13.11: Work-Kinetic Energy Theorem in Three Dimensions
- Page ID
- 26939
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Recall our mathematical result that for one-dimensional motion
\[m \int_{i}^{f} a_{x} d x=m \int_{i}^{f} \frac{d v_{x}}{d t} d x=m \int_{i}^{f} d v_{x} \frac{d x}{d t}=m \int_{i}^{f} v_{x} d v_{x}=\frac{1}{2} m v_{x, f}^{2}-\frac{1}{2} m v_{x, i}^{2} \nonumber \]
Using Newton’s Second Law in the form \(F_{x}=m a_{x}\), we concluded that
\[\int_{i}^{f} F_{x} d x=\frac{1}{2} m v_{x, f}^{2}-\frac{1}{2} m v_{x, i}^{2} \nonumber \]
Equation (13.11.2) generalizes to the y - and z -directions:
\[\int_{i}^{f} F_{y} d y=\frac{1}{2} m v_{y, f}^{2}-\frac{1}{2} m v_{y, i}^{2} \nonumber \]
\[\int_{i}^{f} F_{z} d z=\frac{1}{2} m v_{z, f}^{2}-\frac{1}{2} m v_{z, i}^{2} \nonumber \]
Adding Equations (13.11.2), (13.11.3), and (13.11.4) yields
\[\int_{i}^{f}\left(F_{x} d x+F_{y} d y+F_{z} d z\right)=\frac{1}{2} m\left(v_{x, f}^{2}+v_{y, f}^{2}+v_{z, f}^{2}\right)-\frac{1}{2} m\left(v_{x, i}^{2}+v_{y, i}^{2}+v_{z, i}^{2}\right) \nonumber \]
Recall (Equation (13.8.24)) that the left hand side of Equation (13.11.5) is the work done by the force \(\overrightarrow{\mathbf{F}}\) on the object
\[W=\int_{i}^{f} d W=\int_{i}^{f}\left(F_{x} d x+F_{y} d y+F_{z} d z\right)=\int_{i}^{f} \overrightarrow{\mathbf{F}} \cdot d \overrightarrow{\mathbf{r}} \nonumber \]
The right hand side of Equation (13.11.5) is the change in kinetic energy of the object
\[\Delta K \equiv K_{f}-K_{i}=\frac{1}{2} m v_{f}^{2}-\frac{1}{2} m v_{0}^{2}=\frac{1}{2} m\left(v_{x, f}^{2}+v_{y, f}^{2}+v_{z, f}^{2}\right)-\frac{1}{2} m\left(v_{x, i}^{2}+v_{y, i}^{2}+v_{z, i}^{2}\right) \nonumber \]
Therefore Equation (13.11.5) is the three dimensional generalization of the work-kinetic energy theorem
\[\int_{i}^{f} \overrightarrow{\mathbf{F}} \cdot d \overrightarrow{\mathbf{r}}=K_{f}-K_{i} \nonumber \]
When the work done on an object is positive, the object will increase its speed, and negative work done on an object causes a decrease in speed. When the work done is zero, the object will maintain a constant speed.
Instantaneous Power Applied by a Non-Constant Force for Three Dimensional Motion
Recall that for one-dimensional motion, the instantaneous power at time t is defined to be the limit of the average power as the time interval \([t, t+\Delta t]\) approaches zero,
\[P(t)=F_{x}^{a}(t) v_{x}(t) \nonumber \]
A more general result for the instantaneous power is found by using the expression for dW as given in Equation (13.8.23),
\[P=\frac{d W}{d t}=\frac{\overrightarrow{\mathbf{F}} \cdot d \overrightarrow{\mathbf{r}}}{d t}=\overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathbf{v}} \nonumber \]
The time rate of change of the kinetic energy for a body of mass m is equal to the power,
\[\frac{d K}{d t}=\frac{1}{2} m \frac{d}{d t}(\overrightarrow{\mathbf{v}} \cdot \overrightarrow{\mathbf{v}})=m \frac{d \overrightarrow{\mathbf{v}}}{d t} \cdot \overrightarrow{\mathbf{v}}=m \overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{v}}=\overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathbf{v}}=P \nonumber \]
where the we used Equation (13.8.9), Newton’s Second Law and Equation (13.11.10).