PHE-02/BPHE-102 Solved Assignment For IGNOU BSC 2018
Course Title: Oscillations and Waves
This Solution Is Valid Till: 31st December 2018
1. a) 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.
b) A body of mass 0.15 kg executes SHM described by the equation
x(t) = 2sin(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.
2. a) 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( ωdt−φ)
where symbols have their usual meanings.
b) The equation of motion of a damped harmonic oscillator is given by
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.
3. a) 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
b) 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 35.1 ms .
−1 If the Young’s modulus of elasticity of the solid is 2×1011 N m−2, calculate its mass density.
4. a) A sinusoidal wave is described by
y(x,t) = 0.3 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.
b) Two waves, travelling along the same direction, are given by Suppose that the values of ω1 and 1 k 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. (5+5)
5. a) 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 .
b) 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 = 41.0 s.