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oct. 8, 2012 1
Simple Harmonic MotionBy
Dr. Karamjit SinghSenior Lecturer
Govt. Polytechnic College For Girls Patiala
Email: [email protected]
Mobile:- 9914029020; 9501029020
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Simple Harmonic Motion
Nature follows periodicity. The day and night, life and death, everything is repetitive. Periodic motion is the motion that repeats itself after a fixed interval of time. There are many type of periodic motion and the simplest of them is Simple harmonic motion. In SHM, displacement, velocity and acceleration all are sinusoidal.
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Any vibrating system where the restoring force is proportional to the negative of the displacement is in simple harmonic motion (SHM), and is often called a simple harmonic oscillator (SHO).
SHM is a motion that repeats itself after regular time interval such that the force acting on it is directed towards its mean position or any other fixed point in its path and the force is also proportional to displacement of the particle from that fixed point
Defn.
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Periodic Motion
Periodic motion of a body is that motion which repeats itself over and over again after a fixed interval of time. The fixed or regular interval of time after which the periodic motion is repeated again is called its time period.
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Examples of periodic motion
• The rotation of earth about its axis with period of rotation of one day.
• The motion of moon around the earth with a time period of 27.3 days.
• The phases of moon .
• Revolution of earth around sun with a period of one year.
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Oscillatory motionA motion that repeats itself over and over again about its mean position, such that it remains confined within two well defined limits called extreme positions on either side of the mean position that is a fixed point, in a definite interval of time.A periodic and bounded motion of a body about its mean position is called an oscillatory or vibratory motion. It follows that all oscillatory motion are periodic motions but all periodic motions are not oscillatory.
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Examples of Oscillatory Motion
• When a liquid in a U – tube is displaced, it executes oscillatory motion.
• When a load attached to a spring is pulled once a little from its fixed point and left.
• Motion of pendulum of a wall clock.
• A glass ball dropped along walls of a semi- hemispherical bowl and released.
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Oscillatory motion
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HOOKE'S LAW
The restoring force of an ideal spring is given by,
where k is the spring constant and x is the displacement of the spring from its unstrained length. The minus sign indicates that the restoring force always points in a direction opposite to the displacement of the spring.
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Hooke's LawOne of the properties of elasticity is that it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke's law.
Hooke's LawOne of the properties of elasticity is that it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke's law.
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Simple Harmonic Motion
When there is a restoring force, F = -kx, simple harmonic motion occurs.
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Displacement
sin
sin
sin
sin
OM
OPorOM OP
y r
or tt
y r t
OO
MM PP
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Velocity in SHM
2 2 2 2
2 2 2 2
2 2
cos
cos cos
cos
PN
VPN V V t
OP OQ r yOQ
OP OP r
r y r r yPN V
r r
v r y
OO
MM PP
θθ
θθKK NN
VV
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Velocity is maximum when y is minimum and velocity is minimum when y is maximum.
So, when y=0, v = ωr, i.e. it is maximum.
And when y = r, v=0 i.e. it is minimum
2 2 2 2
2 2 2 2
2 2
cos
cos cos
cos
PN
VPN V V t
OP OQ r yOQ
OP OP r
r y r r yPN V
r r
v r y
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Position VS. Time graph
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Uniform Circular Motion and SHM
Rotational vector
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x, v, and a in SHM
)cos( tAx
)2
cos(
)sin(
tv
tAv
m
)cos(
)cos(2
ta
tAa
m
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The velocity and acceleration for simple harmonic motion can be found by differentiating the displacement:
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Period, T
For any object in simple harmonic motion, the time required to complete one cycle is the period T.
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Expression for Time Period
2
2
2
12 2
1
.
a y
a
y
a
y
a
T y
yT
aay
Tacc perunitdisplacement
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Oscillation diagram
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Frequency, f
The frequency f of the simple harmonic motion is the number of cycles of the motion per second.
2
1 1
22
yT
a
a
yya
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Oscillating Mass
.2k
mT
Consider a mass m attached to the end of a spring as shown.
If the mass is pulled down and released, it will undergo simple harmonic motion.
The period depends on the spring constant, k and the mass m, as given below,
2
2
4kT
m
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• Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke’s law. The motion is sinusoidal in time and demonstrates a single resonant frequency.
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• The motion equation for simple harmonic motion contains a complete description of the motion, and other parameters of the motion can be calculated from it.
The velocity and acceleration are given by
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The motion equations for simple harmonic motion provide for calculating any parameter of the motion if the others are known.
The motion equations for simple harmonic motion provide for calculating any parameter of the motion if the others are known.
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A mass on a spring will trace out a sinusoidal pattern as a function of time, as will any object vibrating in simple harmonic motion . One way to visualize this pattern is to walk in a straight line at constant speed while carrying the vibrating mass. Then the mass will trace out a sinusoidal path in space as well as time.
A mass on a spring will trace out a sinusoidal pattern as a function of time, as will any object vibrating in simple harmonic motion . One way to visualize this pattern is to walk in a straight line at constant speed while carrying the vibrating mass. Then the mass will trace out a sinusoidal path in space as well as time.
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Oscillations of a Spring
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If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a useful model for a periodic system.
Oscillations of a Spring
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Mass on a Spring
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Position vs Time Graph of Mass on Spring
(Sine Curve)
a)What is the amplitude?
b)What is the period?
x
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Period, frequency, and angular frequency
Period is a time when one complete oscillation undergoing.
o
T2
Frequency is numbers of oscillation in unit time.
2
1
T
2
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Amplitude
Amplitude is the magnitude of the maximum displacement.
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Amplitude, phase, and phase constant
Amplitude A is the maximum distance of an oscillator from its equilibrium position.
