Oscillations

PART I
-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