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 A mass stands on a platform that executes simple harmonic oscillation in a vertical
direction at a frequency of
. Show that the mass loses contact with the platform
when the displacement exceeds
. [From Pain 1999.]
 A small body rests on a horizontal diaphragm of a loudspeaker
that is supplied with an alternating current of constant amplitude but variable
frequency. If the diaphragm executes simple harmonic oscillation in the vertical
direction of amplitude
, at all frequencies, find the greatest
frequency (in hertz) for which the small body stays in contact with the diaphragm.
 Two light springs have spring constants
and
, respectively, and are used in a vertical
orientation to support an object of mass
. Show that the angular frequency of small amplitude oscillations about the equilibrium state
is
if the springs are connected in parallel, and
if the springs are connected in series.
 A body of uniform crosssectional area
and mass density
floats in a liquid
of density
(where
), and at equilibrium displaces a volume
. Making use of Archimedes' principle (that the buoyancy force acting on a partially
submerged body is equal to the mass of the displaced liquid), show
that the period of small amplitude oscillations about the equilibrium position is
 A particle of mass
slides in a frictionless semicircular depression in the
ground
of radius
. Find the angular frequency of small amplitude oscillations
about the particle's equilibrium position, assuming that the oscillations
are essentially onedimensional, so that the particle passes through
the lowest point of the depression during each oscillation cycle.
 If a thin wire is twisted through an angle
then a restoring
torque
develops, where
is known as the torsional
force constant. Consider a socalled torsional pendulum,
which consists of a horizontal disk of mass
, and moment of inertia
, suspended at its
center from a thin vertical wire of negligible mass and length
, whose other end is attached to a fixed
support. The disk is free to rotate about a vertical axis passing through the suspension point, but such rotation twists the wire. Find the frequency of torsional oscillations of the disk about its
equilibrium position.
 A circular hoop of diameter
hangs on a nail. What is the period of its small amplitude oscillations? [From French 1971.]
 A compound pendulum consists of a uniform bar of length
that pivots about one of its
ends. Show that the pendulum has the same period of oscillation as a simple pendulum of
length
.
 A compound pendulum consists of a uniform circular disk of radius
that is
free to turn about a horizontal axis perpendicular to its plane. Find the position
of the axis for which the periodic time is a minimum.
 A laminar object of mass
has a moment of inertia
about a perpendicular axis passing
through its center of mass. Suppose that the object is converted into a compound pendulum by
suspending it about a horizontal axis perpendicular to its plane. Show that the minimum effective
length of the pendulum occurs when the distance of the suspension point from the
center of gravity is equal to the radius of gyration,
.
 A particle of mass
executes onedimensional simple harmonic oscillation such that its instantaneous
coordinate is
Find the average values of
,
,
, and
over a single cycle of the
oscillation. Find the average values of the kinetic and potential energies of the
particle over a single cycle of the oscillation.
 A particle executes twodimensional simple harmonic oscillation such that its instantaneous coordinates in the

plane are
Describe the motion when (a)
, (b)
, and (c)
.
In each case, plot the trajectory of the particle in the

plane.
 An LC circuit is such that at
the capacitor is uncharged and a
current
flows through the inductor. Find an expression for the
charge
stored on the positive plate of the capacitor as a function of time.
 A simple pendulum of mass
and length
is such that
and
. Find the subsequent motion,
, assuming that
its amplitude remains small. Suppose, instead, that
and
. Find the subsequent motion. Suppose, finally, that
and
. Find the subsequent motion.
 Demonstrate that
is a constant of the motion of a simple pendulum whose time evolution equation
is given by Equation (50). (Do not make the small angle approximation.) Hence, show
that the amplitude of the motion,
, can be written
Finally, demonstrate that the period of the motion is determined by
where
is the period of small angle oscillations. Verify that
as
. Does the period increase, or decrease, as the amplitude
of the motion increases?
Next: Damped and Driven Harmonic
Up: Simple Harmonic Oscillation
Previous: Compound Pendula
Richard Fitzpatrick
20130408