Chapter 5: Work, Power, and Energy
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{co