1)cos( t
AxA
Since
We get
Phase: t
Phase constant (or initial phase) or epoch =
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is angular frequency determined by the oscillation system; Amplitude A and phase constant are determined from the initial conditions:
o
o
vv
xxt ,0
We get
sin
cos
Av
Ax
o
o
Therefore,
2
22
o
ov
xA
o
o
x
vtan
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Phase difference
Compare the phase difference of two oscillations with:
i) If 2 1 > 0, SHM-2 is in
before SHM-1;ii) If 2 1 < 0, SHM-2 is in
after SHM-1;iii) If 2 1 = 0, SHM-2 is in
synchronization with SHM-1 (or in synchronous phase);
iv) If 2 1 = , SHM-2 is in
anti-phase with SHM-1.
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ENERGY IN SHM :- We already know that the potential energy of a spring is given by:
The total mechanical energy is then:
The total mechanical energy will be conserved, as we are assuming the system is frictionless.
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If the mass is at the limits of its motion, the energy is all potential.
If the mass is at the equilibrium point, the energy is all kinetic.
We know what the potential energy is at the turning points:
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The total energy is, therefore,
And we can write:
This can be solved for the velocity as a function of position:
where
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Energy in SHM:- A particle executing SHM possesses both potential and Kinetic energy
Potential EnergyPotential Energy2
2
2
2 2 21
2
a y
F m y
dw Fdy
dw m ydy
Totalworkdone dw m ydy m y
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Kinetic Energy:- If v is the velocity of the particle executing SHM, when displacement is y, then
2 2 2 2
2 2 2
1 1. . ( )
2 2K E mv m r y
v r y
Total Energy = Kinetic Energy + Potential EnergyTotal Energy = Kinetic Energy + Potential Energy
2 2 2 2 2
2 2
1 1( )
2 21
2
m r y m y
E m r
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This graph shows the potential energy function of a spring. The total energy is constant.
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By means of total energy
kEAx /2max
mEAv /2max
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A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible.
The Simple Pendulum
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In order to be in SHM, the restoring force must be proportional to the negative of the displacement. Here we have:which is proportional to sin θ and not to θ itself.
However, if the angle is small, sin θ ≈ θ.
The Simple Pendulum
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2
22
2
22
2
,
sin , ;
sin ,
0 .
2
2
I
F mg I m
dmg m
dt
d g g
dt
g
l
g
T l
lT
g
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Therefore, for small angles, we have:
where
The period and frequency are:
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So, as long as the cord can be considered massless and the amplitude is small, the period does not depend on the mass.
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If two component frequencies are very close and their difference is small. Therefore, the average frequency is much larger than modulating frequency. The phenomenon that the composite amplitude will change periodically is named a beat.
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Free, Forced and Resonant vibrations
When a body capable of vibrating in the absence of any dissipative force under the action of a linear restoring force executes simple harmonic motion with a single natural frequency depending upon its dimensions and elastic constants, the period of motion is independent of of amplitude and the body vibrates indefinitely with constant amplitude. This type of vibrations are called free or undamped vibrations.
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• The vibrations of electric and magnetic fields in the electromagnetic wave propagating in free space is the best possible example of this type.
2
2
d ym ky
dt
Its solution is of the form y = a sin (wt + φIts solution is of the form y = a sin (wt + φ
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Damped harmonic motion is harmonic motion with a frictional or drag force. If the damping is small, we can treat it as an “envelope” that modifies the undamped oscillation.
If
then
Damped Harmonic Motion
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This gives
If b is small, a solution of the form
will work, with
Damped Harmonic Motion
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If b2 > 4mk, ω’ becomes imaginary, and the system is over damped (C).
For b2 = 4mk, the system is critically damped (B) —this is the case in which the system reaches equilibrium in the shortest time.
Damped Harmonic Motion
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There are systems in which damping is unwanted, such as clocks and watches.
Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings.
Damped Harmonic Motion
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Forced vibrations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system.
If the frequency is the same as the natural frequency, the amplitude can become quite large. This is called resonance.
Forced Oscillations; Resonance
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Forced vibrationsIn case of forced vibrations:
1. The body vibrates with the frequency of applied force and not with its natural frequency.
2. The amplitude is finite and constant. It depends on the frequencies of applied force, body and damping. Lesser the difference in frequencies and lesser the damping, greater will be the amplitude of vibration.
3. The resulting displacement of the body is not in phase with the applied force. It will lag or lead the applied force as the frequency of applied force is lesser or greater than the natural frequency of the body.
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Forced Oscillations; Resonance
The equation of motion for a forced oscillator is:
The solution is:
where
and
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Resonance
It is a special case of forced vibrations. If a body is set into vibration by an external periodic force whose frequency is equal to the natural frequency of the body, The amplitude of vibration increases at each step and becomes very large. Such vibrations are called resonant vibrations nd the phenomenon is called resonance.
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Examples of resonance
• If a tuning fork is vibrated, another object in neighbourhood starts vibrating.
• A vibrating tuning fork held near the mouth of a resonance column.
• A vibrating tuning fork held near a stretched string.
• Soldiers are not allowed to march in steps while passing over a suspension bridge.
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The sharpness of the resonant peak depends on the damping. If the damping is small (A) it can be quite sharp; if the damping is larger (B) it is less sharp.
Like damping, resonance can be wanted or unwanted. Musical instruments and TV/radio receivers depend on it.
Forced Oscillations; Resonance
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Forced Oscillations; Resonance
The width of the resonant peak can be characterized by the Q factor:
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• For SHM, the restoring force is proportional to the displacement.
• The period is the time required for one cycle, and the frequency is the number of cycles per second.
• Period for a mass on a spring:
• SHM is sinusoidal.
• During SHM, the total energy is continually changing from kinetic to potential and back.
Summary
