Continuing to Payment will take you to apayment page

Purchasing
SparkNotes PLUS
for a group?

Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99$18.74/subscription + tax

Subtotal $37.48 + tax

Save 25%
on 2-49 accounts

Save 30%
on 50-99 accounts

Want 100 or more?
Contact us
for a customized plan.

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews June 6, 2023May 30, 2023

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20%Discount

This is not a valid promo code.

Discount Code(one code per order)

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

No URL

Copy

Members will be prompted to log in or create an account to redeem their group membership.

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

Up to this point we have only examined the special case in which the net
force
on an oscillating particle is always proportional to the displacement of
the
particle. Oftentimes, however, there are other forces in addition to
this
restoring
force,
which
create
more complex oscillations. Though much of the study of this motion lies
in the
realm of differential equations, we will give at least an introductory
treatment
to the topic.

Damped Harmonic Motion

In most real physical situations, an oscillation cannot go on
indefinitely.
Forces such as friction and air resistance eventually dissipate energy
and
decrease both the speed and amplitude of oscillation until the system is
at rest
at its equilibrium point. The most common dissipative force encountered
is a
damping force, which is proportional to the velocity of the object,
and
always acts in a direction opposite the velocity. In the case of the
pendulum,
air resistance always works against the motion of the pendulum,
counteracting
the gravitational force, shown below.

We denote the force as F_{d}, and relate it to the velocity of the
object:
F_{d} = - bv, where b is a positive constant of proportionality,
dependent on
the
system. Recall that we generated the differential equation for simple
harmonic
motion using Newton's Second
Law:

- kx = m

We must add our damping force to the left side of this equation:

- kx - b = m

Unfortunately generating a solution to this equation requires more
advanced
mathematics than just calculus. We will simply state the final solution
and
discuss its implications. The position of the damped oscillating
particle
is
given by:

x = x_{m}e^{-bt/2m}cos(σ^{â≤}t)

Where

σ^{â≤} =

Clearly this equation is a complicated one, so let's take it apart piece
by
piece. The most notable change from our simple harmonic equation is
the
presence of the exponential function, e^{-bt/2m}. This function
gradually
decreases the amplitude of the oscillation until it reaches zero. We
still have
our cosine function, though we must calculate a new angular
frequency.
As we can
tell
by
our equation for σ^{â≤}, this frequency is smaller than with
simple
harmonic motion--the damping causes the particle to slow down,
decreasing
the
frequency and increasing the period. Shown below is a graph of typical
damped
harmonic motion:
We can see from the graph that the motion is a superposition of an
exponential
function and a sinusoidal function. The exponential function, on both
the
positive and negative sides, acts as a limit for the amplitude of the
sinusoidal
function, resulting in a gradual decrease of oscillation. Another
important
concept from the graph is that the period of the oscillation does not
change,
even though the amplitude is constantly decreasing. This property
allows
grandfather clocks to work: the pendulum of the clock is subject to
frictional
forces, gradually decreasing the amplitude of the oscillation but, since
the
period remains the same, it can still accurately measure the passage of
time.

The study of damped harmonic motion could be a chapter in and of itself;
we have
simply given an overview of the concepts that give rise to this complex
motion.

Resonance

The second example of complex harmonic motion we will examine is that of
forced
oscillations and resonance. Up to this point we have only looked at
natural
oscillations: cases in which a body is displaced and then released,
subject only
to natural restoring and frictional forces. In many cases, however, an
independent force acts on the system to drive the oscillation. Consider
a
mass
spring system in which the mass oscillates on the spring (as usual) but
the wall
to which the spring is attached oscillates at a different frequency, as
shown
below:

Usually the frequency of the external force (in this case the wall)
differs from
the frequency of the natural oscillation of the system. As such, the
motion is
quite complex, and can sometimes be chaotic. Considering the
complexity,
we
will omit the equations governing this motion, and simply examine the
special
case of resonance in forced oscillations.