1. Answer any two questions : 10 × 2 = 20
a) i) Establish the equation of motion for torsional oscillation. Derive an expression for the natural frequency of free torsional vibration.
ii) When sound wave travels through a medium, the temperature of various points does not remain same. Why ?
iii) What was the necessity of Laplace correction ?
b) i) Calculate the total energy of a lightly damped oscillator.
ii) A cylindrical test tube with a metallic ball inside is floating vertically in water. It was observed that the length of the tube inside water is 15 cm. If now the tube is vertically displaced and released, what will be the time period of vertical oscillation ? The resistance of water is negligible.
c) i) How are stationary waves formed in a stretched string ? State the conditions for generation of harmonics.
ii) Find the changed frequency due to Doppler effect when the source is stationary but the observer is moving.
iii) Evaluate the power carried by a progressive wave.
d) i) Two equal masses are connected by a spring with spring constant k and free ends of the two masses are connected to rigid support by two springs with spring constant k l . The masses are displaced from their equilibrium positions by 1x and 2x respectively. Establish the equations of motion for the two masses and solve them.
ii) What is acoustic interference ?
iii) What is Q-factor ?
2. Answer any three questions : 6 × 3 = 18
a) Find an expression for the time period of torsional oscillation.
b) Explain the working of a tape-recorder.
c) The volume of a room is 3000 m3 and area of all the surfaces is 1300 m2 . After using a 0·5 W sound source it was found that sound reached audible limit after 2 s. If a 100 W source is used the time taken is 3 s. What is the total absorption by the room ? What is the average sound absorption coefficient of the surfaces of the room ?
d) Determine the kinetic energy of a string vibrating in the transverse mode.
e) Equation for a progressive wave is y = 6 sin π ( 3·50 t – 0·03 x ). What are the amplitude, frequency and velocity of this wave ? If the distance between two points in the direction of
propagation be 20 cm, what is the phase difference at any instant ?
f) Determine the expression for resistance offered by a gaseous medium against the propagation of sound.
3. Answer any four questions : 3 × 4 = 12
a) Show that the discharging of a capacitor in an L-C circuit follows SHM.
b) Discuss the expected properties of a good quality microphone.
c) Describe the working of a human ear.
d) What are the uses of ultrasonics ?
e) A string under tension is plucked at a distance 41 th of its length from the fixed end. Which frequencies will be absent in the resulting sound ?
f) Determine the velocity of sound in air after taking Laplace correction into consideration.
g) Define resonance. What is sharpness of resonance ?
h) Sound travels faster on a rainy day than in a dry day. Why ?