June 2004

 

CLASSICAL MECHANICS

 

PROBLEMS FROM THE UNIVERSITY OF VIRGINIA PH.D. PHYSICS QUALIFYING EXAMINATIONS

 

 

1.         Protons, mass m and charge e, are given (nonrelativistic) energy E and sent as a beam to scatter from much heavier nuclei of charge Ze. The experiment shows that the differential cross section agrees with the Rutherford cross section for scattering angles less than some critical angle qc, but departs rapidly from it for larger angles. This is due to the presence of a strong force between the incoming proton and the nucleus from which it scatters; this force is of short range and effectively only comes into play when the proton touches the nucleus. Assuming that the nucleus is spherical, find its effective radius in terms of the given parameters. [Hint: The equation for the orbit of a particle of mass m and angular momentum L moving under the influence of a central force of magnitude k/r2 is
                       
where q¢ is an arbitrary constant that specifies the orientation of the orbit.]

 

2.         A bead of mass m can slide without friction along a horizontal rod fixed in place inside a large box. The bead is connected to the walls of the box by two large identical massless springs of spring constant k as sketched in the figure, and the entire box is rotated about a vertical axis through its center with angular speed w.

                       


(a)        Write down the Lagrangian using the distance r from the bead to the center of the rod as a generalized coordinate.

(b)        What is the condition for the bead to be in equilibrium off the center of the rod? Please comment on whether this equilibrium is stable, neutral, or unstable.

(c)        Compute the time dependence of the radial position of the bead r(t) assuming that r(0) = l (still within the box) and dr(0)/dt = 0 (the bead starts at a given value of r with no initial radial velocity). Note that there are two possible regimes, so please state precisely the corresponding conditions of validity.

 

3.        A pendulum consists of a thin rod of length  and mass m suspended from a pivot  in the figure to the right. The bob is a cube of side L and mass M, attached to the rod so that the line of the rod extends through the center of the cube, from one corner to the diametrically opposite corner (dashed line).

                          

(a)        Locate the distance of the center of mass from the point of support.

(b)        Find the moment of inertia I of the (entire) pendulum about the pivot point.
(Hint: obviously it is too hard to find the moment of inertia of a uniform cube about an arbitrary axis through its center of mass by integrating directly, so there must be some simple trick…)

(c)        Write down the equation of motion in terms of I and any other relevant parameters.

(d)        Find the frequency of small oscillations.

 

4.         A bead of mass m slides without friction, under the influence of gravitation, down a wire that has the form of a simple curve in a vertical plane—say the  plane. The bead starts at the point  and ends at the point .

Find the curve (brachistochrone) joining these points for which the bead descends in the least time, if it starts with zero initial velocity.

 

5.         A space station orbits Earth on a circular trajectory. At some moment the captain decides to change the trajectory by turning on the rocket engine for a very short period of time. During the time the engine was on, it accelerated the station in its direction of motion. As a result, the station speed increased by a factor of a. Provide the conditions, in terms of a, that the new trajectory is elliptic, parabolic or hyperbolic. Justify your answers.

 

6.         A particle of mass m, total energy E, and angular momentum L is moving in a central potential of the form
.                         
where
a and b are positive constants. What is the condition for the particle motion to be bounded? For the case that the motion is bounded, compute the angular displacement Dj between two subsequent passages of the perihelion (the point r = rmin). What is the most general condition in terms of a and b are, m, E, and L for the particle trajectory to be closed? (A trajectory is said to be closed if the radius-vector of the particle visits its original position more than once.)

 

7.         An Earth satellite of mass m is placed in a circular orbit.  Due to the fact that space is not an ideal vacuum, the satellite is subject to an extra frictional force F, which we assume is linear in the satellite velocity v, i.e. F = -Av where A is a constant.  This force dissipates the satellite energy so that eventually the spacecraft hits the ground, which determines its lifetime (in reality the drag constant A is a function of altitude and satellites often burn in upper layers of atmosphere, but we will ignore this).  Assuming that the energy dissipated by friction during one full revolution is much smaller than the total energy, compute the lifetime of the satellite.  Assume the Earth can be modeled by a sphere, the initial radius of the orbit is 10 times as big as the Earth radius, and the satellite is much lighter than the Earth.

 

8.         Two identical point masses m are connected by a spring of constant k and unstretched/ uncompressed length a.

(a)        Write the Lagrangian of the system in terms of the 3-dimensional coordinates of the masses.

