# 6.4E: Motion Under a Central Force (Exercises)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## Q6.4.1

Find the equation of the curve

$r={\rho\over 1+e\cos(\theta-\phi)} \tag{A}$

in terms of $$(X,Y)=\left(r\cos(\theta-\phi),r\sin(\theta-\phi)\right)$$, which are rectangular coordinates with respect to the axes shown in Figure 6.4E.1 . Use your results to verify that (A) is the equation of an ellipse if $$0<e<1$$, a parabola if $$e=1$$, or a hyperbola if $$e>1$$. If $$e<1$$, leave your answer in the form

${(X-X_0)^2\over a^2}+{(Y-Y_0)^2\over b^2}=1, \nonumber$

and show that the area of the ellipse is

$A={\pi\rho^2\over(1-e^2)^{3/2}}. \nonumber$

Then use Theorem 6.4.1 to show that the time required for the object to traverse the entire orbit is

$T={2\pi\rho^2\over h(1-e^2)^{3/2}}. \nonumber$

(This is Kepler’s third law; $$T$$ is called the period of the orbit.)

## Q6.4.2

Suppose an object with mass $$m$$ moves in the $$xy$$-plane under the central force

${\bf F}(r,\theta)=-{mk\over r^2}(\cos\theta\,{\bf i}+\sin\theta\,{\bf j}), \nonumber$

where $$k$$ is a positive constant. As we shown, the orbit of the object is given by

$r={\rho\over 1+e\cos(\theta-\phi)}. \nonumber$

Determine $$\rho$$, $$e$$, and $$\phi$$ in terms of the initial conditions

$r(0)=r_0,\quad r'(0)=r_0', \quad\text{and}\quad \theta(0)=\theta_0,\quad \theta'(0)=\theta_0'. \nonumber$

Assume that the initial position and velocity vectors are not collinear.

## Q6.4.3

Suppose we wish to put a satellite with mass $$m$$ into an elliptical orbit around Earth. Assume that the only force acting on the object is Earth’s gravity, given by

${\bf F}(r,\theta)=-mg\left(R^2\over r^2\right)(\cos\theta\,{\bf i}+\sin\theta\,{\bf j}), \nonumber$

where $$R$$ is Earth’s radius, $$g$$ is the acceleration due to gravity at Earth’s surface, and $$r$$ and $$\theta$$ are polar coordinates in the plane of the orbit, with the origin at Earth’s center.

1. Find the eccentricity required to make the aphelion and perihelion distances equal to $$R\gamma_1$$ and $$R\gamma_2$$, respectively, where $$1<\gamma_1<\gamma_2$$.
2. Find the initial conditions $r(0)=r_0,\quad r'(0)=r_0',\quad \text{and} \quad \theta(0)=\theta_0,\quad \theta'(0)=\theta_0' \nonumber$ required to make the initial point the perigee, and the motion along the orbit in the direction of increasing $$\theta$$. HINT: Use the results of Exercise 6.4.2

## Q6.4.4

4. An object with mass $$m$$ moves in a spiral orbit $$r=c\theta^2$$ under a central force

${\bf F}(r,\theta)=f(r)(\cos\theta\,{\bf i}+\sin\theta\,{\bf j}). \nonumber$

Find $$f$$.

## Q6.4.5

An object with mass $$m$$ moves in the orbit $$r=r_0e^{\gamma\theta}$$ under a central force

${\bf F}(r,\theta)=f(r)(\cos\theta\,{\bf i}+\sin\theta\,{\bf j}). \nonumber$

Find $$f$$.

## Q6.4.6

Suppose an object with mass $$m$$ moves under the central force

${\bf F}(r,\theta)=-{mk\over r^3}(\cos\theta\,{\bf i}+\sin\theta\,{\bf j}), \nonumber$

with

$r(0)=r_0,\quad r'(0)=r_0', \quad\text{and}\quad \theta(0)=\theta_0,\quad \theta'(0)=\theta_0', \nonumber$

where $$h=r_0^2\theta_0'\ne0$$.

1. Set up a second order initial value problem for $$u=1/r$$ as a function of $$\theta$$.
2. Determine $$r=r(\theta)$$ if (i) $$h^2<k$$; (ii) $$h^2=k$$; (iii) $$h^2>k$$.

This page titled 6.4E: Motion Under a Central Force (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.