June 2004
CLASSICAL MECHANICS
PROBLEMS FROM THE
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
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
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
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.