(b)        Transform the Lagrangian to an appropriate set of generalized coordinates.

(c)        List the conserved quantities, giving defining expressions.

(d)        If the system has large internal angular momentum, what is the equilibrium separation between the masses?

(e)        Find the frequency of small oscillations about this equilibrium in the limit of large angular momentum.

 

9.         A thin uniform rod of length a and mass m slides without friction with its two ends in contact with the inside of a vertical hoop of diameter  d (a<d) in the gravitational field of the earth.

                          

(a)        Write the Lagrangian.

(b)        What is the angular frequency for small oscillations about equilibrium?  How does it behave as a/d®0 and as a/d®1?

 

10.       A free uniform disk lying on a frictionless horizontal surface is rotating about its (vertical) symmetry axis with angular velocity w. Its center of mass is at rest.  Suddenly a point on its circumference is fixed.  Calculate the subsequent angular velocity.

 

11.       A spacecraft is in a circular low-Earth orbit directly above the Equator, mean altitude 300 Km above the Earth's surface.  The orbit must be transformed to a circular geosynchronous orbit (that is, one that keeps the spacecraft directly above the same point on the Equator).

(a)        What is the radius of the geosynchronous orbit?

(b)        The pilot wishes to attain the orbit change by two applications of the rockets (the minimum possible).  What sort of intermediate orbit will the spacecraft be in, after the first period of acceleration?

(c)        What is the change-of velocity that must be applied to transform the initial circular orbit to the intermediate orbit?  That is, what change of speed is necessary and in what direction?

(d)        What is the change-of-velocity necessary to transform the intermediate orbit into the final circular geosynchronous orbit?  (Again specify change-of-speed and direction.)
You may, if you wish, assume the periods of acceleration have the form of instantaneous impulses.

 

12.       A spherical pendulum consists of a mass m hung by a massless string of length R from a fixed point.  q is the angle the string makes from the vertical, where q = 0 is down, and f is the azimuthal angle of the string.

(a)        Write the Lagrangian for the motion of the mass.  You may assume that the string is always taut.

(b)        With what velocity and in what direction must the mass be set in motion to make a circular orbit with q = 30°?

(c)        If the mass is launched with a slight error and it oscillates about q = 30° what is the angular frequency of the oscillation in q ?

 

13.       When a car collides with a fixed obstacle or another car, the collision is highly inelastic (by design).  The force required to crush a typical car to a depth x behaves something like a "one-way spring",
                          
The ratio of spring constant to mass, k/m, is roughly constant across the spectrum of passenger cars and has the typical value 385 sec-2.
Now imagine that a car of mass m travelling East at 10 m/s collides head-on with one of mass 2m travelling West at 10 m/s.  Assume they are on dry pavement with coefficient of sliding friction
m = 1, and that both have locked their brakes prior to the crash.

(a)        Calculate the forces on each car, including friction, assuming the motion remains collinear.

(b)        Write down the equations of motion of both cars.

(c)        How long does it take for the cars to come to rest relative to each other?

(d)        How long does it take for the cars to come to rest relative to the road, and how far do they slide?

 

14.       In terms of the (vector) displacement field  describing elastic deformations from equilibrium, the equation of motion of an isotropic, elastic solid is

                          

where Y is Young's modulus and ? is the Poisson ratio (both are positive quantities).

(a)        Show that this equation describes three distinct types of wave and give expressions for each type, in terms of  and the standard vector operators.

(b)        Derive expressions for the phase velocities of each type of wave.

(c)        From the results of b) find the group velocities of each type of wave.

(d)        A remote Pacific island, aboutlongitude west of Perth, Australia is the antipode of Charlottesville, VA.  A submerged volcanic explosion at this location produces seismic waves that travel through the Earth and are detected at the Environmental Sciences Department at UVa.  Given that the Earth's core is liquid, characterize the type(s) of wave detected here, and compare qualitatively their time(s) of arrival.

 

15.       A bead of mass m is free to slide along a smooth rigid circular wire which is forced to rotate with a constant angular velocity about the vertical diameter.  Consider a small displacement from the position (not on axis of rotation) of stable equilibrium for an angular velocity  and obtain the condition for, and frequency of simple harmonic motion.

                          

 

16.       A square lamina, sides 2a, mass m, is lying on a table when struck on a corner by a bullet of mass m with velocity v parallel to one of the edges of the lamina (inelastic).  Find the subsequent angular velocity of the lamina.

