}=\frac{1}{2} m v_{\mathrm{f}}^{2}+m g h_{\mathrm{f}}. \nonumber\]

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 \(v_{\mathrm{f}}\) and substituting known values gives

\[v_{\mathrm{f}}=\sqrt{\frac{(2)\left(\mathrm{PE}_{\mathrm{si}}-m g h_{\mathrm{f}}\right)}{m}}=\sqrt{\frac{0.0472 \mathrm{~J}}{0.100 \mathrm{~kg}}}=0.687 \mathrm{~m} / \mathrm{s} \nonumber\]

###### 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 \(\PageIndex{1}\). 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.

### Conservation of Total Energy

Energy, as we have noted, is conserved, making it one of the most important physical quantities in nature. The **law of conservation of energy** can be stated as follows:

*Total energy is constant in any process. It may change in form or be transferred from one system to another, but the total remains the same.*

We have explored some forms of energy and some ways it can be transferred from one system to another, in the case of work done by conservative forces transforming between potential energy and kinetic energy. Together, these make up the mechanical energy (**KE+PE**), and the mechanical energy is not always conserved, because it can be transformed into other forms of energy through work done by non-conservative forces. We will explore other types of energy in the next section.

### Section Summary

- A conservative force is one for which work depends only on the starting and ending points of a motion, not on the path taken.
- We can define potential energy (
**PE**) for any conservative force, just as we defined \(\mathrm{PE}_{\mathrm{g}}\) for the gravitational force. - Mechanical energy is defined to be
**KE+PE**for a conservative force. - When only conservative forces act on and within a system, the total mechanical energy is conserved, \(\Delta \mathrm{KE}+\Delta \mathrm{PE}=0\).
- The law of conservation of energy states that the total energy, including the mechanical energy and other forms of energy, is constant in any process. Energy may change in form or be transferred from one system to another, but the total remains the same.

### Glossary

**conservative force**- a force that is a function of position alone, with the result that the work done by the force depends only on the starting and ending points of a motion and not on the particular path taken

**potential energy**- energy due to position, shape, or configuration

**conservation of mechanical energy**- the rule that the sum of the kinetic energies and potential energies remains constant if only conservative forces act on and within a system

**mechanical energy**- the sum of kinetic energy and potential energy

**law of conservation of energy**- the general law that total energy is constant in any process; energy may change in form or be transferred from one system to another, but the total remains the same