Download Abby AII1 C083 Ye, Zi Topic: Energy in Simple harmonic motion

Abby
Ye, Zi
C083
AII1
Topic: Energy in Simple harmonic motion; damped,
forced, free oscillations and resonance
A. Introduction
Simple harmonic motion (SHM) is the motion of a simple harmonic oscillator, a
motion that is neither driven nor damped. The motion is periodic - as it repeats
itself at standard intervals in a specific manner - and sinusoidal, with constant
amplitude; the acceleration of a body executing SHM is directly proportional to
the displacement of the body from the equilibrium position and is always directed
towards the equilibrium position.
The motion is characterized by its amplitude (which is always positive), its period,
the time for a single oscillation, its frequency, the reciprocal of the period (i.e. the
number of cycles per unit time), and its phase, which determines the starting
point on the sine wave. The period and frequency are constants determined by
the overall system, while the amplitude and phase are determined by the initial
conditions (position and velocity) of that system.
A typical example of a system that undergoes simple harmonic motion is a
spring-mass system, which is a mass attached to a spring.
Simple harmonic motion shown both in real space and phase space. The orbit is
periodic. (Here the velocity and position axes have been reversed from the
standard convention in order to align the two diagrams)
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If the spring is unstretched, there is no net force on the mass - in other words, the
system is in equilibrium. However, if the mass is displaced from equilibrium, the
spring will exert a restoring force, which is a force that tends to restore it to the
equilibrium position. In the case of the spring-mass system, this force is the
elastic force, which is given by Hooke's Law, F = − kx, where F is the restoring
force, x is the displacement, and k is the spring constant.
Any system that undergoes simple harmonic motion exhibits two key features.
1. When the system is displaced from equilibrium there must exist a restoring
force that tends to restore it to equilibrium.
2. The restoring force must be proportional to the displacement, or
approximately so.
The spring-mass system satisfies both.
Once the mass is displaced it experiences a restoring force, accelerating it,
causing it to start going back to the equilibrium position. As it gets closer to
equilibrium the restoring force decreases; at the equilibrium position the restoring
force is 0. However, at x=0, the mass has some momentum due to the impulse of
the force that has acted on it; this causes the mass to shoot past the equilibrium
position, in this case, compressing the spring. The restoring force then tends to
slow it down, until the velocity reaches 0, whereby it will attempt to reach
equilibrium position again.
As long as the system does not lose energy, the mass will continue to oscillate
like so; thus, the motion is termed periodic motion. Further analysis will show that
in the case of the spring-mass system the motion is simple harmonic.
B. Analysis
Energy in Simple harmonic motion: E = (1/2)kA2
The energy E in the system is proportional to the square of the amplitude. It is a
continuously changing mixture of kinetic energy and potential energy.
For any object executing simple harmonic motion with angular frequency , the
restoring force F = -m2x obeys Hooke's law, and therefore is a conservative
force. We can define a potential energy U = (1/2) m2x 2, and the total energy of
the object is given by E = (1/2)m2A2
The simple harmonic motion of a mass on a spring is an example of an energy
transformation between potential energy and kinetic energy. In the example
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below, it is assumed that 2 joules of work has been done to set the mass in
motion.
Damped oscillations: produced by resistive forces which dissipate the energy of
the vibrating system
Damping is a condition in which external force operates in such a manner that it
impedes the motion of oscillatory body. As a matter of fact, all real time harmonic
motions that we consider to be simple are
actually damped SHM. We consider them
SHM only as an approximation. The motion
of a block hanging from a spring, for
example, is not SHM as air works to
oppose the motion – at every instant. As a
consequence, the span (amplitude) of the
motion keeps decreasing every cycle.
Diminishing amplitude is the characterizing
feature of damped oscillation. A typical
displacement – time plot looks as shown
here.
We know that in reality, a spring won't oscillate for ever. Frictional forces will
diminish the amplitude of oscillation until eventually the system is at rest.
We will now add frictional forces to the mass and spring. Imagine that the mass
was put in a liquid like molasses. Your lab instructor will not like it when they see
their nice metal weight coated with a thick layer of ants in the morning. Be that as
it may, when the mass is inside the molasses, it'll hardly oscillate at all.
Here's a plot of of an example of such a function
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An example of light damping: the amplitude of vibration of the oscillation to
decrease gradually. Critical damping causes the displacement possible, without
any oscillation of the object. Over-damping also causes an exponential reduction
in displacement, but over a greater time than for critical damping.
Forced oscillations: a periodic driving force is applied to a system which is
capable of vibration.
The oscillations we have discussed up to now are free oscillations in which the
system is given some energy, then left alone. For instance, you could pull a child
on a swing up to a certain height, then let go and wait for the motion to die away.
But this is not the only possibility; we could also repeatedly push the swing at any
frequency we like and watch what happens. In this case we say that we
have forced oscillations. There are now two frequencies in the problem:
the natural frequency of the free oscillations, and the driving frequency of
the forced oscillations. This means that you will have to resist the urge to use the
formula
whenever you encounter a frequency. If the frequency in
question is the driving frequency, there is no formula for it; it is simply set by the
design of the driving circuit.
It is clear from the discussion so far that most of artificial oscillation system tends
to cease as damping is part of the natural set up. There can be various sources
of damping force, but friction is one common source. There can be air resistance
or resistance at the fixed hinge from which oscillating part is hung.
It is imperative that we supply appropriate energy (force) to compensate for the
loss of energy due to damping. To meet this requirement, the oscillating system
is subjected to oscillatory external force.