 

17.       Two rings of equal mass M and radius R are rigidly fastened together at a point on their periphery so that their diameters form an angle a.  If they are free to swing      as a pendulum in a vertical plane, find the torque  tending to change the angle a for small motion about the position of equilibrium.

                          

 

18.       A bead is constrained to move along the smooth, uniform, conical spiral as shown.  Obtain the equation of motion of the bead if the spiral is rotated about its own axis with a constant angular velocity .  For what value of  will the particle maintain its vertical position?

                          

 

19.       A solid cylinder of radius b has a cylindrical hole of radius  cut out of it.  The hole is centered at a distance c from the center of the cylinder.  This cylinder is at rest on top of a large perfectly rough, fixed cylinder of radius R as shown.  For what values of R is the equilibrium position shown stable, and what will be the frequency of small oscillations about this equilibrium position?

                          

 

20.       A particle of mass m is placed in a smooth uniform tube of mass M and length .  The tube is free to rotate about its center in a vertical plane.  The system is started from rest with the tube horizontal and the particle a distance  from the center of the tube. For what length of the tube will the particle leave the tube when  is a maximum and ?  Your answer should be in terms of w and  .

                          

 

21.       Two particles of equal mass m interact according to the potential,

                           V = 0,  r > a

                           V = -  r < a.

Initially the particles are separated by some distance r > a, one is at rest and the other has velocity v0.  Calculate the differential cross section for scattering.

 

22.       (a)           State Euler's equations of motion, defining all terms precisely.

(b)        Define Euler's angles for rigid body motion and express the body components of angular velocity in terms of them.

(c)        A symmetrical top, (the moments of inertia are  ) of mass M spins with one point fixed in the earth's gravitational field.  Its center of mass is a distance b from the fixed point.  Express Euler's equations for the top in terms of Euler's angles and impose the solution of a uniformly precessing top without nutation, i.e. the angle between the figure axis and the vertical direction remains constant.  Substituting this solution into the equations of motion, obtain a condition between  for this solution to be valid.

 

23.       Derive Rutherford's formula for the center of mass differential scattering cross section of two particles interacting under a repulsive Coulomb force.  What qualitative differences would you expect if one of the particles were an impenetrable charged sphere of finite radius?

 

24.       A double pendulum has equal lengths, but the upper mass is much greater than the lower.  Obtain the exact Lagrangian for motion in a vertical plane, and then make the approximation of small motion.  What are the resonant frequencies of the system?  What is the resultant motion if the system initially at rest is subjected at time t = 0 to a small impulsive force applied horizontally to the upper mass?

 

25.       (a)        In terms of its relationship to the kinetic energy, derive an expression for the inertia tensor of a rigid body relative to an arbitrary Cartesian coordinate system.  Show how to diagonalize this matrix and prove the reality of the eigenvalues and eigenvectors obtained.

(b)        What is the moment of inertia about the diagonal of a homogeneous cube?

 

26.       Consider a classical treatment of the small oscillations of the atoms about equilibrium in a linear tri-atomic molecule.  Two of the masses are equal and at equilibrium are located a distance, a, from the third.  Making a reasonable assumption about the functional form of the potential for small motions, obtain the frequencies of vibration.  Describe the mode of vibration for each of the eigenfrequencies.

                          

 

27.       A particle of mass m is observed to move in a central force field in a circular orbit of radius  with angular velocity w.

(a)        State the relation between m, r0, w, and , where V(r) is the potential.

(b)        Consider small radial perturbations from the circular orbit, and describe them by defining variables r and f so that the polar coordinates of the particle are r = r0 + r and q = wt + f. Expand the potential function in a Taylor series in r and write the Lagrangian, ignoring terms higher than second order in the perturbations r and f and their time derivatives.

(c)        Derive the equations of motion and make use of part a) to show that stable oscillations in r will result if

                       

 

28.       A rigid body rotates freely about its center of mass.  There are no torques.  Show by means of Euler's equations that. if all three principal moments of inertia are different, then the body will rotate stably about either the axis of greatest moment of inertia or the axis of least moment of inertia, but that rotation about the axis of intermediate moment of inertia is unstable.

 

29.       If a particle is projected vertically upward from a point on the earth's surface at northern latitude l, show that it strikes the ground at a point  to the west (neglect air resistance and consider only small vertical heights).

