Oscillations Formula's Summary


-If a particle moves such that it retraces its path regularly after regular interval of time,its motion is said to be periodic Ex-Motion of earth around Sun

-If a body in periodic motion moves back and forth over the same path then the motion is said to be oscillatory motion

-Simple harmonic motion is simplest form of oscillatory motion

-SHM is a kind of motion in which the restoring force is propotional to the displacement from the mean position and opposes its increase.Mathematically restoring force is
F=-Kx
Where K=Force constant
x=displacement of the system from its mean or equilibrium position
Diffrential Equation of SHM is
d2x/dt2 + ω2x=0
Solutions of this equation can both be sine or cosine functions .We conveniently choose
x=Acos(ωt+φ) where A,ω and φ all are constants

-Quantity A is known as amplitude of SHM which is the magnitude of maximum value of displacement on either sides from the equilibrium position

-Time period (T) of SHM the time during which oscillation repeats itself i.e, repeats its one cycle of motion and it is given by
T=2π/ω where ω is the angular frequency

-Frequency of the SHM is the number of the complete oscillation per unit time i.e, frequency is reciprocal of the time period
f=1/T
Thus angular frequncy
ω=2πf

-Velocity of a system executing SHM as a function of time is
v=-ωAsin(ωt+φ)

-Acceleration of particle executing SHM is
a=-ω2Acos(ωt+φ)

So a=-ω2x

This shows that acceleration is proportional to the displacement but in opposite direction

-At any time t KE of system in SHM is
KE=(1/2)mv2
=(1/2)mω2A2sin2(ωt+φ)
which is a function varying periodically in time

-PE of system in SHM at any time t is
PE=(1/2)Kx2
=(1/2)mω2A2cos2(ωt+φ)

-Total Energy in SHM
E=KE+PE
=(1/2)mω2A2
and it remain constant in absense of dissapative forces like frictional forces

1) Some system Executing SHM

a)Oscillations of a Spring mass system

-In this case particle of mass m oscillates under the influence of hooke's law restoring force given by F=-Kx where K is the spring constant

Angular Frequency ω=√(K/m)

Time period T=2&#960√(m/K)

And frequency is =(1/2&#960)√(K/m)

Time period of both horizontal ans vertical oscillation are same but spring constant have diffrent value for horizontal and vertical motion

b) Simple pendulum

-Motion of simple pendulum oscillating through small angles is a case of SHM with angular frequency given by
ω=√(g/L)
and Timeperiod
T=2&#960√(L/g)
Where L is the length of the string.

-Here we notice that period of oscillation is independent of the mass m of the pendulum

c) Compound Pendulum

- Compound pendulum is a rigid body of any shape,capable of oscillating about the horizontal axis passing through it.
-Such a pendulum swinging with small angle executes SHM with the timeperiod

T=2&#960√(I/mgL)

Where I =Moment of inertia of pendulum about the axis of suspension
L is the lenght of the pendulum

(2) Damped Oscillation

-SHM which continues indefinitely without the loss of the amplitude are called free oscillation or undamped and it is not a real case

- In real physical systems energy of the oscillator gradually decreases with time and oscillator will eventually come to rest.This happens because in acutal physical systems,friction(or damping ) is always present

-The reduction in amplitude or energy of the oscilaltor is called damping and oscillation are call damped

(3) Forced Oscillations and Resonance.

- Oscillations of a system under the influence of an external periodic force are called forced oscillations

- If frequency of externally applied driving force is equal to the natural frequency of the oscillator resonance is said to occur

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