Revision: Oscillations Physics HSC Science (General) 12th Standard Board Exam Maharashtra State Board

Oscillation refers to a complete 'to and fro' motion of the body along its course.

A fixed time period after which a motion repeats itself is called the time period.

Any motion which repeats itself after a definite interval of time is called periodic motion.

A mathematical function which represents a periodic motion is called a periodic function.

A particular type of periodic motion in which a particle moves to and fro under the influence of restoring force about a mean position, where the acceleration is directly proportional to displacement from the mean position and its direction is always towards the mean position, is called Linear Simple Harmonic Motion.

Linear simple harmonic motion (S.H.M.) is defined as the linear periodic motion of a body, in which the restoring force (or acceleration) is always directed towards the mean position and its magnitude is directly proportional to the displacement from the mean position.

Linear S.H.M. is defined as the linear periodic motion of a body in which the restoring force (or acceleration) is always directed towards its mean position and its magnitude is directly proportional to the displacement from the mean position.
Consider a particle ‘P’ moving along the circumference of a circle of radius 'a' and centre O, with uniform angular speed of 'ω' in anticlockwise direction as shown.
Particle P along the circumference of the circle has its projection particle on diameter AB at point M.

The distance an object in SHM travels from its initial position as a function of time is called displacement.

The rate of change of velocity with respect to time at any instant of a particle executing SHM is called acceleration.

The rate of change of displacement with respect to time at any instant by a particle executing SHM is called velocity.

The maximum displacement of a particle performing SHM from its mean position is called amplitude.

The time taken by the particle performing SHM to complete one oscillation is called period of SHM.

A way to analyse simple harmonic motion (SHM) by using a circle as reference is called the reference circle method.

The state of oscillation of a particle performing SHM, represented by the angular displacement θ, is called phase in SHM.

The initial phase of a particle i.e., the phase at time t=0t=0 is called epoch.

The pictorial representation of variation of displacement, velocity, and acceleration of a particle performing SHM with respect to time (or ωtωt) is called graphical representation of SHM.

The sum of kinetic energy and potential energy of a particle performing SHM is called total energy.

The energy possessed by a particle performing SHM due to its displacement from the mean position, which is maximum at the extreme position and minimum at the mean position, is called potential energy.

The energy possessed by a particle performing SHM due to its motion, which is maximum at the mean position and minimum (zero) at the extreme position, is called kinetic energy.

A simple pendulum whose period is two seconds is called a second's pendulum.

A simple pendulum whose period of oscillation is exactly two seconds is called a second’s pendulum.

An ideal simple pendulum consists a point mass suspended from a perfectly rigid support by weightless, inextensible and perfectly flexible fibre.

An ideal simple pendulum is a heavy particle suspended by a massless, inextensible, flexible string from a rigid support.

A heavy but small sized metallic bob suspended by a light, inextensible and flexible string, which performs oscillatory motion, is called a simple pendulum.

A type of oscillatory motion where an object undergoes periodic angular displacement about an equilibrium position due to a restoring torque, which is the rotational analogue of linear SHM, is called angular SHM.

The oscillations of a body whose amplitude goes on decreasing with time due to the presence of dissipative forces are called damped oscillations.

The oscillations of a body whose amplitude remains the same throughout the time are called undamped oscillations.

The oscillation of a body under the influence of an external periodic force is called forced oscillation.

The oscillations in which a body or system oscillates with its own natural frequency without being acted upon by an external force are called free oscillations.

The phenomenon that occurs when an external periodic force matches the natural frequency of a system, leading to maximum amplitude of oscillation, is called resonance.

Any motion which repeats itself after a definite interval of time is called periodic motion.

A body performing periodic motion goes on repeating the same set of movements. The time taken for one such set of movements is called its period or periodic time.

Simple harmonic motion is an oscillatory motion in which the restoring force (or acceleration) is directly proportional to the displacement from the equilibrium position and is directed towards it.

Another type of periodic motion in which a particle repeatedly moves to and fro along the same path is the oscillatory or vibratory motion.

Linear S.HM. is defined as the linear periodic motion of a body, in which force (or acceleration) is always directed towards the mean position and its magnitude is proportional to the displacement from the mean position.

The smallest interval of time after which the to and fro motion is repeated is called its period (T).

The number of oscillations completed per unit time is called the frequency (n) of the periodic motion.

The maximum displacement of a particle performing S.H.M from its mean position is called the amplitude of S.H.M.

The time taken by the particle performing S.H.M. to complete one oscillation is called the period of S.H.M.

Mathematically,

T = 2π\[\sqrt{\frac{m}{k}}\]

The number of oscillations performed by a particle performing S.H.M. per unit time is called the frequency of S.H.M.

Mathematically,
n = \[\frac {1}{T}\] = \[\frac {ω}{2π}\] = \[\frac{1}{2\pi}\sqrt{\frac{k}{m}}\]

Phase in S.H.M. is the angular quantity θ = (ωt + ϕ) which specifies the state of oscillation of a particle at a given instant.

