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,
  $$W=\text{Fd}\text{cos}\theta \text{.}$$
• The SI unit for work and energy is the joule (J), where $1\text{J}=1\text{N}\cdot \text{m}=\text{1 kg}\cdot {\text{m}}^2{\text{/s}}^2$.
• 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.
• In a uniform gravitational field, such as that near Earth’s surface, the change in gravitational potential energy of a body of mass m, $$\Delta P{E}_g$$, is $$\Delta P{E}_g=mgh$$, 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, ${\text{ΔPE}}_{\mathrm{g}}$, have physical significance.
• As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increasing speed, so that $\text{ΔKE}\text{= −}{\text{ΔPE}}_{\text{g}}$.
• In a gravitational field described by Newton’s universal law of gravitation, the change in gravitational potential energy of an object of mass m that occurs when it is brought from infinitely far away to a distance r from the center of mass of an object of mass M is $$\Delta P{E}_g=-G\frac{mM}{r}$$, where G is the gravitational constant.
• A scalar field associated with an object of mass M, called the gravitational potential field, can be defined as the change in gravitational potential energy per unit mass of a test object moved from infinitely far away to a given point in space. The potential is given by $$V_g=-G\frac{M}{r}$$, where r is the distance between the object’s center of mass and a given point in space.
• When more than one object is present, the gravitational potential at a point in space is the sum of the individual gravitational potentials at that point.
• Characteristics of a gravitational potential field—represented, for example, by isolines on a contour plot—can be used to make inferences about the number, relative size, and location of the sources.

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,

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

7.5 Nonconservative Forces

• A nonconservative force is a force 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, $W_{\text{nc}}=\text{Δ}\text{KE}+\text{Δ}\text{PE}$ 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 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 do different activities is called the metabolic rate; the corresponding rate when at rest is called the basal metabolic rate (BMR).
• The energy included in the BMR 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 percent of food calories are used to sustain basic body functions included in the BMR.
• Since digestion is basically a process of oxidizing food, people's energy consumption during various activities can be determined by measuring their oxygen use.

7.9 World Energy Use

• The relative use of different fuels to provide energy has changed over the years. Fuel use is currently dominated by oil, although natural gas and solar contributions are increasing.
• Although nonrenewable sources dominate, some countries meet a sizeable percentage of their electricity needs from renewable resources.
• The United States obtains only about 10 percent of its energy from renewable sources, mostly hydroelectric power.
• Economic well-being is dependent upon energy use. In most countries, higher standards of living—as measured by GDP (Gross Domestic Product) per capita—are matched by high 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.