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CHAPTER
RESISTED MOTION AND
4 DAMPED FORCE
OSCILLATOR
Motion in a Resisting Medium
In practice an object is acted upon not only by is weight but by other forces as
well. An important class of forces are those which tend to oppose the motion of an
object and reduce the magnitude of successive oscillations about the equilibrium
position. Such forces, which generally arises because of motion in some medium
such as air or water, are often called resisting, damping or dissipative force and
the corresponding medium is said to be a resisting, damping or dissipative
medium. A useful approximated damping force is given as follows;
⃗ ⃗ ̂ ̂
Where the descript D stands for the damping force and is the positive constant
called the damping coefficient. Note the ⃗ and ⃗ are in opposite direction.
Friction Force
Friction forces play an important role in damping or retarding motion initiated by
other forces friction force between two bodies results from the interaction between
the surface molecules of the two bodies and involves a very large number of such
iteration. The phenomenon is therefore complex and depends on factor such as the
condition and nature of the surfaces and their relative velocity.
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Some Useful Definitions
Simple Harmonic Motion and Simple Harmonic Oscillator: SHM occur
when the net force is directly proportional to the displacement from the mean
position and is always directed towards the mean position. The body
executing SHM is called Simple Harmonic Oscillator. The motion of simple
pendulum and the motion of mass spring system is SHM.
Simple Harmonic Motion is an oscillatory motion that occurs whenever a
force acts on a body in the opposite direction to its displacement from its
equilibrium position , with the magnitude of the force , proportional to the
magnitude of the displacement. i.e. ⃗ 𝑥 or ⃗ 𝑥
Where is the constant of proportionality often called the spring constant,
elastic constant, stiffness factor or modulus of elasticity
Restoring Force: A force that always pushes of pulls the object performing
oscillatory motion towards the mean position.
Vibration: One complete round trip of a vibrating body about its mean
position is called one vibration.
Time Period: The time taken by a vibrating body to complete one vibration
is called time period.
Frequency: The number of vibrations or cycles of a vibrating body in one
second is called its frequency. It is reciprocal of time period.
Amplitude: The maximum displacement of a vibrating body on either side
from its mean position is called its amplitude.
Oscillations/Vibrations: A body is said to be vibrating (oscillating) if it
moves back and forth or to and fro about a point.
Damped forced oscillations/ Damped oscillations: The oscillations of a
system in the presence of some resistive force.
Linear frequency: The amount of vibrations completed in unit time is
called linear frequency. Its SI unit is called hertz (Hz).
Angular frequency: The amount of rotations completed in unit time is called
linear frequency. The linear frequency and the angular frequency are
related as
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Equation of Motion of Simple Harmonic Oscillator
Consider a block of mass m is attached with one end of a string. The other end of
spring is fixed to a support. The block is free to move to and fro over a frictionless
horizontal surface as shown in figure.
The point x = 0 when block is at rest is called mean position because spring is not
exerting any force on the block. The block attached with spring having constant k
takes to and fro motion under restoring force F given as
⃗ 𝑥 ……………….(1)
⃗ ⃗ ……………….(2) by Newton‟s 2nd Law
Comparing (1) and (2) we have
𝑥̈ 𝑥 Or 𝑥̈ 𝑥
This is called equation of motion of simple harmonic oscillator or linear Harmonic
Oscillator. This type of motion is often called Simple Harmonic Motion.
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Damped Harmonic Oscillator
The oscillator which moves in a resistive medium under a restoring force is called
the Damped Harmonic Oscillator and equation of motion of the harmonic
oscillator is given as
𝑥 or 𝑥
We may write it as follows;
𝑥̈ 𝑥̇ 𝑥 using
Remark
Damped Harmonic Oscillation 𝑥̈ 𝑦𝑥̇ 𝑥 represent over damped
motion if . i.e. and in this case equation
𝑥̈ 𝑦𝑥̇ 𝑥 has the general solution of the following form
𝑥 ( ) where √
And A,B are arbitrary constants can be found from the initial conditions.
Damped Harmonic Oscillation 𝑥̈ 𝑦𝑥̇ 𝑥 represent critically
damped motion if . i.e. and in this case equation
𝑥̈ 𝑦𝑥̇ 𝑥 has the general solution of the following form
𝑥 ( )
And A,B are arbitrary constants can be found from the initial conditions.
Damped Harmonic Oscillation 𝑥̈ 𝑦𝑥̇ 𝑥 represent under
damped or damped oscillatory motion if . i.e. and in
this case equation 𝑥̈ 𝑦𝑥̇ 𝑥 has the general solution of the
following form
𝑥 ( ) ( ) where √
And where √ called the amplitude, and called the phase
angle or epoch, can be determined from the initial conditions.
