Section Summary

5.1 Work

• Work is the transfer of energy by a force acting on an object as it is displaced.
• The work [latex]W[/latex] that a force [latex]\mathbf{F}[/latex] does on an object is the product of the magnitude [latex]F[/latex] of the force, times the magnitude [latex]d[/latex] of the displacement, times the cosine of the angle [latex]\theta[/latex] between them. In symbols,
  [latex]W=\text{Fd}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\theta \text{.}[/latex]
• The SI unit for work and energy is the joule (J), where [latex]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}[/latex].
• 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.

5.2 Kinetic Energy and the Work-Energy Theorem

• The net work [latex]{W}_{\text{net}}[/latex] 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 [latex]m[/latex] moving at speed [latex]v[/latex] is [latex]\text{KE}=\frac{1}{2}{\text{mv}}^{2}[/latex].
• The work-energy theorem states that the net work [latex]{W}_{\text{net}}[/latex] on a system changes its kinetic energy, [latex]{W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}-\frac{1}{2}{m{v}_{0}}^{2}[/latex].

5.3 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, [latex]\Delta {\text{PE}}_{\text{g}}[/latex], is [latex]{\text{ΔPE}}_{g}=\text{mgh}[/latex], with [latex]h[/latex] being the increase in height and [latex]g[/latex] 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, [latex]{\text{ΔPE}}_{g}[/latex], have physical significance.
• As an object descends without friction, its gravitational potential energy changes into kinetic energy corresponding to increasing speed, so that [latex]\text{ΔKE}\text{= −}{\text{ΔPE}}_{\text{g}}[/latex].

5.4 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 [latex]\left(\text{PE}\right)[/latex] for any conservative force, just as we defined [latex]{\text{PE}}_{g}[/latex] for the gravitational force.
• Mechanical energy is defined to be [latex]\text{KE}+\text{PE}[/latex] for a conservative force.
• When only conservative forces act on and within a system, the total mechanical energy is constant. In equation form,
  [latex]\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}}[/latex]
  where i and f denote initial and final values. This is known as the conservation of mechanical energy.
• For a conservative force, the infinitesimal work is an exact differential. This implies conditions on the derivatives of the force’s components.
• The component of a conservative force, in a particular direction, equals the negative of the derivative of the potential energy for that force, with respect to a displacement in that direction.
• For a single-particle system, the difference of potential energy is the opposite of the work done by the forces acting on the particle as it moves from one position to another.
• Since only differences of potential energy are physically meaningful, the zero of the potential energy function can be chosen at a convenient location.
• The potential energies for Earth’s constant gravity, near its surface, and for a Hooke’s law force are linear and quadratic functions of position, respectively.
• The potential energy of a spring is [latex]{\text{PE}}_{s}=\frac{1}{2}{\text{kx}}^{2}[/latex], where [latex]k[/latex] is the spring’s force constant and [latex]x[/latex] is the displacement from its undeformed position.

5.5 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 [latex]{W}_{\text{nc}}[/latex] done by a nonconservative force changes the mechanical energy of a system. In equation form, [latex]{W}_{\text{nc}}=\text{Δ}\text{KE}+\text{Δ}\text{PE}[/latex] or, equivalently, [latex]{\text{KE}}_{\text{i}}+{\text{PE}}_{\text{i}}+{W}_{\text{nc}}={\text{KE}}_{\text{f}}+{\text{PE}}_{\text{f}}[/latex].
• 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.
• A form of the work-energy theorem says that the change in the mechanical energy of a particle equals the work done on it by non-conservative forces.
• If non-conservative forces do no work and there are no external forces, the mechanical energy of a particle stays constant. This is a statement of the conservation of mechanical energy and there is no change in the total mechanical energy.

5.6 Conservation of Energy

5.7 Power

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

5.8 Work, Power and Energy 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.