# 7: Work, Energy, and Energy Resources

- Page ID
- 1432

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

There is no simple, yet accurate, scientific definition for energy. Energy is characterized by its many forms and the fact that it is conserved. We can loosely define energy as the ability to do work, admitting that in some circumstances not all **energy** is available to do work. Because of the association of energy with work, we begin the chapter with a discussion of work. Work is intimately related to energy and how energy moves from one system to another or changes form.

- 7.0: Prelude to Work, Energy, and Energy Resources
- Energy plays an essential role both in everyday events and in scientific phenomena. You can no doubt name many forms of energy, from that provided by our foods, to the energy we use to run our cars, to the sunlight that warms us on the beach. You can also cite examples of what people call energy that may not be scientific, such as someone having an energetic personality. Not only does energy have many interesting forms, it is involved in almost all phenomena, and is one of the most important con

- 7.1: Work- The Scientific Definition
- Work is the transfer of energy by a force acting on an object as it is displaced. The work \(W\) that a force \(F\) does on an object is the product of the magnitude \(F\) of the force, times the magnitude \(d\) of the displacement, times the cosine of the angle \(\theta\) between them. In symbols, \[W = Fd \space cos \space \theta. \] The SI unit for work and energy is the joule (J), where \(1 \space J = 1 \space N \cdot m = 1 \space kg \space m^2/s^2\). The work done by a force is zero if the

- 7.2: Kinetic Energy and the Work-Energy Theorem
- The net work \(W_{net}\) is the work done by the net force acting on an object. Work done on an object transfers energy to the object. The translational kinetic energy of an object of mass \(m\) moving at speed \(v\) is \(KE = \frac{1}{2}mv^2\). The work-energy theorem states that the net work \(W_{net} \) on a system changes its kinetic energy, \(W_{net} = \frac{1}{2}mv^2 - \frac{1}{2}mv_0^2\).

- 7.3: Gravitational Potential Energy
- Work done against gravity in lifting an object becomes potential energy of the object-Earth system. The change in gravitational potential energy \(\Delta PE_g\), is \(\Delta PE_g = mgh\), with \(h\) being the increase in height and \(g\) the acceleration due to gravity. The gravitational potential energy of an object near Earth’s surface is due to its position in the mass-Earth system. Only differences in gravitational potential energy, \(\Delta PE_g\), have physical significance. As an obje

- 7.4: Conservative Forces and Potential Energy
- 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 \(PE_g\) for the gravitational force. The potential energy of a spring is \(PE_s = \frac{1}{2}kx^2\), where \(k\) is the spring’s force constant and |(x\) is the displacement from its undeformed position. Mechanical energy is defined to be \(KE = PE\) for conservative force.

- 7.5: Nonconservative Forces
- A nonconservative force is one for which work depends on the path. Friction is an example of a nonconservative force that changes mechanical energy into thermal energy. Work \(W_{nc}\) done by a nonconservative force changes the mechanical energy of a system. In equation form, \(W_{nc} = \Delta KE + \Delta PE \) or, equivalently, \(KE_i + PE_i + W_{nc} = KE_f + PE_f .\) When both conservative and nonconservative forces act, energy conservation can be applied and used to calculate motion in terms

- 7.6: Conservation of Energy
- The law of conservation of energy states that the 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. When all forms of energy are considered, conservation of energy is written in equation form as \[KE_i + PE_i + W_{nc} + OE_i = KE_f + PE_f + OE_f ,\] where \(OE\) is all other forms of energy besides mechanical energy.

- 7.7: Power
- Power is the rate at which work is done, or in equation form, for the average power \(P\) for work \(W\) done over a time \(t\), \(P = W/t\). The SI unit for power is the watt (W), where \(1 \space W = 1 \space J/s\). The power of many devices such as electric motors is also often expressed in horsepower (hp), where \(1\space hp = 746 \space W.\)

- 7.8: Work, Energy, and Power in Humans
- The human body converts energy stored in food into work, thermal energy, and/or chemical energy that is stored in fatty tissue. The rate at which the body uses food energy to sustain life and to do different activities is called the metabolic rate, and the corresponding rate when at rest is called the basal metabolic rate (BMR) The energy included in the basal metabolic rate is divided among various systems in the body, with the largest fraction going to the liver and spleen, and the brain.

- 7.9: World Energy Use
- The relative use of different fuels to provide energy has changed over the years, but fuel use is currently dominated by oil, although natural gas and solar contributions are increasing. Although non-renewable sources dominate, some countries meet a sizeable percentage of their electricity needs from renewable resources. The United States obtains only about 10% of its energy from renewable sources, mostly hydroelectric power.

Thumbnail: One form of energy is mechanical work, the energy required to move an object of mass m a distance d when opposed by a force F, such as gravity. Image use with permission (CC-SA-BY-NC -3.0; anonymous).