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.
The normal mode of the oscillatory system is a motion scheme in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The normal mode is
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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.
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.
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 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.
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.
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)
Explain the formation of wave packet.