 

30.       A thin hoop of radius R and mass M is allowed to oscillate in its own plane (a vertical plane) with one point of the hoop fixed.  Attached to the hoop is a small mass M which is constrained to move (in a frictionless manner) along the hoop.  Consider only small oscillations and show that the eigen frequencies are

                          

 

31.       Show that the angular deviation e of a plumb line from the true vertical at a point on the earth's surface at latitude l is

                          

where  is the radius of the earth.

 

32.       Two discs of radius R and mass M are connected by an axle of radius r and mass m.  A pendulum of length  with a bob of mass m is suspended from the midpoint of the axle.  If the wheels and the axle can roll without changing the length of the pendulum, find the equations of motion of the system if it rolls down a slope of angle a.

 

33.       A point on the circumference of a uniform rotating disc, whose center is at rest, is suddenly fixed.  Show that its new angular speed is one-third of its former value.

 

34.       Two bodies of equal mass m are connected by a smooth, inextensible string which passes through a hole in a table.  One body can slide without friction on the table; the other hangs below the hole, and moves only along a vertical line through the hole.  Using polar coordinates (r,q) for the body on the table write the Lagrange equations for the system, reduce them to a single second order differential equation, and integrate it once to obtain an energy equation.  Find the equation whose roots yield the maximum and minimum values of r. Imagine the string is long enough so that the hanging body does not hit the hole.

 

35.       A bar of length 2L, mass m, slides without friction on a horizontal plane.  Its velocity is perpendicular to the axis of the bar.  It makes a fully elastic impact (energy conserved) with a fixed peg at a distance a from the center of the bar.  Using conservation of energy, momentum and angular momentum, find the final velocity of the center of mass.

 

36.       A uniform hoop of mass M and radius a can roll without sliding on a horizontal floor.  A small particle of mass m is constrained to slide without friction on the inside rim of the hoop.  Introduce two coordinates that specify the instantaneous state of the system and calculate the Lagrangian in terms of these coordinates.  One of the coordinates should be chosen so that it is zero when the particle is at floor level.  Assuming that the particle never rises far off the floor, write down the equations of motion and show that the general motion consists of an oscillation of period

                          

superimposed on a uniform translation.

 

37.       A cone, symmetrical about its axis and with principal moments of inertia A, A, and C, is supported at its vertex and is spinning about its axis with period  and its axis is rotating about the vertical with a fixed angle of inclination a, and period . Determine the instantaneous angular velocity of the cone and its angular momentum.  Show that if the center of mass is at a distance l from the vertex and m is the mass of the cone the above motion is only possible if .  Determine the reaction at the support of the vertex

 

38.       A uniform heavy rod is held vertically by a pivot at its lower end and there is a red spot painted on its upper end.  A very slight disturbance makes it swing down around the pivot point, and it is released just as it reaches the downward vertical position.  Describe the subsequent motion in detail. How far has the center of mass dropped at each time when the bar returns to a vertical orientation with the red spot up?

 

39.       A particle is projected with velocity v on a smooth horizontal plane at latitude l on the earth which has angular velocity w. Show that the particle will travel in a circle and find the radius of the circle and the period of the motion in terms of the given parameters.  Assume that v is small enough that the radius of the orbit is negligible compared to the radius of the earth.

 

40.       A particle moves under the action of an attractive central force which is proportional to the displacement from the origin.  The Lagrangian in cylindrical coordinates is:



(a)        What are two constants of the motion?

(b)        Find the maximum and minimum values of r as functions of the constants of motion.

(c)        Under what conditions is the orbit a circle?

(d)        Under what conditions is the orbit a straight line?

(e)        Write down a formal solution to the problem.

(f)         What is the frequency of the motion?

 

41.       A pendulum consists of a weightless rod of length  and bob of mass . The support of the pendulum is connected to a mass .  The mass  is connected by a spring (spring constant = k) to a rigid wall.  The mass  slides on a frictionless, horizontal plane. The pendulum swings in a plane perpendicular to the wall.  Find the frequencies of the normal modes and discuss the method of finding the motion corresponding to each of the normal modes.
                          

 

42.       A heavy particle is constrained to move on the inside surface of a smooth spherical shell of inner radius a. Gravity is acting vertically downward.  Initially the particle is projected with a horizontal velocity  from a point which is at a depth b below the center of the sphere.  Find the depth z below the center of the sphere when the particle is again moving horizontally.  Briefly describe the motion.