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SIMPLE HARMONIC OSCILLATION
Prepared by;
Dr. Rajesh Sharma
Assistant Professor
Dept of Physics
P.G.G.C-11, Chandigarh
Email: [email protected]
• Periodic Motion: any motion of system which
repeats itself at regular, equal intervals of time.
• Oscillatory or vibratory motion: A periodic to and
fro motion of a particle or body about a fixed
point is called oscillatory or vibratory motion.
• The trigonometric functions sines and cosines are
periodic as well as bounded, so the oscillatory
motions can be expressed in terms of sine and
cosine functions or harmonic functions.
• All the oscillatory motions are periodic motion
but all periodic motions are not oscillatory.
SIMPLE HARMONIC MOTION:SHM
• Simple Harmonic Motion (SHM) refers to a certain kind of
oscillatory, or wave-like motion that describes the behavior of
many physical phenomena:
– a pendulum
– a bob attached to a spring
– low amplitude waves in air (sound), water, the
ground
– the electromagnetic field of laser light
– vibration of a plucked guitar string
– the electric current of most AC power supplies
– …etc
S.H.M.
• SHM is a special type of oscillatory motion in which a
particle or body moves to an fro repeatedly about a mean
(or equilibrium) position under the influence of a restoring
force which is always directed towards the mean position
and whose magnitude at any instant of time is directly
proportional to the displacement of the particle from the
mean position at that instant.
• Let x be the displacement of a particle of, executing SHM,
from its mean position at any instant of time, then the
resorting force acting on the particle on that instant is given
by
F = - kx
• Where k is known as the force constant or stiffness
constant and the –ve sign shows that the restoring force is
always directed towards the mean position.
Spring Constant, K
The constant k is called the spring
constant.
SI unit of k = N/m.
Simple Harmonic Motion
When there is a restoring force, F = -kx, simple harmonic
motion occurs.
Geometrical interpretation of SHM
•
•
•
•
A particle P moving on a circle of radius r with
uniform angular velocity w. This is known as
Circle of Reference. M is the projection of
the particle P on the diameter YOY’.
When the particle P completes the circle from
XY X’ Y’ X, the projection of the
particle P i.e. M will move from O Y O X’
Y’ O. so, during this time in which the
particle completes one revolution, its
projection M completes one oscillation.
This motion of the projection M, on the
diameter YOY’ is called Simple Harmonic
Motion.
Thus, the geometrical definition of SHM is
the projection of uniform circular motion on
any of the diameters of the circle of
reference.
Y
P
M
y
r
wt
O
Y’
Fig. 1
X
Position VS. Time graph :
Displacement
Displacement
• It is defined as the distance of the oscillating
particle in a particular direction from the
mean position at any instant of time. In figure
1, the displacement of the particle at any
Y
instant of time t is given by y
P
M
• In OMP
OM
 sin wt
OP
OM  y  OP sin wt
y  r sin wt
y
r
wt
X’
O
Y’
X
Amplitude
Amplitude is the magnitude of the maximum displacement on either
side of the mean position.
Velocity
• It is defined as the rate of change of the
displacement with respect to time at any
instant of time. dy d
v

r sin wt 
dt dt
v  rw cos wt
Acceleration
• It is defined as the rate of change of the
velocity with respect to time at any instant of
dv d
time.
a
 rw cos wt 
dt dt
a  rw 2 sin wt  w 2 y
Period, T
For any object in simple harmonic motion, the time
required to complete one cycle is the period T.
Time 
T
angular displaceme nt
angular ve locity
2
w
but
a  w 2 y
a
w2 
y
w
a
y
then,
2
y
T
 2
a
a
y
T  2
Displaceme nt
Accelerati on
(taking magnitude only)
Frequency, f
The frequency f of the simple harmonic motion is the
number of cycles of the motion per second.
Phase
• It determines the status of the particle as
regards its position and direction of motion. It
is expressed either in terms of the angle swept
by the radius vector of the particle since it
crossed its mean position or as the fraction of
the time interval that has lapsed since the
particle crossed the mean position.
• wt, (wt+f) or (wt-f) are called phase angles.
Initial phase or Epoch
• It is the phase of an oscillating particle at time
t = 0. if a particle has initial phase a or - a
then
y = r sin (wt-a)
or
y = r sin (wt+a)
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.
• A constant value of the stiffness restricts the
displacement x to small values (this is Hooke’s
Law of Elasticity). The stiffness s is obviously
the restoring force per unit distance (or
displacement) and has the dimensions
Force
MLT 2

Distance
L
Differential equation of SHM
• When an oscillator is displaced from it mean position, a
resorting force is developed in the system, which tries
to regain the mean position of the oscillator. This
restoring force is directly proportional to the
displacement of the oscillator and is always directed
towards the mean position (Hook’s law)
• The equation of motion of such a disturbed system is
given by the dynamic balance between the forces
acting on the system, which by Newton’s Law is
mass X acceleration = restoring force
since, Restoring force F   - S  Displaceme nt
where, S is the stiffness constant.
Now,
d2y
F  my  m 2
dt
 my   Sy
S
y    y  0
m
S
2
  has the dimensions of T ,
m


square of the angular frequency w 2 , also has the same dimensions .
So, putting
we get,
y  w 2 y  0
S
 w2
m
• This is the standard second degree differential
equation of SHM
• The general solution of the above eq. is
y = r sin (wt+a)