OSCILLATION
SIMPLE HARMONIC MOTION
meaning
repetitive movement back and forth through an equilibrium, or central, position, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side
how to get equation x = A cos ωt
Substituting F = kx into Newton’s second
law gives the equation of motion
by differentiating the displacement, we can get velocity
− A ω sin ( ω t + ϕ )
by differentiating the velocity, we get acceleration
−ω^2A sin(ωt+ϕ)
initial phase angle
in phase, oscillating together with identical motion
antiphase, opposite direction and amplitude at the same time
SIMPLE HARMONIC MOTION RELATED TO CIRCULAR MOTION
simple pendulum
consist of a mass at the end of a lightweight cord
ENERGY IN SIMPLE HARMONIC MOTION
potential energy
1/2 mω^2 x^2
total energy
E= U+K= 1/2 kx^2+ 1/2 mv^2
kinetic energy
1/2 mω^2 (A^2-x^2)
energy variation with displacement
energy variation with time
DAMPED OSCILLATION
meaning
harmonic motion with a frictional or drag force.
if
then
value of the damping ratio ζ critically determines the behavior of the system
Critically damped (ζ = 1)
Underdamped (ζ < 1)
Overdamped (ζ > 1)
The system returns (exponentially decays) to steady state without oscillating
The system returns to steady state as quickly as possible without oscillating (although overshoot can occur if the initial velocity is nonzero)
The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero
In order to be in SHM
the restoring force must be proportional to the negative of the displacement.
F= -mg sin θ
period and frequency
period does not depend on the mass as we consider the cord is massless and the amplitude is small