Exercises

1. A particle is projected vertically upward from the Earth's surface with a velocity that would, were gravity uniform, carry it to a height $h$. Show that if the variation of gravity with height is allowed for, but the resistance of air is neglected, then the height reached will be greater by $h^{\,2}/(R-h)$, where $R$ is the Earth's radius. (From Lamb 1923.)

2. A particle is projected vertically upward from the Earth's surface with a velocity just sufficient for it to reach infinity (neglecting air resistance). Prove that the time needed to reach a height $h$ is
   $$\frac{1}{3}\left(\frac{2\,R}{g}\right)^{1/2}\,\left[\left(1+\frac{h}{R}\right)^{3/2}-1\right],$$
   where $R$ is the Earth's radius, and $g$ its surface gravitational acceleration. (From Lamb 1923.)

3. Assuming that the Earth is a sphere of radius $R$, and neglecting air resistance, show that a particle that starts from rest a distance $R$ from the Earth's surface will reach the surface with speed $\sqrt{R\,g}$ after a time $(1+\pi/2)\sqrt{R/g}$, where $g$ is the surface gravitational acceleration. (Modified from Smart 1951.)

4. Demonstrate that if a narrow shaft were drilled though the center of a uniform self-gravitating sphere then a test mass moving in this shaft executes simple harmonic motion about the center of the sphere with period
   $$T = 2\pi\sqrt{\frac{R}{g}},$$
   where $R$ is the radius of the sphere, and $g$ the gravitational acceleration at its surface.

5. Consider an isolated system consisting of $N$ point objects interacting via gravity. The equation of motion of the $i$th object is
   $$m_i\,\ddot{\bf r}_i = \sum_{j=1,N}^{j\neq i} G\,m_i\,m_j\,\frac{{\bf r}_j-{\bf r}_i}{\vert{\bf r}_j-{\bf r}_i\vert^{\,3}},$$
   where $m_i$ and ${\bf r}_i$ are the mass and position vector of this object, respectively. Moreover, the total potential energy of the system takes the form
   $$U= -\frac{1}{2}\sum_{i,j=1,N}^{i\neq j} \frac{G\,m_i\,m_j}{\vert{\bf r}_j-{\bf r}_i\vert}.$$
   Write an expression for the total kinetic energy, $K$. Demonstrate, from the equations of motion, that $K+U$ is constant in time.

6. Consider a function of many variables $f(x_1,x_2,\cdots,x_n)$. Such a function that satisfies
   $$f(t\,x_1, t\,x_2,\cdots,t\,x_n) = t^{\,a}\,f(x_1,x_2,\cdots,x_n)$$
   for all $t>0$, and all values of the $x_i$, is termed a homogeneous function of degree $a$. Prove the following theorem regarding homogeneous functions:
   $$\sum_{i=1,n} x_i\,\frac{\partial f}{\partial x_i} = a\,f$$

7. Consider an isolated system consisting of $N$ point particles interacting via attractive central forces. Let the mass and position vector of the $i$th particle be $m_i$ and ${\bf r}_i$, respectively. Suppose that magnitude of the force exerted on particle $i$ by particle $j$ is $k_i\,k_j\,\vert{\bf r}_i-{\bf r}_j\vert^{-n}$. Here, $k_i$ measures some constant physical property of the $i$th particle (e.g., its electric charge). Show that the total potential energy $U$ of the system is written
   $$U = -\frac{1}{2}\,\frac{1}{n-1}\sum_{i,j=1,N}^{j\neq i}\frac{k_i\,k_j}{\vert{\bf r}_j-{\bf r}_i\vert^{\,n-1}}.$$
   Is this a homogeneous function? If so, what is its degree? Demonstrate that the equation of motion of the $i$th particle can be written
   $$m_i\,\ddot{\bf r}_i = -\frac{\partial U}{\partial {\bf r}_i}.$$
   (This is shorthand for $m_i\,\ddot{x}_i=-\partial U/\partial x_i$, $m_i\,\ddot{y}_i=-\partial U/\partial y_i$, etc., where the $x_i$, $y_i$, $z_i$, for $i=1, N$, are treated as independent variables.) Use the mathematical theorem from the previous exercise to show that
   $$\frac{1}{2}\frac{d^{\,2} {\cal I}}{dt^{\,2}} = 2\,K + (n-1)\,U,$$
   where ${\cal I}=\sum_{i=1,N} m_i\, r_i^{\,2}$, and $K$ is the total kinetic energy. This result is known as the virial theorem. Demonstrate that when $n\geq 3$ the system possesses no virial equilibria (i.e., states for which ${\cal I}$ does not evolve in time) that are bounded.

8. Demonstrate that the gravitational potential energy of a spherically symmetric mass distribution of mass density $\rho(r)$ that extends out to $r=R$ can be written
   $$U = - 16\,\pi^{\,2}\,G\int_0^R \int_0^r r'^{\,2}\,\rho(r')\,r\,\rho(r)\,dr'\,dr.$$
   Hence, show that if the mass distribution is such that
   $$\rho(r) = \left\{\begin{array}{lcc} \rho_0\,r^{-\alpha}&\mbox{\hspace{1cm}}& r \leq R\\ [0.5ex] 0&& r>R\end{array}\right.,$$
   where $\alpha<5/2$, then
   $$U = - \frac{(3-\alpha)}{(5-2\,\alpha)}\,\frac{G\,M^{\,2}}{R},$$
   where $M$ is the total mass.

9. A globular star cluster can be approximated as an isolated self-gravitating virial equilibrium consisting of a great number of equal mass stars. Demonstrate, from the virial theorem, that
   $$K = - \frac{1}{2}\,U$$
   for such a cluster. Suppose that the stars in a given cluster are uniformly distributed throughout a spherical volume. Show that
   $$\skew{3}\bar{\varv} = \sqrt{\frac{3}{10}}\,\varv_{\rm esc},$$
   where $\skew{3}\bar{\varv}$ is the mean stellar velocity, and $\varv_{\rm esc}$ is the escape speed (i.e., the speed a star at the edge of the cluster would require in order to escape to infinity.) See Section 3.8, Exercise 7.

10. A star can be through of as a spherical system that consists of a very large number of particles, of mass $m_i$ and position vector ${\bf r}_i$, interacting via gravity. Show that, for such a system, the virial theorem implies that
    $$\frac{d^{\,2} {\cal I}}{dt^{\,2}} = -2\,U + c,$$
    where $c$ is a constant, ${\cal I}=\sum m_i\,r_i^{\,2}$, and the $r_i$ are measured from the geometric center. Hence, deduce that the angular frequency of small-amplitude radial pulsations of the star (in which the radial displacement is directly proportional to the radial distance from the center) takes the form
    $$\omega = \left(\frac{\vert U_0\vert}{{\cal I}_0}\right)^{1/2},$$
    where $U_0$ and ${\cal I}_0$ are the equilibrium values of $U$ and ${\cal I}$. Finally, show that if the mass density within the star varies as $r^{-\alpha}$, where $r$ is the radial distance from the geometric center, and where $\alpha<5/2$, then
    $$\omega = \left(\frac{5-\alpha}{5-2\,\alpha}\,\frac{G\,M}{R^{\,3}}\right)^{1/2},$$
    where $M$ and $R$ are the stellar mass and radius, respectively. See Section 3.8, Exercises 7 and 8.