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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

    $\displaystyle \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

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle 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:

    $\displaystyle \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

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle \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

    $\displaystyle 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

    $\displaystyle \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

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle \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 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

    $\displaystyle \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

    $\displaystyle \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

    $\displaystyle \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 Exercises 7 and 8.


next up previous
Next: Keplerian orbits Up: Newtonian gravity Previous: Potential due to uniform
Richard Fitzpatrick 2016-03-31