Prof. Nabila.M.Hassan

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Faculty of Computer and Information Basic
Science department 2012/2013
Prof. Nabila.M.Hassan
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• Aims of Course
• The graduates have to know the nature of
  vibration wave motions with emphasis on their
  mathematical descriptions and superposition.
• The fundamental ideas can be introduced with
  reference to mechanical systems which are easy to
  visualize.
• The graduates have to know the nature of
  vibration and wave motions with emphasis on their
  mathematical description and superposition
  Developing the graduate's skills and creative
  thought needed to meet new trends in science.
• Supplying graduates with basic attacks and
  strategies for solving problems.

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1- A particle oscillates with simple harmonic
motion, so that its displacement varies according
to the expression x (5 cm)cos(2t p/6) where x
is in centimeters and t is in seconds. At t 0
find(a) the displacement of the particle,(b)
its velocity, and(c) its acceleration.(d) Find
the period and amplitude of the motion.
Solution The displacement as a function of
time is x(t) A cos(?t f). Here ? 2/s, f
p/6, and A 5 cm. The displacement at t 0 is
x(0) (5 cm)cos(p/6) 4.33 cm. (b) The velocity
at t 0 is v(0) -?(5 cm)sin(p/6) -5
cm/s. (c) The acceleration at t 0 is a(0)
-?2(5 cm)cos(p/6) -17.3 cm/s2. (d) The period
of the motion is T p sec, and the amplitude is
5 cm.  
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1- An oscillator consists of a block of mass 0.50
kg connected to a spring. When set into
oscillation with amplitude 35 cm, it is observed
to repeat its motion every 0.50 s. The maximum
speed is (a) 4.4 m/s ,(b) 44.0 m/s ,( c) 44.0
m/s 2- A particle executes linear harmonic
motion about the point x 0. At t 0, it has
displacement x 0.37 cm and zero velocity. The
frequency of the motion is 0.25 Hz. The max speed
of the motion equal (a) 0.59 cm/s ,(b) 5.9 cm/s
,( c) 0.059 cm/s 3- An oscillating block-spring
system has a mechanical energy of 1.0 J,
amplitude of 0.10 m, and a maximum speed of 1.2
m/s. The force constant of the spring is, (a)
100 N/m ,(b) 200 N/m ,( c) 20 N/m 4- An
oscillating block-spring system has a mechanical
energy of 1.0 J, amplitude of 0.10 m, and a
maximum speed of 1.2 m/s. The mass of the block
is, (a) 1.4 kg ,(b) 14.0 kg ,( c) .140 kg
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Content Part II Waves Chapter 1 Oscillation
Motion - Motion of a spring - Energy of the
Simple Harmonic Oscillator - Comparing SHM with
uniform motion - The simple pendulum - Damped
Oscillations - Forced Oscillation
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• Objectives
• Student will be able to
• - Define the damped motion
• - Define the resonance.
• -Compare between free, damped and derived
  oscillations

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Damped Oscillations
Where the force is proportional to the speed of
the moving object and acts in the direction
opposite the motion. The retarding force can be
expressed as R - bv ( where b is a constant
called damping coefficient) and the restoring
force of the system is kx, then we can write
Newton's second law as
When the retarding force is small compared with
the max restoring force that is, b is small the
solution is,
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represent the position vs time for a damped
oscillation with decreasing amplitude with time
The fig. shows the position as a function in time
of the object oscillation in the presence of a
retarding force, the amplitude decreases in time,
this system is know as a damped oscillator. The
dashed line which defined the envelope of the
oscillator curve, represent the exponential factor
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• The fig. represent position versus time
• under damped oscillator
• critical damped oscillator
• - Overdamped oscillator.

as the value of "b" increase the amplitude of the
oscillations decreases more and more rapidly.
When b reaches a critical value bc (
), the system does not oscillate and is said
to be critically damped. And when
the system is overdamped.
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Forced Oscillation
For the forced oscillator is a damped oscillator
driven by an external force that varies
periodically Where
where ? is the angular frequency of the driving
force and Fo is a constant From the Newton's
second law
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is the natural frequency of the un-damped
oscillator (b0).
The last two equations show the driving force
and the amplitude of the oscillator which is
constant for a given driving force. For small
damping the amplitude is large when the frequency
of the driving force is near the natural
frequency of oscillation, or when ?? ?o the is
called the resonance and the natural frequency
is called the resonance frequency.
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Amplitude versus the frequency, when the
frequency of the driving force equals the natural
force of the oscillator, resonance occurs. Note
the depends of the curve as the value of the
damping coefficient b.
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Summary of the chapter 1- The acceleration of
the oscillator object is proportional to its
position and is in the direction opposite the
displacement from equilibrium, the object moves
with SHM. The position x varies with time
according to,
2- The time for full cycle oscillation is defined
as the period,
. For block spring moves as SHM on the
frictionless surface with a period
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and
3- The frequency is defined as the number of
oscillation per second, is the inverse of the
period
4- The velocity and the acceleration of SHM as a
function of time are
We not that the max speed is A? , and the max
acceleration is A?2 . The speed is zero when the
oscillator is at position of x A , and is a max
when the oscillator is at the equilibrium
position at the equilibrium position x0.
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5- The kinetic energy and potential energy for
simple harmonic oscillator are given by,
The total energy of the SHM is constant of the
motion and is given by
6- A simple pendulum of length L moves in SHM for
small angular displacement from the vertical, its
period is
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7- For the damping force R - bv, its position
for small damping is described by
8 - If an oscillator is driving with a force
it exhibits resonance, in which the amplitude is
largest when driving frequency matches the
natural frequency of the oscillator.
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What is the effect on the period of a pendulum of
doubling its length?
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Useful website http//cnx.org/content/m15880/late
st/ http//www.acs.psu.edu/drussell/Demos/SHO/mas
s-force.html