1.4: Summary and further problems
- Page ID
- 58560
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Do not add apples to oranges: Every term in an equation or sum must have identical dimensions! This restriction is a powerful tool. It helps us to evaluate integrals without integrating and to predict the solutions of differential equations. Here are further problems to practice this tool.
Problem 1.11 Integrals using dimensions
Use dimensional analysis to find \(\int_{0}^{\infty} e^{-ax}\)dx and \(\int\) \(\frac{dx}{x^{2}+a^{2}}\). A useful result is
\[\int \frac{dx}{x^{2}+1} = arctanx + C. \label{1.20} \]
Problem 1.12 Stefan–Boltzmann law
Blackbody radiation is an electromagnetic phenomenon, so the radiation intensity depends on the speed of light c. It is also a thermal phenomenon, so it depends on the thermal energy \(k_{B}T\), where \(T\) is the object’s temperature and \(k_{B}\) is Boltzmann’s constant. And it is a quantum phenomenon, so it depends on Planck’s constant h. Thus the blackbody-radiation intensity I depends on \(c\), \(k_{B}T\), and \(h\). Use dimensional analysis to show that \(I ∝ T^{4}\) and to find the constant of proportionality σ. Then look up the missing dimensionless constant. (These results are used in Section 5.3.3.)
Problem 1.13 Arcsine integral
Use dimensional analysis to find \(\int \sqrt{1 - 3x^{2}}\)dx. A useful result is
\[\int \sqrt{1-x^{2}} dx =\frac{\arcsin x}{2}+\frac{x \sqrt{1-x^{2}}}{2}+C \label{1.21} \]
Problem 1.14 Related rates
Water is poured into a large inverted cone (with a \(90^{◦}\) opening angle) at a rate dV/dt = \(10m^{3}s^{-1}\). When the water depth is \(h = 5\)m, estimate the rate at which the depth is increasing. Then use calculus to find the exact rate.
Problem 1.15 Kepler’s third law
Newton’s law of universal gravitation the famous inverse-square law says that the gravitational force between two masses is
\[F = -\frac{Gm_{1}m_{2}}{r^{2}}, \label{1.22} \]
where G is Newton’s constant, \(m_{1}\) and \(m_{2}\) are the two masses, and r is their separation. For a planet orbiting the sun, universal gravitation together with Newton’s second law gives
\[m\frac{d^{2}r}{dt^{2}} = -\frac{GMm}{r^{2}} \hat{r}, \label{1.23} \]
Where M is the mass of the sun, m the mass of the planet, r is the vector from the sun to the planet, and \(\hat{r}\) is the unit vector in the r direction.
How does the orbital period τ depend on orbital radius r? Look up Kepler’s third law and compare your result to it.