SAPERE
ACADEMY
NOTES ON OSCILLATIONS
AND WAVES#1
PREPARED BY:@SAPERE_AUDE7
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 Periodic motion is a type of motion that repeats: it returns to the
same initial position after some time interval t.
 In Mechanical systems, when the force on the object is
proportional to the position(x) with respect to some reference
position(Equilibrium position) and directed towards the equilibrium
position, it Is called a SIMPLE HARMONIC MOTION(SHM)
Consider the Spring-Mass systems in the above figure.
 When the mass is neither stretched nor compressed it is
found at the equilibrium position(B) where the origin is taken
as reference(x=0) and no horizontal forces are acting on it→
It’s in Equilibrium
 If the mass is displaced from this position it will oscillate back
and forth until it return to its equilibrium position. Since the
only force acting on it is by the spring and it is conservative
hypothetically such motion will continue forever.
 When the block is displaced to the right, the force(Fs) acts to
the left to return it to the equilibrium position(a)
 When the block is displaced to the left, the force(Fs) acts to
the right to return it to the equilibrium position(c)
 Using Hooke’s law:
Fs = -kx………………………….(1)
Where: Fs is the Restoring force
K is the spring constant or stiffness(Unit: N/m)
X is the position relative to the equilibrium position
From Hooke’s law we can conclude:
 The Force exerted by the spring on the block is proportional to the block’s
position→ Larger force is needed to return a block to its initial position if it’s
stretched way further and the converse is also true.
 The direction of Fs is always opposite to x
Applying Newton’s 2nd law:
Fx = max but the only force on the X -axis is Fs → Fs= max
From Hooke’s law: -kx=max → ax =(
−𝑘
𝑚
)x……………………………………………………….(1,1)
The acceleration is proportional to the position of the block, and its direction is
Opposite the direction of the displacement from equilibrium.
Systems that behave in this way are said to exhibit simple harmonic motion. An
object moves with simple harmonic motion whenever its
 acceleration is proportional to its position and is
 Oppositely directed to the displacement from equilibrium.
NOTE: THE ABOVE CONCLUSION ARE THE SAME FOR A
VERTICAL SPRING-MASS SYSTEM AS WELL. → THE NET FORCE
ACTING ON BOTH CASES IS FS
If the block starts form initial potion A then x=A the equation 1.1 becomes:
ax =(
−𝑘
𝑚
)A
When the block passes the origin X=O, ax becomes 0 → Here, speed is
maximum because acceleration in changing sign and V is a Sine function (detail
later).
QUIZ 1 :
A block on the end of a spring is pulled to position x =A and released. In one
full cycle of its motion, through what total distance does it travel?
(a) A/2 (b)A (c) 2A (d) 4A
Let us rewrite equation 1.1 as : ax= -ω 2 x, where ω 2 is
𝑘
𝑚
.
Using differential calculus:
𝑑𝑥
𝑑𝑣
𝑑 𝑑𝑥
= v but a = , ax = ( )
𝑑𝑡
𝑑𝑡
𝑑𝑡 𝑑𝑡
ax =
𝑑2 x
𝑑𝑡 2
= -ω2x
Solving the above 2nd degree differential yield the position of the block as a
function of time :
x(t)= Acos(ωt + Φ)
Where A: Amplitude(Maximum or minimum initial position)
ω: Radial Frequency(Number of oscillations per unit time measured in
rad per second)
Φ:Phase Constant or Initial phase angle : determined based on A and
initial time(t=0)
 The quantity (ωt+Φ) is called the phase of the motion.
Note that the function x(t) is periodic and its value is the same each time ωt
increases by 2πradians.
Graph of x vs t for a SHM is periodic .
A: the graph of a SHM with random
Φ
B: graph of SHM with x=A at t=0:
from x(t)= Acos(ωt + Φ),
x(0)=Acos(ωt+Φ) but x(0)=A
A=Acos(ωt+Φ) thus cos(ωt+Φ)=1,
since t=o, cos(Φ)=1, Φ=0
𝛳
From circular motion: ω= for full revolution ϴ = 2π, thus
𝑡
2𝜋
2𝜋
𝑇
𝜔
ω= = T=
Period: Time taken to do a full oscillation.
We know that -ω 2 is
−𝑘
𝑚
where T is the period.
𝑘
, implying ω=√ ,
𝑚
√
𝐤
𝐦
=
𝟐𝛑
𝐓
It follows: T=
𝟐𝛑
√
𝐦
𝐤
𝐦
= 𝟐𝛑√
𝐤
The inverse of the period is called the frequency f of the motion. Whereas the
period is the time interval per oscillation, the frequency represents the number
of oscillations that the particle undergoes per unit time interval:
f=
1
𝑇
=
1
2𝜋
𝜔
=2𝜋 or f= 𝜔 or
𝜔
2𝜋
1
2𝜋√
=
𝐦
1
2𝜋
𝐤
.√ V
𝐦
𝐤
WE IDENTIFY TWO KINDS OF FREQUENCY FOR A SIMPLE HARMONIC
OSCILLATOR— f, CALLED SIMPLY THE FREQUENCY, IS MEASURED IN HERTZ,
AND THE ANGULAR FREQUENCY(ω) IS MEASURED IN RADIANS PER SECOND.
The period and frequency depend only on the mass of the particle
and the force constant of the spring, and not on the parameters of
the motion, such as A or Φ.As we might expect, the frequency is
larger for a stiffer spring (larger value of k) and decreases with
increasing mass of the block.