Madhava Publications
Introduction
In our daily life, we come across various kinds of motions. You have already learned about some of them, e.g., rectilinear motion and motion of a projectile. Both these motions are non-repetitive. We have also learned about the uniform circular motion and orbital motion of planets in the solar system. In these cases, the motion is repeated after a certain interval of time, that is, it is a periodic motion.
The study of oscillatory motion is basic to Physics. In musical instruments, like the sitar, guitar and violin. We come across vibrating strings that produce pleasing sounds. The vibration of air molecules makes the propagation of the sound possible.
In this chapter, we will learn about oscillatory motion or oscillations. Any motion which repeats itself at regular intervals of time is known as periodic motion. If a body moves back and forth repeatedly about its mean position then it is said to be in oscillatory motion. For example, The to and fro movement of the pendulum, jumping on a trampoline, a child swinging on a swing.
Periodic and Oscillatory Motion
Periodic Motion: A motion is called periodic motion when it repeats itself after equal intervals of time. The interval of time is called the Time period of periodic motion. Example: The rotational motion of the earth about its axis is periodic motion with a time period of 24 hours.
Oscillatory Motion: An oscillatory or vibratory motion is defined as a periodic and bounded motion about a fixed point. In other words, Oscillations are defined as to and fro motion which repeat itself after regular intervals of time.In oscillations, the frequency of vibrations is comparatively less.
For example, the Motion of the Pendulum of the wall clock, the motion of the bob of a simple pendulum displaced once from its mean position.
Every oscillatory motion is periodic motion, that is every oscillatory motion repeats itself after a fixed interval of time. But every periodic motion is not oscillatory.For e.g.:- Motion of planets around the sun is periodic but is not oscillatory motion.
Simple Harmonic Motion
Simple harmonic motion is the simplest form of oscillatory motion. This motion arises when the force on the oscillating body is directly proportional to its displacement from the mean position, which is also the equilibrium position. Further, at any point in its oscillation, this force is directed towards the mean position.
Simple harmonic motion can be considered as a specific type of oscillatory motion, in which:
- The particle moves in a single dimension
- The particle oscillates to and fro about a fixed mean position (where Fnet=0).
- The net force on the particle always gets directed towards the equilibrium position
- The magnitude of the net force is always proportional to the displacement of the particle from the equilibrium position at that instant.
Equilibrium Position
Oscillating bodies come to rest at their equilibrium positions. When a bob is suspended from rigid support it goes to extreme positions and then comes to its mean position which is also known as equilibrium position.
Equilibrium Position is that position where an object tends to come at rest when no external force is applied.
To and fro motion of the pendulum oscillating from its mean position B to its either extreme positions A and C respectively.
Period/Time period (T)
The time is taken by an oscillating body to complete one cycle of oscillation. This means the to and fro motion of the body gets repeated after a fixed interval of time.
It is denoted by T. and the unit is second.
The above image describes the motion of the pendulum, it goes from B to A and then back to B from A. Similarly The motion of pendulum from B to C.
Frequency (f): The number of repetitions in one second of a periodic motion is called Frequency (ν). Its unit is Hertz (Hz). The reciprocal of T gives the number of repetitions that occur per unit of time.
Displacement: We defined the displacement of a particle as the change in its position vector. Displacement in periodic motion can be represented by a function which is periodic which repeats after a fixed interval of time.
In the above image, we can see that motion of an oscillating simple pendulum can be described in terms of angular displacement θ from the vertical. And In the above right image, we can see that there is a block whose one end is attached to a spring and another is attached to a rigid wall.x is the displacement from the wall.
SIMPLE HARMONIC MOTION
Simple Harmonic Motion (SHM) is a periodic motion in the body that moves to and fro about its mean position. The restoring force on the oscillating body is directly proportional to its displacement and is always directed towards its mean position.
In the above image, we can see that a particle is vibrating to and fro within the limits –A and +A.
Mathematically,
So, acceleration 
where k is known as force constant
However ,In SHM we know that acceleration
This equation is known as the differential equation of S.H.M.
where ω is known as angular frequency here 
The general expression for solution satisfying the equation d
The oscillatory motion is said to be SHM if the displacement x of the particle from origin varies with time t:
The above graph shows displacement as a continuous function of time.
Now let's consider this x (t) = A cos (ωt + Φ) Where
x (t) : displacement x as a function of time
A= amplitude, It is defined as the magnitude of maximum displacement of the particle from its mean position.
ωt +Φ = phase angle (time-dependent) , ω= angular frequency and Φ = phase constant
SHM is a periodic motion in which displacement is a sinusoidal function of time.
If we plot the graph between displacement versus time we can conclude that displacement is a continuous function of time.
Phase
It is that quantity that determines the state of motion of the particle.
1. Its value is (ωt + Φ)
2. It is dependent on time.
The value of phase at time t=0 is termed as Phase Constant. When the motion of the particle starts it goes to one of the extreme positions at that time phase is considered as 0.
Let x (t) = A cos (ωt) where we are taking (Φ = 0)
1. Mean Position (t= 0)
2. x (0) = A cos (0) = A (cos 0=1)
3. t=T/4, t= T/2, t=3T/4, t=T and t=5T/4
The above figures depict the location of the particle in SHM at different values of t=0, T/4, T/2, 3T/4, T, and 5T/4.The time after which motion repeats is T. The speed is maximum for zero displacements (x=0) and zeroes at the extremes of motion.
In the above graph the curves (3) and (4) are for φ = 0 and -π/4 respectively but the amplitude is the same for both.
Angular Frequency (ω)
Angular frequency refers to the angular displacement per unit time. It can also be defined as the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves). Angular frequency is larger than frequency f (in cycles per second, also called Hz), by a factor of 2π.
Mathematically
Oscillations due to spring
Consider a block if it is pulled on one side and is released, and then it executes to and fro motion about a mean position.
In the above image a block, is on a frictionless surface when pulled or pushed and released, executes simple harmonic motion.
F (x) = –k x (expression for restoring force)
‘K’ is known as spring constant and its value is governed by the elastic properties of the spring.
- The above expression is the same as the force law for SHM and therefore the system executes a simple harmonic motion. Therefore,
- Angular frequency ‘w’ is given by
- The time period of oscillation is
Vertical spring ( loaded spring).
When the spring is suspended vertically from a fixed point and carries the block at its other end as shown, the block will oscillate along the vertical line.
- In first figure we have an unstretched spring of length L
- In the second figure, we load a mass ‘m’ with the spring and it gets stretched by a distance s, this is its equilibrium position when gravity (mg) and restoring force ( ks ) balance each other.
- In the third one we displaced the mass ‘m’ from its equilibrium position by a distance ‘x’. And then the loaded pendulum will start oscillating about its mean/ equilibrium position.
Time period of oscillation 
= 
Combination of springs:
1. Springs in series:
Consider two springs of force constants K1 and K2 respectively, connected in series as shown. They are equivalent to a single spring of force constant K which is given by 
2. Spring in parallel : For a parallel combination as shown, the effective spring constant is K=K1+ K2
