Mechanics

UNIT 2: OSCILLATIONS
Oscillatory motion: A motion which repeats itself after regular intervals of time is called a periodic
motion. If a body in periodic motion executes to and fro motion about a fixed reference point then it is
called oscillatory motion.
Simple Harmonic Motion (SHM): When a body moves such that its acceleration is always directed
towards a fixed point and varies directly as its distance from that point, the body is said to execute SHM.
For SHM to take place, the force acting on the body should be directed towards the fixed point and should
also be proportional to displacement. The function of the force is to bring the body back to its equilibrium
position and hence this force is often known as restoring force.
It should be noted that all periodic motions are not SHM, for example, the motion of the earth
around the sun and the motion of the moon around the earth are periodic but not simple harmonic.
However all SHM are periodic since SHM has a definite frequency and hence has fixed time period
which makes it periodic.
Differential equation of SHM:
Consider a particle of mass m executing simple harmonic motion. If y be the displacement of the particle
from equilibrium position at any instant of time t, then the restoring force F acting on the particle is given
by
Fy or, F   sy ............(1)
where s is the force constant of proportionality or stiffness or spring constant. The negative sign
indicates that the direction of restoring force is opposite to the direction of increasing displacement.
F
From (1), in terms of magnitude, we get s
y
Thus force constant s is defined as the restoring force per unit displacement. Its unit is N/m.
d2y
From Newton’s law of motion, we have, F  ma  m 2
dt
2
d y
 m 2   sy
dt
d2y
Or, m 2  sy  0
dt
d2y s
  y0
dt 2 m
s 2
But,  2  where    2 is the angular frequency
m T
d2y
 2
 2 y  0
dt
Which gives the differential equation of motion for SHM.
Solution of differential equation of SHM:
The differential equation representing SHM is given by
d2y
 2 y  0
dt 2
dy
Multiplying 2 on both sides, we get
dt
dy d 2 y dy
2 2
 2 2 y  0
dt dt dt
2
d  dy  d 2 2
Or,     y   0
dt  dt  dt
Integrating with respect to t, we get,
2
 dy 
    y  C
2 2
(2)
 dt 

, where C is a constant of integration.
When the displacement is maximum, i.e., at y = a where a is the amplitude, velocity is zero
dy
 0
dt
dy
Substituting y = a and  0 in equation (2), we get
dt
0   2 a 2  C
 C   2a 2
Substituting the value of C in equation (2), we get
2
 dy 
    y   a
2 2 2 2

 dt 
2
 dy 
    a  y
2 2
 2

 dt 
dy

dt

  a2  y2  ............(3)
which gives the velocity of the particle executing SHM at a time t when displacement is y.
dy
Also,  dt
 a  y2
2

y
Integrating, we get, sin 1  t   where ϕ is another constant of integration.
a
y
  sin t   
a
 y  a sint    .............(4)
The term t    represent the total phase of the particle at time t and ϕ is known as the initial
phase or phase constant. ϕ will be zero, if the time is recorded at the instant when y = 0 and increasing.
Other solutions:
1. y  a cost   
2. y  A sin t  B cos t where A  a cos  and B  a sin 
3. y  Aeit  Be  it where A  a  ib and B  a  ib , i.e., they are complex conjugates
4. y  aei (t  )
Velocity and acceleration of Simple Harmonic Oscillator:
The displacement of a simple harmonic oscillator is y  a sint    ........(1)
dy d
Therefore velocity, v   a sin t     a cost    ........(2)
dt dt
y
But from (1), sin t    
a
2
y
 sin 2 t     2
a
y2 a2  y2
 cos2 t     1  sin 2 (t   )  1  2 
a a2
a2  y 2
 cost     ................(3)
a2
a2  y2
From (2) and (3), we get v  a 2
  a2  y2
a
Velocity is maximum at mean position i.e., when y = 0.