Oscillations

Periodic Motion
Periodic Motion is the motion that repeats itself in equal intervals of time.

Oscillatory Motion
To and fro motion is called as Oscillatory Motion.

Harmonic Oscillations
Harmonic oscillations are those, which can be represented as a sine or cosine function of an angle.

Linear Simple Harmonic Motion
Th motion of an object moving to and fro along a straight line is called linear simple harmonic motion.

F = -kx where F is the restoring Force ; k is the spring constant of force ; x is the displacement from equilibrium.

a = -kx / m

k/m = ω2

Spring Constant
a) Parallel Connection : k = k1 + k2

b) Series Connection : 1/k = 1/k1 + 1/k2

Differential Equation of Linear SHM
m [d2x / dt2] = 0

Kinetics of SHM
Displacement

x = asin (ωt + α)

☀α is a constant of integration.

Velocity

v = ω sqrt[a2 - x2]

Acceleration

d2x/dt2 = - ω2x

Period

ω = 2π / T

... T = 2π / ω

but, ω = sqrt [k/m]

... T = 2π / sqrt [k/m]

T = 2π / sqrt [a/x]

T = 2π / sqrt [acceleration per unit distance]

Frequency

ω = 2πn

... n = ω / 2π

n = 1/T

Phase

Displacement is given by :- x = a sin(ωt + α)

... Phase Angle is given as :-  ωt + α

Epoch

Epoch is the phase angle of a SHM when t = 0

S.H.M. and U.C.M.
x = R sin (wt + δ)

R = sqrt [a12 + a22 + 2a1a2cos(α1-α2)]

δ = tan-1[a1sinα1 + a2sinα2 / a1cosα1 + a2cosα2]

Energy of a Particle performing S.H.M.
Kinetic Energy = 1/2 k (a2- x2)

Potential Energy = 1/2 kx2

Total Energy = 1/2 mω2a2

Resultant S.H.M.
During the superposition of two oscillations, their displacements at the point of superposition are added up to give the resultant Amplitude.

x = Rsin

Simple Pendulum
F = -mqsinθ

θ = x/L

acceleration = -gx/L = ω2x

T = 2π√(L/g)

f = 1/T

Laws of Simple Pendulum
1) Law of length : The period of a simple pendulum is directly proportional to square root of it's length.

2) Law of Acceleration due to gravity : The period of a simple pendulum is inversely proportional to square root of acceleration due to gravity.

3) Law of Mass : The period of a simple pendulum does not depend upon it's mass.

4) Law of isochronous : The period of a simple pendulum does not dpend upon its amplitude.

Damped Oscillations
Periodic oscillations of gradually decreasing amplitude are called damped harmonic oscillations.

A' = Ae(-BT/2m)