Free oscillations: no resistive forces acting on the oscillation system.
Harmonic oscillation is the periodic process in which the parameter of interest is
varied as sinus or cosine. If there is no time-dependent force applied to the
oscillator, then it is called a free oscillator.
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Resonance: Resonance occurs when the natural frequency of vibration of an
object is equal to the driving frequency, giving a maximum amplitude of vibration.
The resonance is an interesting feature of oscillation. This phenomenon attracts
interest as it makes possible to achieve extra-ordinary result (material failure of
large structure) with small force! Resonance also explains why earthquake
causes differentiating result to different structures – most devastating where
resonance occurs!
The condition for maximum amplitude is obtained by differentiating amplitude
function with respect to applied frequency “ωe” and setting the resulting
expression equal to zero.
If damping is absent, then the amplitude function is maximum when applied
frequency is equal to natural frequency. The amplitude is infinite in such case.
However, damping is always present in actuality and as such resonance
amplitude is finite. For small value of damping constant, the resonance frequency
is close to natural frequency and resonance is sharply defined. As the damping
increases, resonance amplitude is reduced. The plots of amplitude – frequency
curves outline the features of resonance amplitude as shown in the figure. Note
that resonance becomes flatter as damping force increases.
Resonance underlines certain fundamental aspects of oscillatory system. First, it
demonstrates that oscillation is a storing mechanism of energy. Second, it
demonstrates that the energy can be supplemented in a constructive manner to
increase the energy of the oscillating system to an extra-ordinary level. We can
experience both these aspects easily by observing motion of swings in a nearby
park. Ask a very strong adult to apply the most of his/her power to swing in one
go and compare the result with that of a child who produces large swings with no
such power. What is the difference? The child begins with small swing and
synchronizes subsequent jerk with the oscillation. Each time the energy of the
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system is increased by the small amount associated with each jerk. The key,
here, is the synchronization (timing of external force) and periodicity with which
the jerk is applied to the swing.
Resonance is not specific to mechanical oscillatory system. The concept is
equally applicable to electrical system and waves in general. Tunning of a
particular radio station at a specific frequency is one of the most striking
applications of this phenomenon.
Many vibrating systems like atomic arrangements, vibrating strings etc have
multiple natural frequencies. This means that these systems can be subjected to
resonance at more than one frequency.
C. Examples
Example 1
Question: A block of mass m=4 kg is attached to a spring, and undergoes simple
harmonic motion with a period of T=0,35s. The total energy of the system is
E=2.5J. What is the force constant of the spring? What is the amplitude of the
motion?
Answer: The angular frequency of the motion is
Now,
for a mass on a spring. Rearrangement of this formula yields
The total energy of a system executing simple harmonic motion is
Rearrangement of this formula gives
Thus, the force constant is
and the amplitude is
.
.
Example 2
A particle that hangs from a spring oscillates with an angular frequency of 2
rad/s. The spring is suspended from the ceiling of an elevator car and hangs
motionless (relative to the car) as the car descends at a constant speed of 1.5
m/s. The car then suddenly stops. Neglect the mass of the spring.
(a) With what amplitude does the particle oscillate?
(b) What is the equation of motion for the particle? (Choose the upward direction
to be positive.)
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Solution:
(a) When traveling in the elevator at constant speed, the total force on the mass
is zero. The force exerted by the spring is equal in magnitude to the gravitational
force on the mass, the spring has the equilibrium length of a vertical
spring. When the elevator suddenly stops, the end of the spring attached to the
ceiling stops. The mass, however has momentum, p = mv, and therefore starts
stretching the spring. It moves through the equilibrium position of the vertical
spring with its maximum velocity vmax = 1.5m/s.
Its velocity as a function of time is v(t) = -Asin(t+).
Since vmax = A and = 2/s, the amplitude of the amplitude of the oscillations is
A = 0.75m.
(b) The equation of motion for the particle is d2x/dt2 = -(k/m)x = -2x.Its solution is
x(t) = Acos(t+) = 0.75mcos(2/st+). If we choose the t = 0 to be the time the
elevator stops and let the upward direction be positive, then x(0) = 0, and v(0) = 1.5m/s. We therefore need  to be /2.
Example 3
A mass-spring system oscillates with an amplitude of 3.5cm. If the force constant
of the spring of 250N/m and the mass is 0.5kg, determine
(a) the mechanical energy of the system,
(b) the maximum speed of the mass, and
(c) the maximum acceleration
Solution:
(a) We have m = 0.5kg, A = 0.035m, k = 250N/m, 2 = k/m = 500/s2, = 22.36/s.
The mechanical energy of the system is E = (1/2)kA2 = 0.153J.
(b) The maximum speed of the mass is vmax = A = 0.78m/s.
(c) The maximum acceleration is amax = 2A = 17.5m/s2.
D. Bibliography
http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html
http://farside.ph.utexas.edu/teaching/301/lectures/node146.html
http://electron9.phys.utk.edu/phys135d/modules/m9/oscillations.htm
http://en.wikipedia.org/wiki/Simple_harmonic_motion#Introduction
http://physics.ucsc.edu/~josh/6A/book/harmonic/node18.html
http://sysdyn.clexchange.org
http://www.walter-fendt.de/ph11e/resonance.htm
http://maxwell.byu.edu/~spencerr/websumm122/node100.html
http://www.acoustics.salford.ac.uk/feschools/waves/shm3.html
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http://cnx.org/content/m15880/latest/
http://www.answers.com/topic/forced-oscillation
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