OSCILLATIONS AND WAVES
In this blog, you will study all about oscillations and waves. Also, we will study simple harmonic motion, differential equation, simple harmonic oscillator and many more.
- Review of Simple harmonic motion
- Characteristics of SHM
- The differential equation of motion
- Mechanical simple oscillator
- A mass suspended to a spring
- The physical significance of free constant
- Period and frequency of oscillation
- Equivalent force constant for springs in series
- Equivalent force constant for springs in parallel
- Complex notation and Phasor Representation
- Free Oscillations
- Damped Oscillations
- Forced Vibrations
What is the need of studying oscillations or waves ?
A stone thrown into a pool of water will generate a circular expansion of waves moving away from the location of the disturbance. In this instance, a vertical motion (the falling stone) produces a horizontal motion (a traveling wave). The traveling (or propagating) wave has mechanical energy. One of the main reasons that people want to learn about waves is because generating a wave brings into being a means for quickly sending energy from one place to another. And, since this energy can travel faster than a person can, it can be used as a rapid signaling means.
Review of Simple Harmonic Motion
Before talking about simple harmonic motion, first of all, we need to know what is periodic motion.
Periodic Motion: The motion which repeats at regular intervals of time is called periodic motion.
Ex: Oscillation of simple pendulum, Vibration of tuning fork.
What is Simple Harmonic motion?
Simple Harmonic Motion is a periodic motion in which restoring force(F) is directly proportional to the displacement and directed opposite to the displacement of the body towards the equilibrium point.
If ‘ F ‘ is the restoring force
‘x’ is the displacement
F ∝ x
F = – kx
Where ‘k’ is the force or spring constant. Here minus sign indicated force ‘F’ is acting opposite to the displacement ‘x’.
Equilibrium position: It is the point where net force acting on the system is zero.
Types of SHM (simple harmonic motion)
Linear SHM: Here the motion repeats periodically in one direction or vertical oscillation. Ex: Motion of sping
Angular SHM: Here the motion repeats periodically in circular direction or rotatory oscillation. Ex: Simple pendulum
Characteristics of SHM
- SHM is a periodic motion.
- There is a constant restoring force acting on the system/body.
- SHM can be represented in terms of sine or cosine function.
- The acceleration due to restoring force is directly proportional to displacement towards the equilibrium point.
Force ∝ displacement
F ∝ x
F = – kx
- During SHM, the velocity of the particle is maximum at the mean position and minimum at the extreme position.
- The restoring force and acceleration are directed opposite to the displacement towards the equilibrium point.
Terminologies of Simple Harmonic Motion
Displacement ( x ): Distance traveled by the body from the mean position (equilibrium position) during SHM.
Amplitude ( A ): It is the maximum distance traveled by the body from the mean position during SHM or It is the maximum displacement.
Frequency ( f ): Number of oscillations per unit time.
f = 1 / T
Period ( T ): During SHM, It is the time taken to complete one oscillation or cycle.
T = 1 / f
Angular displacement ( θ ): It is the angle covered by the radius vector in time ( t ).
Angular Velocity ( ω ): It is the rate of change of angular displacement with respect to time ( t ) or It is the ratio of angular displacement by time.
ω = ω = 2π / T
Differential Equation of motion for SHM
Mechanical Simple Harmonic Oscillator
The body which executes or performs Simple Harmonic motion is known as a simple harmonic oscillator.
In other words, the simple harmonic oscillator is basically a system where if we displace the object by ‘x’ distance then it will experience a restoring force ‘F’ which opposite to the displacement.
In our future discussions, we are considering an ideal spring as a simple harmonic oscillator and we are going to assume,
- The spring is light or the mass of the spring is very small compared to the suspended mass. Hence, the mass of the spring is neglected.
- The dissipative forces (forces tending to decrease the motion of spring) acting on the oscillations are not taken into an account.
- Restoring force exerted by the spring is directly proportional to its displacement.
F ∝ x
F = – kx
Where k is spring constant. And spring constant or force constant is defined as the force required to produce unit extension or compression. It mainly indicates the stiffness of spring.
But note one thing these conditions can’t be realized completely well in actual practice.
Physical Significance of spring constant ( k )
- The spring constant is a measure of the stiffness of the spring.
- If ‘k’ is large then stiffness is large or it is a strong spring. If ‘k’ is small then it is a weak spring.
An expression for a period of spring
Spring constant for a series combination of springs
Spring constant for a parallel combination of springs
Representation of SHM as a complex numbers
The complex number ‘z’ can be represented in terms of cartesian co-ordinate as ‘ z= x + iy ‘.
Where ‘x’ is a real part and projection ‘z’ along X-axis.
‘iy’ is an imaginary part and is projection of ‘z’ along the Y-axis.
Hence representation of cartesian co-ordinates in terms of complex numbers is called Argand Diagram.
In polar co-ordinate, the Argand diagram can be written as,
z = Acos ωt + i Acos ωt
Representation of ‘z’ in exponential form
From Euler’s therom,
ei θ = cos θ + i sin θ
z = Aeiωt
If particle has already made an angle ‘ ϕ ‘ then,
z = Aei (ωt + ϕ )
z = Acos (ωt + ϕ) + i Asin (ωt + ϕ)
Different cases of simple harmonic motion
- Free Oscillation
- Damped Oscillation
- Forced Oscillation
If an oscillating body oscillates with a constant amplitude at its own natural frequency without the help of an external force is called free oscillation.
But, do you know what does natural frequency mean?
Natural frequency is the rate at which an object vibrates when it is not disturbed by an outside force.
Examples for free oscillations :
- The vertical oscillation of a loaded spring suspended from a rigid support.
- Motion of needle of a sewing machine.
- Motion of pendulum.
- Vibration of tuning fork
Equation of motion for free oscillation
The equation of motion of a free oscillation is given by,
Where, m – mass of oscillating body
k – force constant
x – displacement
The oscillation of the body, whose amplitude goes on decreases with time due to the presence of resistive forces is called damped oscillation.
Examples of damped oscillation :
- Mechanical oscillation of a simple pendulum
- A swing left free to oscillate after being pushed