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 characterized 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. 


Oscillations:

  1. 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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Characteristics of SHM:

  • Each cycle period is constant

  • Repetitious motions through a central equilibrium point

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

  • Equipoise of maximum displacement.  

  • F is proportional to the displacement from the equilibrium. 


Acceleration = -ω2 * Displacement


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

x + ω2x = 0

This aforementioned equation can be written in several ways:

  • x(t) = A cos (ωt + φ)

  • x(t) = A sin ωt + B cos ωt

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


Simple Harmonic Oscillator:

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The restoring force will be F = -kx.

The simple harmonic oscillator is denoted as supposed spring is 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:


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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. 

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Graphical representation of SHM: 

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

  • 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 ω 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 ω2.


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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The Wave Equation:     

Let y(x,t) = f(x ± vt) = f(u)

  • According to chain rule, 


∂y/∂x = ∂f/∂u * ∂u/∂x = ∂f/∂x

∂y/∂t = ∂f/∂u *  ∂u/∂t = ± v∂f/∂u


  • The second derivatives,


2y/∂x2 = ∂2f/∂u2

2y/∂t2 = v22f/∂u2 = v22y/∂x2

2y/∂x2 = 1/v22y/∂t2



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)=Acos(ωt+φ)

  •  Thus, the formation of the harmonic wave happens. 


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Here wavelength is λ: then x=λand kx=kλ=2π

Wavenumber will be: k=2π/λ [k = radians per unit length]


MCQ:

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%

A: (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

A: (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

A: (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

A: (b) Decrease.