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Equation of Motion of Damped Harmonic Oscillator
Consider a block of mass m is attached with one end of a string. The other end is
connected with a mass less vane. The block is free to move to and fro over a
frictionless horizontal surface as shown in figure.
Now displace the block towards right through some displacement and release. The
block attached with spring having constant k takes to and fro motion under
restoring force F given as
⃗ 𝑥
The damping force experienced by vane when it moves in resistive medium is
⃗ ⃗
⃗ ⃗ ⃗ 𝑥 ⃗ ……………….(1)
⃗ ⃗ ……………….(2) by Newton‟s 2nd Law
Comparing (1) and (2) we have
𝑥 or 𝑥
We may write it as follows;
𝑥̈ 𝑥̇ 𝑥 using
This is called equation of motion of damped harmonic oscillator . This type of
motion is often called damped Harmonic Motion.
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Simple Pendulum
The metallic bob suspended by a weightless inextensible string is called simple
pendulum. The distance between point of suspension and center of bob is called
length of simple pendulum. The bob at rest when no resultant force acts on it is
called mean position or equilibrium position.
Equation of motion of a Simple Pendulum
Consider a bob of mass m attached with a string. The string is hanged vertically
from a support as shown in figure;
Pull the pendulum from mean position to position A such that string makes a small
angle with vertical. The bob starts moving toward mean position under restoring
force when released. It gets maximum velocity at mean position and does not stop
due to inertia but continues to move towards extreme position B. The velocity of
bob becomes zero at position B due to restoring force.
The path followed by bob when it moves from mean position to position A
is called an arc of circle having radius . The arc length S and chord length x are
approximately equal for small angle.
The forces acting on bob when it is at position A are
Weight of bob acting vertically downward
Tension acting along the string
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Resolving weight force into components we get ⃗
The negative sign means direction of ⃗ is opposite to direction of increasing
and for small amplitude we have
⃗ ……………….(1)
⃗ ⃗ ……………….(2) by Newton‟s 2nd Law
Comparing (1) and (2) we have
⃗ ⃗
The relation 𝑟 for circular path gives 𝑥 then
⃗ ( ) ⃗ ( )𝑥 ( )𝑥
𝑥̈ ( ) (Equation of motion of a Simple Pendulum)
Resonance / Resonance Frequency
Resonant frequency is the oscillation of a system at its natural or unforced
resonance. Resonance occurs when a system is able to store and easily transfer
energy between different storage modes, such as Kinetic energy or Potential
energy as you would find with a simple pendulum. A familiar example is a
playground swing, which acts as a pendulum.
Forced Vibrations
Forced vibration occurs when motion is sustained or driven by an applied periodic
force in either damped or undamped systems. Vibration of vehicles during the
running on uneven roads, vibration of air compressors and musical instruments etc.
are some of the examples for forced vibrations.
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Question
Determine the motion of simple pendulum of length and mass m assuming small
vibrations and no resisting force.
Solution
Let the position of m at any time be determined by s,
the arc length measured from the equilibrium position O.
Let be the angle made by the pendulum string with the
vertical. If ⃗⃗ is a unit tangent vector to the circular path of
the pendulum bob m, then by Newton‟s second law
⃗ ⃗ ⃗⃗ ……………….(1)
Resolving force into components we get ⃗
The negative sign means direction of ⃗ is opposite to direction of increasing
and for small amplitude we have
⃗ ……………….(2)
Comparing (1) and (2) we have
⃗⃗ ( )
Which has solution √ √
Using initial conditions at we get
√ . Here is time period √
Energy of a Simple Harmonic Oscillator
If T is the kinetic energy, V the potential energy and E = T + V the total energy of
a simple harmonic oscillator then we have
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Question
Prove that the force ⃗ 𝑥 ̂ acting on a simple harmonic oscillator is
conservative.
Solution: Given that ⃗ 𝑥 ̂ then
̂ ̂ ̂
⃗⃗ ⃗ | | . Thus the force ⃗ ⃗ 𝑟⃗ is conservative.
𝑥
Question
Find the potential energy of a simple harmonic oscillator.
Solution
In this case the potential or potential energy is given as ⃗
𝑥̂ 𝑥̂ ̂ ̂ ̂ 𝑥̂
𝑥 ( ) ( ) ( )
( ) 𝑥
𝑥 using for 𝑥 we get
Question
Express in symbol the principal of conservation of energy for a simple harmonic
oscillator.
Solution
We know that 𝑥
𝑥 𝑥 𝑥
𝑥 after integration
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