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# 1. a) Derive expressions for potential energy and kinetic energy of an oscillating spring-masssystem. (5+5)b) The displacement of a simple harmonic oscillator is given by p+p=2 4( ) 2sintx twhere x is measured in cm and t in second. Calculate the (i) period of oscillation (ii)maximum velocity, and (iii) maximum acceleration. (3+3+4)2. Two collinear harmonic oscillations x1 = 8 sin (100 pt) and x2 = 12 sin (96 pt) aresuperposed. Calculate the (i) maximum and minimum amplitudes, and (ii) the frequency ofamplitude modulation. (5+5)3. For a damped harmonic oscillation, the equation of motion is0, 22+ g + kx =dtdxdtd xmwith m = 0.25 kg, g = 0.07 kgs−1 and k = 85 Nm−1. Calculate (i) the period of motion,(ii) number of oscillations in which its amplitude will become half of its initial value,(iii) the number of oscillations in which its mechanical energy will drop to half of its initialvalue, (iv) its relaxation time, and (v) quality factor. (4×5 = 20)4. A spring is stretched 5 × 10−2 m by a force of 5 × 10−4 N. A mass of 0.01 kg is placed on thelower end of the spring. After equilibrium has been reached, the upper end of the spring ismoved up and down so that the external force acting on the mass is given by F(t) = 20 cos wt.Calculate (i) the position of the mass at any time, measured form the equilibrium position and(ii) the angular frequency for which resonance occurs. (5+5)5. a) The displacement of an oscillator is written asy (t) = a sin (wt ± f)whereas the equation of a wave is written asy (x, t) = a sin (wt − kx)Highlight the differences between the two. (5)4b) The equation of transverse wave on a rope isy (x, t) = 5 sin (4.0t −0.02x)where y and x are measured in cm and t is expressed in second. Calculate the maximumspeed of a particle on the rope. (5)6. The oscillations of two points x1 and x2 at x = 0 and x = 1 m respectively are modelled asfollows:y1 = 0.2 sin 3ptand y2 = 0.2 sin (3pt +8p)Calculate the wavelength and speed of the associated wave. (5+5)7. Two waves, travelling along the same direction, are given byy1 = a sin (w1t − k1x)and y2 = a sin (w2t − k2x)Suppose that w1 and k1 are respectively slightly greater than w2 and k2. (i) Derive anexpression for the resultant wave obtained by their superposition, and (ii) explain theformation of wave packet. (5+5)8. a) A stretched string is observed to vibrate with frequency 30 Hz in its fundamental modewhen the supports are 60 cm apart. The amplitude at the antinode is 3 cm. The string hasa mass of 30g. Calculate the speed of propagation of the wave and the tension in thestring. (7)b) State whether the variation in pressure at nodes in a stationary wave is maximum or zero.Justify your answer.

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