Sunday, January 3, 2010

8-Oscillations Notes


Free Oscillation – oscillations which occur at the natural frequency of a body when displaced from the equilibrium position and allowed to oscillate freely without the application of any external periodic force. (Note: In real life situations, damping forces will reduce the amplitude of the free oscillation).

Period – the time required to complete one oscillation or cycle of motion.

Frequency – the number of complete oscillations or cycles per unit time.

Displacement – the linear distance of the oscillating body from its equilibrium position at any instant in a specified direction.

Amplitude is the magnitude of the maximum value of the displacement.

Equilibrium position - position at which no net force acts on the oscillation mass.

Simple Harmonic Motion is defined as motion taking place in which the acceleration of the object is directly proportional to its displacement from a fixed point and in the opposite to the displacement.

Defining equation for SHM: a = - ω2 x

where a = acceleration, in m s-1 , ω = angular frequency, in rad s-1,

x = displacement, in m

ω = angular frequency = 2πf = 2π/T

For a body moving with simple harmonic motion and with the displacement given by the acceleration are given by

x = xo sin ωt, the velocity and
in terms of time velocity acceleration v = x0 ω cos ωt a = - x0 ω2 sin ωt
in terms of displacement v = ± ω√(x02 – x2) a = - ω2 x
v is max at the equilibrium position. vmax = ω x0 where x0 is the amplitude
a is max at the amplitude positions. amax = ω2 x0

The kinetic energy of a body undergoing SHM is ½ m ω2 (x02 – x2).

The potential energy of a body undergoing SHM is ½ m ω2 x2.

Damped Oscillations – oscillations in which the amplitude diminishes with time as a result of dissipative forces that reduce the total energy of the oscillations.

Critical Damping
– the displacement is reduced to zero in the minimum time possible without any oscillations occurring.

Forced Oscillations - oscillations which results from the application of an external periodic force. A periodic force is applied to sustain the oscillations, doing work against the dissipative forces or replacing the energy lost to dissipative forces.

Resonance – a phenomenon that occurs when the frequency at which an object is being made to vibrate (the driving frequency) is equal to its natural frequency which causes it to vibrate with a maximum amplitude.
Increased damping will reduce the maximum amplitude, i.e. the peak of the curve will be lower and to the left.


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