Giving the necessary mathematical expressions, discuss the transient and steady state of a weakly damped forced oscillator. Show that the average power absorbed by a forced oscillator is given by
time average over one complete cycle of the power, i.e, F(t) v, given above, and using the relations
Using the expressions, for <E> and <P> ,we can now obtain, by using Eqn.(6.42), the expression for the quality factor, i.e.
A more useful definition of the quality factor can be given in terms of the amplitudes. The quality factor can be defined as the ratio of the amplitude, at resonance, to that at low frequencies ( i.e. as → 0).
We have seen above that the amplitude at resonance [see Eqn.(6.39],
and the amplitude at low frequencies has been obtained as
(See Eqn.(6.34))
Thus, the quality factor can be expressed as
which can be further simplified for the case when damping is small when
. In such a case, neglecting b2 in Eqn.(6.48), we get
Using the above relations, we can re-express Amax in terms of the quality factor. Thus
Substituting ,from Eqn.(6.49) for
, the amplitude at resonance can
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What is the effect of damping in an oscillatory system? Differentiate between heavy and critical damping. Show that the displacement of a weakly damped oscillator is given by
x(t ) = a0 exp(−bt ) cos(ωd t − φ)
where symbols have their usual meanings.
The equation of motion of a damped harmonic oscillator is given by
with m = 0.25 kg, b = 0.14 s −1
and ω0 = 18.4 s −1.
Calculate i) the time period; ii) number of oscillations in which its amplitude will become half of its initial value; and iii) number of oscillations in which its
mechanical energy will reduce to half of its initial value.
The amplitude of an oscillator is 8 cm and it completes 100 oscillations in 80s.
i) Calculate its time period and angular frequency. ii) If the initial phase is π/4, write expressions for its displacement and velocity. iii) Calculate the values of maximum
velocity and acceleration
An ambulance siren has frequency 250 Hz. The ambulance is headed towards an
accident site with a speed of 90 km h−1, Two police officers on separate motor
cycles head for the same accident site: one follows the ambulance with a speed of
80 km h−1 and the other approaches the accident site from the other direction with a
speed of 80 km h−1. What frequency does ambulance siren has for each of the police
officers? Take the speed of sound equal to 340 ms−1.
Giving the necessary mathematical expressions, discuss the transient and steady state of a weakly damped forced oscillator. Show that the average power absorbed by a forced oscillator is given by
What do you understand by the normal modes of a coupled oscillator? If a coupled
system has many normal modes, do all normal modes have the same frequency? In a
solid, the speed of elastic longitudinal wave is 1.35 ms−1 If the Young’s modulus of
elasticity of the solid is 2 x 1011 N m −2 ,calculate its mass density.
A harmonic wave on a rope is described by
i) Calculate the wavelength and time period of the wave. ii) Determine the
displacement and acceleration of the element of the rope located at x = 0.58 m at
time, t = 0.41 s.