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.
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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 body of mass 0.15 kg executes SHM described by the equation
x(t ) = 2 sin(πt + π / 4)
where x is in meters and t is in seconds.
i) Determine the amplitude and time period of the oscillation.
ii) Calculate the initial values of displacement and velocity.
iii) Calculate the values of time when the energy of the oscillator is purely kinetic.
Two waves, travelling along the same direction, are given by
y1(x, t) = asin (ω1t − k1x)
and y2 (x, t) = asin (ω2t − k2x)
Suppose that the values of ω1 and k1 are respectively slightly greater than ω2 and
k2 . i) Obtain an expression for the resultant wave due to their superposition. ii)
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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.
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.
A sinusoidal wave is described by
y(x, t) = 3.0 sin (5.95t − 4.20x) cm
where x is the position along the wave propagation. Determine the amplitude, wave
number, wavelength, frequency and velocity of the wave.