Oscillation and Waves - Detailed Guide

Oscillation is usually known as the periodic fluctuation between two things, and according to the comprehensive sense, oscillation may occur in most of the cases starting from a person’s decision-making process to tides and finally the pendulum of a clock. More precisely, an oscillator is a device where the oscillation is observed. In the case of this pendulum, the potential energy is readily converted into kinetic energy. The process of oscillation is the back and forth movement of the pendulum. 

The wave is denoted as the physical phenomenon characterised by its amplitude, wavelength, and frequency. Wave possesses a velocity where they transfer energy from one space to another. Well, this session of oscillation and wave MCQ will significantly help you to easily crack NEET examinations. 

History of Inventor

Christiaan Huygens, commonly known as Christian Huyghens, was born in 1629 and passed away in 1695. He was a Dutch mathematician, astronomer, and physicist who developed the wave theory of light, revealed the exact structure of Saturn's rings, and contributed significantly to the science of dynamics.

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Christiaan Huygens

While recuperating from a sickness in 1665, he became the first to record the phenomena of linked oscillation in two pendulum clocks (which he invented) in his bedroom.

Oscillations

Simple Harmonic Motion:

Simple Harmonic Motion happens when the restoring force (the force which remains stable towards equilibrium point) is proportional to the prolapse from equilibrium. 

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Diagram showing simple harmonic motion of a pendulum bob.

Characteristics of SHM:

• Each cycle period is constant.

• Repetitious motions through a central equilibrium point.

• The motion is caused by the force directed towards the equilibrium point.

• Equipoise of maximum displacement.  

• $F$ is proportional to the displacement from the equilibrium. 

• Acceleration = $-\omega^2 \times \text{Displacement}$

The time reliance on a single dynamic variable that satisfies this differential equation:

$x + \omega^2 x = 0$

This aforementioned equation can be written in several ways:

• $x(t) = A \cos (\omega t + φ)$

• $x(t) = A \sin \omega t + B \cos \omega t$

• $x(t) = re^{i(ωt + φ)}= (re^{i})e^{iωt} = ce^{iωt}$

Simple Harmonic Oscillator:

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Simple harmonic oscillator

The restoring force will be $F = -kx$. The simple harmonic oscillator is denoted as a supposed spring resting on a frictionless and horizontal surface. Well, if the spring maintains the Hooke's law where the force is proportional to the extension, then it will be called a simple harmonic oscillator. The motion will be called simple harmonic motion. 

Two springs with diverse Amplitude

The diagram given below shows the combination of two springs having spring constants $k_1$, $k_2$ and different magnitude of displacement i.e, $x_1$ and $x_2$. Since the magnitude of displacement is different for both the springs hence they have a diverse value of amplitude. 

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Combination of two springs with different displacement.

Simple Harmonic Motion period is independent of the Amplitude

SHM is produced by the small angular displacements. The period of a pendulum does not necessarily count on the ball’s mass, but only depends on the string’s length. So, with the help of the small angles, the period and frequency of the pendulum are independent of the amplitude. And, the expression of time period of a simple pendulum is,

$T=2\pi \sqrt{\dfrac{l}{g}}$

Here, $l$ is the length of the pendulum and $g$ is the acceleration due to gravity.

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Diagram showing motion of a massive bob.

Graphical representation of SHM

The graphical representation of acceleration, velocity, and displacement of a particle moving in simple harmonic motion with respect to time is illustrated below. 

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Graphical representation of SHM

• The aforementioned displacement curve is a sinusoidal curve and the maximum particle’s displacement will be: $y = ±a$.

• The vibrating particle’s velocity will be highest at the mean position, $v = ± a \omega$ and it will be null at the extreme position. 

• The vibrating particle’s acceleration will be null at the mean position and highest at the extreme position, $∓a \omega^2$.

Wave Motion

The characteristics of wave motion are:

• The motion will be periodic in nature.

• There will be no single dynamical variable present.

One Dimensional Wave Motion

• It is known as a mechanical wave which certainly is described as a transfer of energy, a disturbance, and the deviation of equilibrium is a function of time and position. 

• Examples: Transverse Springs and Longitudinal Springs

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Diagram showing the one dimensional travelling wave.

