- 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.
- A mass is attached to the mid-point of a stretched string of negligible mass, length ,
and uniform tension . Let be the transverse displacement of the mass from its
equilibrium position. Show that the displacement executes simple harmonic oscillation at the
- 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
if the springs are connected in parallel, and
if the springs are connected in series.
- A mass is suspended at the end of a uniform spring of unstretched length and spring constant . If the
mass of the spring is and the velocity of an element of its length is proportional to its
distance from the fixed end of the spring, show that the kinetic energy of this element
where is the velocity of the suspended mass. Hence, by integrating over the length of the spring, show
that its total kinetic energy is
. Finally, deduce, from energy conservation arguments, that the
angular oscillation frequency of the system is given by
[From Pain 1999.]
- A body of uniform cross-sectional 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 weight of the displaced liquid), show
that the period of small amplitude oscillations about the equilibrium position is
- A U-tube of constant cross-sectional area consists of a horizontal section connected at either end to two vertical sections.
Suppose that the tube is filled with an incompressible liquid of mass density . Let the total length of the liquid column be . (Where
exceeds the length of the horizontal section.) Suppose that the surface of the liquid in one of the vertical sections is initially displaced (vertically) a
small distance from its equilibrium position. Show that the surface displacement subsequently executes simple harmonic oscillation at the
, where is the acceleration due to gravity.
- A particle of mass slides in a frictionless semi-circular depression in the
of radius . Find the angular frequency of small amplitude oscillations
about the particle's equilibrium position, assuming that the oscillations
are essentially one-dimensional, so that the particle passes through
the lowest point of the depression during each oscillation cycle.
[From Ingard 1988.]
- Imagine a straight tunnel passing through the center of the Earth, which is regarded as
a sphere of radius and uniform mass density. A particle is dropped into the tunnel from the surface.
Show that the particle undergoes simple harmonic oscillation at the angular
, where is the gravitational acceleration at Earth's surface.
(Hint: The gravitational acceleration at a point inside a spherically symmetric mass distribution
is the same as if all of the mass interior to the point were concentrated at the center, and
all of the mass exterior to the point were neglected.) Estimate how long it takes the particle to
reach the other end of the tunnel.
- Assuming that the tunnel is smooth (i.e., ignoring
friction), show that motion is simple harmonic even if the tunnel does not pass through the
center of the Earth, and that the travel time from one end of the tunnel to the other is the same as before.
- A particle executing simple harmonic oscillation in one dimension has speeds and at displacements
and , respectively, from its equilibrium position.
- Show that the period of the motion can be written
- Show that the amplitude of the motion can be written
- If a thin wire is twisted through an angle then a restoring
develops, where is known as the torsional
force constant. Consider a so-called 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
- 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
- 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 uniform disk of mass and radius rolls without slipping over a rough horizontal surface. Suppose that a small
mass is attached to the edge of the disk. Show that the angular frequency of small amplitude oscillations of the disk
about its equilibrium position is
- A body hung at the end of a light vertical spring stretches the spring statically to twice its
original length. The system can be set into motion either as a simple pendulum or as a
mass-spring oscillator. Determine the ratio between the periods of these motions. (In the
pendulum mode of motion, assume the length of the spring to be constant.) [From Ingard 1988.]
- Show that the average speed of a particle executing simple harmonic oscillation is times the maximum speed. [From Ingard 1988.]
- A particle of mass executes one-dimensional simple harmonic oscillation such that its instantaneous coordinate is
- Find the average values of , , , and
over a single cycle of the
- Find the average values of the kinetic and potential energies of the
particle over a single cycle of the oscillation.
- A particle executes two-dimensional simple harmonic oscillation such that its instantaneous coordinates in the -
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
Find the subsequent motion, , assuming that
its amplitude remains small.
- Suppose, instead, that
. Find the subsequent motion.
- Suppose, finally, that
. Find the subsequent motion.
- Demonstrate that
is a constant of the motion of a simple pendulum whose time evolution equation
is given by Equation (1.50). (Do not make the small-angle approximation.)
that the amplitude of the motion,
, can be written
- Demonstrate that the period of the motion is determined by
where is the period of small-angle oscillations.
- Making use of the substitution
, show that the
previous expression transforms to
Hence, deduce that