# 2.2.1: Potential Energy

- Page ID
- 84577

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Potential Energy is any form of energy that can be "stored" by a physical system for instance, the energy of a stretched spring, a body in a gravitational field, or the energies of some physical fields. The name comes from the fact that such energies, essentially, can be stored for an unlimited time, and then converted to work or other energy form at a chosen moment moment of time. One can then think of such systems "pumped" with energy as of those which "have a potential".

## Potential Energy of Stretched Springs

The energy of a stretched or compressed spring can be readily calculated, based on the Hooke's Law that describes the relation between the extension (a.k.a. elongation) of a stretched spring, or the contraction (a.k.a. shortening) of a compressed spring, and the "restoring force" \(\mathrm{F}\), i.e., the force tending to bring the spring back to its original length:

\[

F=-k \cdot \Delta l

\]

where \(\Delta l\) is the change in the spring length (positive for extension, and negative for contraction), and \(k\) is the so-called "stiffness constant" of the spring; the minus sign reflects the fact that the restoring force is always oriented towards the equilibrium position, so its sign is always opposite to the sign of \(\Delta l\).

Knowing the force, we can readily calculate the work that has to be performed for stretching/compressing the spring by \(\Delta l\). For such an action we need to apply a force which overcomes the "restoring force", i.e., which points in the opposite direction. Since the force needed is not constant, but it changes during the process, we cannot use a simple multiplication as in Equation (1). We have to integrate:

\[

\Delta W=\int_{0}^{\Delta l} F(l) d l=\int_{0}^{\Delta l} k l d l

\]

\(=\frac{1}{2} k(\Delta l)^{2}\)

Now, invoking the Work-Energy Principle, we can conclude: *The amount of potential energy \({ }^{1}\) stored in the stretched spring considered is*:

\[

\Delta V=\frac{1}{2} k(\Delta l)^{2} .

\]

The discussion on the energy of a stretched spring is an instructive example demonstrating how to use the Work-Energy Principle. It also shows that a spring can be thought of as an elementary h "energy storing" device. In fact, stretched springs were used in the old days for energizing clocks, watches, phonographs, mechanical toys, small movie cameras and many other portable devices. Today, they have been replaced by batteries. Therefore, we will not talk very much about the energy of stretched springs (or "elastic energy") further on with one exception, namely, when we discuss the the energy of a spring oscillator.

## Potential Energy due to Gravity

Another good example is the potential energy of a lifted object of mass \(m\). The force of gravity acting on such an object is \(F=m g\), where \(g=9.81\) \(\mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to Earth's gravity. The force is constant, we don't need to integrate so if we lift the mass from the height \(H_{i}\) to the height \(H_{f}\), we do work \(\Delta W=m g\left(H_{f}-H_{i}\right)\). So, again invoking the Work Energy principle, we have shown that by lifting the mass we have increased its potential energy by:

\[

\Delta V=m g\left(H_{f}-H_{i}\right)

\]

A lifted object can be then used as an "energy reservoir". Actually, such method is used for storing huge amounts of energy in facilities known as "Pumped Storage Hydroelectric Power Plants", about which we will talk later, in a section titled "Hydroelectric Power". They store energy not by lifting a solid object of mass \(m\), but by pumping water from a lower tank to a higher tank but the general principle is the same.

A lifted object can be then used as an “energy reservoir”. Actually, such method is used for storing huge amounts of energy in facilities known as “Pumped Storage Hydroelectric Power Plants”, about which we will talk later, in a section titled “Hydroelectric Power”. They store energy not by lifting a solid object of mass *m*, but by pumping water from a lower tank to a higher tank – but the general principle is the same.

There is, however, one “tricky issue” with the gravitational potential energy. Suppose that you are standing on a balcony. On the balcony’s floor there is a weight of 1 kg mass. You lift it above your head, so it’s now 2 meters above the balcony’s floor. You say: *Its potential energy is now 1 kg times 2 **meters times g, which is 19.62 Joules*. But your friend standing underneath at the ground level disagrees: *The balcony’s floor is three meters above the **level, so the weight is now at a height of five meters, hence its potential energy **is 49.05 Joules! *Who is right? You both are right and aren’t right. How comes? The thing is that the Eq. 2.8 describes only the **change **of the potential energy, not its absolute value. Giving the potential energy value is a worthless information – one has always specify **relative to what level t**he potential energy is of the value given. So, the correct statements in the above examples should be, respectively: *The potential energy of the weight **relative to the balcony floor is 19.62 Joules*, and *The potential energy of the weight relative to the ground level is 49.05 Joules*.

One may insist that if we reach an “universal agreement” – say, that all potential energies should be given relative to the same level, e.g., the sea level, then it would be possible to skip the *relative to .... *part of the statement. Well, such a convention would be possible, but not very practical – everybody would need to know exactly how high about the sea level the object considered is. Using a GPS would not be a good solution, because the altitude information offered by such devices is known to be not too accurate. And the need of always giving information about the “reference level” is not a major inconvenience – therefore, the idea of using an universal reference such as the sea level never caught up.

Which is not 100% true, though. There are situations in which such an universal reference level becomes very useful. Namely, if we consider potential energies of objects at very high altitudes, such as artificial or natural satellites, spacecrafts, or “near-Earth asteroids”. High above the Earth surface the value of *g *can no longer be taken as a constant. It decreases with the altitude *h *as:

\[g(h)=g_{0} \frac{R^{2}}{(R+h)^{2}}\]

where *g*_{0} = 9*.*81 m/s^{2} is the acceleration due to gravity at the Earth surface, and *R *= 6371*, *000 m is the Earth radius.

When considering the potential energy of an object at *h *such that *g*(*h*) is considerably smaller than *g*_{0}, we use as the “reference level” *h *= , and we assume that the potential energy at this level is... zero. So, it means that at any distance “closer than infinity” to Earth the potential energy of any object is... negative! Yes, in such convention all potential energies are negative. Weird? At first glance it might seem so. But note that there is nothing wrong with physics: for a lower *h*, the potential energy is “more negative than for a larger *h *– so, with increasing *h *the potential energy indeed increases, as it should be.

We will return to this issue later on, when discussing the energy of the Earth-Moon system, and the ocean tides. Moving the masses of tidal waters needs some energy input, and some part of it is “extracted” from the energy of the Moon’s orbital motion – leading to a small increase (yes!) of the Moon-Earth distance, with the rate of 3.8 cm/year.

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^{1}I use *V *as the symbol of potential energy, but in other books or Web sites you may find different symbols. The same is true for many other quantities in physics. The thing is that there are many more quantities of interest in physics than available Roman and Greek letters. Therefore, the same letter is often used for several different things – e.g. *V *is also used for volume, velocity, voltage and perhaps several other quantities.