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6.4E: Motion Under a Central Force (Exercises)

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[exer:6.4.1] Find the equation of the curve

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

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 [figure:6.4.5} . 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

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

and show that the area of the ellipse is

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

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

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

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

[exer:6.4.2] Suppose an object with mass $$m$$ moves in the $$xy$$-plane under the central force

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

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

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

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

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

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

[exer:6.4.3] 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

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

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.

Find the eccentricity required to make the aphelion and perihelion distances equal to $$R\gamma_1$$ and $$R\gamma_2$$, respectively, where $$1<\gamma_1<\gamma_2$$.

Find the initial conditions

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

required to make the initial point the perigee, and the motion along the orbit in the direction of increasing $$\theta$$.

[exer:6.4.4] An object with mass $$m$$ moves in a spiral orbit $$r=c\theta^2$$ under a central force

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

Find $$f$$.

[exer:6.4.5] An object with mass $$m$$ moves in the orbit $$r=r_0e^{\gamma\theta}$$ under a central force

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

Find $$f$$.

[exer:6.4.6] Suppose an object with mass $$m$$ moves under the central force

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

with

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

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

Set up a second order initial value problem for $$u=1/r$$ as a function of $$\theta$$.

Determine $$r=r(\theta)$$ if (i) $$h^2<k$$; (ii) $$h^2=k$$; (iii) $$h^2>k$$.