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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 . 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 , where 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 is

where is the Earth's radius, and its surface gravitational acceleration. (From Lamb 1923.)

3. Assuming that the Earth is a sphere of radius , and neglecting air resistance, show that a particle that starts from rest a distance from the Earth's surface will reach the surface with speed after a time , where 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

where is the radius of the sphere, and the gravitational acceleration at its surface.

5. Consider an isolated system consisting of point objects interacting via gravity. The equation of motion of the th object is

where and are the mass and position vector of this object, respectively. Moreover, the total potential energy of the system takes the form

Write an expression for the total kinetic energy, . Demonstrate, from the equations of motion, that is constant in time.

6. Consider a function of many variables . Such a function that satisfies

for all , and all values of the , is termed a homogeneous function of degree . Prove the following theorem regarding homogeneous functions:

7. Consider an isolated system consisting of point particles interacting via attractive central forces. Let the mass and position vector of the th particle be and , respectively. Suppose that magnitude of the force exerted on particle by particle is . Here, measures some constant physical property of the th particle (e.g., its electric charge). Show that the total potential energy of the system is written

Is this a homogeneous function? If so, what is its degree? Demonstrate that the equation of motion of the th particle can be written

(This is shorthand for , , etc., where the , , , for , are treated as independent variables.) Use the mathematical theorem from the previous exercise to show that

where , and is the total kinetic energy. This result is known as the virial theorem. Demonstrate that when the system possesses no virial equilibria (i.e., states for which does not evolve in time) that are bounded.

8. Demonstrate that the gravitational potential energy of a spherically symmetric mass distribution of mass density that extends out to can be written

Hence, show that if the mass distribution is such that

where , then

where 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

for such a cluster. Suppose that the stars in a given cluster are uniformly distributed throughout a spherical volume. Show that

where is the mean stellar velocity, and 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 and position vector , interacting via gravity. Show that, for such a system, the virial theorem implies that

where is a constant, , and the 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

where and are the equilibrium values of and . Finally, show that if the mass density within the star varies as , where is the radial distance from the geometric center, and where , then

where and are the stellar mass and radius, respectively. See Exercises 7 and 8.

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