### 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 $\mathbf{F}$ 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,
$$W=\text{Fd}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta \text{.}$$
- The SI unit for work and energy is the joule (J), where $1\phantom{\rule{0.25em}{0ex}}\text{J}=1\phantom{\rule{0.25em}{0ex}}\text{N}\cdot \text{m}=\text{1 kg}\cdot {\text{m}}^{2}{\text{/s}}^{2}$.
- The work done by a force is zero if the displacement is either zero or perpendicular to the force.
- The work done is positive if the force and displacement have the same direction, and negative if they have opposite direction.

### 7.2 Kinetic Energy and the Work-Energy Theorem

- The net work ${W}_{\text{net}}$ 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 $\text{KE}=\frac{1}{2}{\text{mv}}^{2}$.
- The work-energy theorem states that the net work ${W}_{\text{net}}$ on a system changes its kinetic energy, ${W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}-\frac{1}{2}{m{v}_{0}}^{2}$.

### 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, $\mathrm{\Delta}{\text{PE}}_{\text{g}}$, is ${\text{\Delta PE}}_{\mathrm{g}}=\text{mgh}$, with $h$ being the increase in height and $\mathrm{g}$ 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, ${\text{\Delta PE}}_{\mathrm{g}}$, have physical significance.
- As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increasing speed, so that $\text{\Delta KE}\text{= \u2212}{\text{\Delta PE}}_{\text{g}}$.

### 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 $(\text{PE})$ for any conservative force, just as we defined ${\text{PE}}_{\mathrm{g}}$ for the gravitational force.
- The potential energy of a spring is ${\text{PE}}_{\mathrm{s}}=\frac{1}{2}{\text{kx}}^{2}$, where $k$ is the spring’s force constant and $x$ is the displacement from its undeformed position.
- Mechanical energy is defined to be $\text{KE}+\text{PE}$ for a conservative force.
- When only conservative forces act on and within a system, the total mechanical energy is constant. In equation form,

$$\left.\begin{array}{cc}& \text{KE}+\text{PE}=\text{constant}\\ \text{or}& \\ & {\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}}={\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}\end{array}\right\}$$

where i and f denote initial and final values. This is known as the conservation of mechanical energy.

### 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 ${W}_{\text{nc}}$ done by a nonconservative force changes the mechanical energy of a system. In equation form, ${W}_{\text{nc}}=\text{\Delta}\text{KE}+\text{\Delta}\text{PE}$ or, equivalently, ${\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}}+{W}_{\text{nc}}={\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}$.
- When both conservative and nonconservative forces act, energy conservation can be applied and used to calculate motion in terms of the known potential energies of the conservative forces and the work done by nonconservative forces, instead of finding the net work from the net force, or having to directly apply Newton’s laws.

### 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 ${\text{KE}}_{\mathrm{i}}+{\text{PE}}_{\mathrm{i}}+{W}_{\text{nc}}+{\text{OE}}_{\mathrm{i}}={\text{KE}}_{\mathrm{f}}+{\text{PE}}_{\mathrm{f}}+{\text{OE}}_{\mathrm{f}}$, where $\text{OE}$ is all
**other forms of energy**besides mechanical energy. - Commonly encountered forms of energy include electric energy, chemical energy, radiant energy, nuclear energy, and thermal energy.
- Energy is often utilized to do work, but it is not possible to convert all the energy of a system to work.
- The efficiency $\text{Eff}$ of a machine or human is defined to be $\text{Eff}=\frac{{W}_{\text{out}}}{{E}_{\text{in}}}$, where ${W}_{\text{out}}$
_{}is useful work output and ${E}_{\text{in}}$_{}is the energy consumed.

### 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 $\text{1 W}=\text{1 J/s}$.
- The power of many devices such as electric motors is also often expressed in horsepower (hp), where $\text{1 hp}=\text{746 W}$.

### 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 coming next.
- About 75% of food calories are used to sustain basic body functions included in the basal metabolic rate.
- The energy consumption of people during various activities can be determined by measuring their oxygen use, because the digestive process is basically one of oxidizing food.

### 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.
- Economic well-being is dependent upon energy use, and in most countries higher standards of living, as measured by GDP (Gross Domestic Product) per capita, are matched by higher levels of energy consumption per capita.
- Even though, in accordance with the law of conservation of energy, energy can never be created or destroyed, energy that can be used to do work is always partly converted to less useful forms, such as waste heat to the environment, in all of our uses of energy for practical purposes.