The Wave Equation:     

Let $y(x,t) = f(x ± vt) = f(u)$

According to chain rule, 

$\dfrac{\partial y}{\partial x}=\dfrac{\partial f}{\partial u}\dfrac{\partial u}{\partial x}=\dfrac{\partial f}{\partial x}$

$\dfrac{\partial y}{\partial t}=\dfrac{\partial f}{\partial u}\dfrac{\partial u}{\partial t}=\pm\dfrac{\partial f}{\partial u}$

The second derivatives,

$\dfrac{\partial^2 y}{\partial x^2}=\dfrac{\partial^2 f}{\partial u^2}$

$\dfrac{\partial^2 y}{\partial t^2}=v^2\dfrac{\partial^2 f}{\partial u^2}=v^2\dfrac{\partial^2 y}{\partial x^2}$

$\dfrac{\partial^2 y}{\partial x^2}=\dfrac{1}{v^2}\dfrac{\partial^2 y}{\partial t^2}$

The general solution would be $y(x,t) =f(x ± vt)$.

Well, the wave equation if the disturbance is made by SHM at one point: 

$y(0,t)=A \cos(\omega t+φ)$

Thus, the formation of the harmonic wave happens. 

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Diagram showing the wave formation of SHM.

Here wavelength is $\lambda$: then $x=\lambda$ and $kx=k \lambda=2\pi$

Wavenumber will be: $k=\dfrac{2 \pi}{\lambda}$.....(k = radians per unit length)

Solved Problem

1. A particle performs SHM with an amplitude of 20 cm and a time period of 2 s. What is the shortest time necessary for the particle to travel 10 cm on each side of the mean position?

Sol:

According to the question, we are provided with the following quantities;

Amplitude of oscillation,$a = 20\,cm$

Time period, $T = 2\,s$

Displacement from the mean position, $y=10\,cm$

If $t$ is the time it takes a particle to travel from the mean position to a location 10 cm to the left or right of the mean position,

$y=a \sin \omega t=a \sin (\dfrac{2\pi t}{T})$

Now putting the values of the given quantities in the above relation, we get;

$10=20 \sin (\dfrac{2\pi t}{2})$

$\dfrac{1}{2}=\sin (\pi t)$

$\sin(\dfrac{\pi}{6})=\sin (\pi t)$...........(because $\sin(\dfrac{\pi}{6})=\dfrac{1}{2}$)

After comparing the angles of the sine from both sides we get;

$\dfrac{\pi}{6}=\pi t$

$t=\dfrac{1}{6}=0.166\,s$

Clearly, the particle's time spent moving between two places 10 cm apart on each side of the mean position will be,

$2t=2\times 0.166=0.332\,s$

Hence, the value of the shortest time necessary for the particle to travel 10cm on both sides is 0.332s. 

Multiple Choice Questions (MCQs)

These Oscillations and waves MCQ for NEET will significantly help you to crack this examination:

1. The length of a pendulum executing SHM is enhanced by 21%. The increase in the percentage in a certain time period of the pendulum of enhanced length is:

1. 21%

2. 10.5%

3. 11%

4. 42%

Ans: (b) 10.5%

2. The pendulum is about to provide the exact time at the equator. What will happen if that pendulum will be taken to the earth’s pole?

1. Unchanged

2. It will gain time

3. It will lose time

4. None of them

Ans: (b) It will gain time.

3.  The equation for the acceleration of a particle is: $a= -k(x+b)$, where $k$ is a positive constant, $x$ is the total distance along the x-axis. The motion of the particle will be:

1. SHM

2. Oscillatory

3. Periodic

4. All of the above

Ans: (d) All of the above. 

4.  A kid is swinging in a sitting position. When he stands up, the time period will be:

1. Increase

2. Decrease

3. Remains exact

4. Increase if the kid is tall and decrease if the kid is short

Ans: (b) Decrease.

Summary

In this post we have mentioned the concept of oscillations and waves. In the oscillation part, we have studied about the concept of simple harmonic motion, how the SHM executes, what are its characteristics and equation. We have also studied the graphical representation of SHM. After that in the wave part, we studied the representation of the wave and its equation of motion. We have also discussed the application part of these concepts through solved problems and MCQs.