An ideal simple pendulum consists of a point mass suspended by a massless, inextensible, and perfectly flexible string from a fixed rigid support.

The distance between the point of suspension and the centre of gravity of the bob is called the length of the pendulum.

A simple pendulum whose period is two seconds is called a second's pendulum. 

Angular S.H.M is defined as the oscillatory motion of a body in which the torque for angular acceleration is directly proportional to the angular displacement and its direction is opposite to that of angular displacement. 

Periodic oscillations of gradually decreasing amplitude are called damped harmonic oscillations, and the oscillator is called a damped harmonic oscillator.

F(t) = F(t + T)

where T is the time period of the periodic function.

F = −kx, a = −\[\frac {kx}{m}\]​

\[\frac{d^2x}{dt^2}+\frac{k}{m}x\] = 0 OR \[\frac{d^2x}{dt^2}+\omega^2x\] = 0

θ = (ωt + ϕ)

where ω = angular frequency, t = time, ϕ = initial phase (epoch).

v_y​ = rω cos(ωt + ϕ), v = rω

U = \[\frac {1}{2}\]kx² = \[\frac {1}{2}\]kA² cos² (ωt + Φ)

E = U + K = \[\frac {1}{2}\]kA² = \[\frac {1}{2}\]mω² A²

K = \[\frac {1}{2}\]mv² = \[\frac {1}{2}\]mω² A² sin2(ωt + Φ) = \[\frac {1}{2}\]KA² sin² (ωt + Φ) = \[\frac {1}{2}\]k(A² - x²)

T = 2π\[\sqrt {\frac {l}{g}}\]

n = \[\frac {1}{2π}\]\[\sqrt {\frac {g}{l}}\]

T = 2π\[\sqrt {\frac {L_s}{g}\] = 2 seconds

\[I\frac{d^2\theta}{dt^2}+C\theta=0\]

ω = \[\sqrt {\frac {k}{m}\]

\[\omega^{\prime}=\sqrt{\frac{k}{m}-\left(\frac{b}{2m}\right)^2}\]

\[\frac{d^2x}{dt^2}+\omega^2x=0\]

\[m\frac{d^{2}x}{dt^{2}}+b\frac{dx}{dt}+kx\] = 0 & emsp; (b = damping constant)

x = A sin(ωt + α)

x = Ae^−bt/2m sin(ω′t + ϕ)

A = Ae^−bt/2m

\[\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=F_0\cos\omega_dt\]

x = A cos(ω_d​t + ϕ)

A = \[\frac{F_0}{\sqrt{m^2(\omega^2-\omega_d^2)^2+\omega_d^2b^2}}\]

ω = ω_0​

where ω = driving angular frequency and ω_0​ = natural angular frequency of the system.

At resonance: A = \[\frac {F_0}{ω_0b}\]

\[\frac{d^2x}{dt^2}+\omega^2x=0\] ⇒ x = ±A cos(ωt)

T = \[\frac {2π}{ω}\] = \[\frac {2\pi}{\sqrt{\text {angular acceleration per unit angular displacement}}}\]

T = \[2\pi\sqrt{\frac{I}{\mu B}}\]

(a) Particle starts from mean position (towards positive direction):

(b) Particle starts from positive extreme position:

• When a particle is subjected simultaneously to two SHMs having the same period and along the same path, the resultant motion is also SHM along the same path.
• The resultant displacement of the particle at any instant is equal to the vector sum of its individual displacements due to both the SHMs at that instant.

• The total mechanical energy of a harmonic oscillation is independent of time, as expected for motion under any conservative force.
• Both kinetic and potential energies peak twice during each period of SHM.
• Period of kinetic energy and potential energy = \[\frac {T}{2}\].

• In S.H.M., displacement, velocity, and acceleration change with time in sine or cosine form.
• Velocity is \[\frac {π}{2}\] ahead of displacement, and acceleration is opposite to displacement.
• Starting from the mean position → sine curve; starting from the extreme position → cosine curve.

• Total mechanical energy in S.H.M. is constant and equals
  E = \[\frac {1}{2}\]kA².
• At the mean position, K.E. is maximum, and P.E. is zero.
  At extreme position: P.E. is maximum, and K.E. is zero.
• During motion, energy continuously converts between kinetic and potential, but total energy remains conserved.

• For a small angular displacement, the restoring force of a simple pendulum is
  F = −\[\frac {mg}{L}\]x,
  hence the motion is simple harmonic.
• The time period of a simple pendulum is
  T = \[2\pi\sqrt{\frac{L}{g}},\]
  which is valid only for small oscillations.
• The period is directly proportional to \[\sqrt {L}\], inversely proportional to \[\sqrt{g}\], and is independent of mass and amplitude (for small angles).