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1. a) Derive expressions for potential energy and kinetic energy of an oscillating spring-mass

system. (5+5)

b) The displacement of a simple harmonic oscillator is given by

p

+

p

=

2 4

( ) 2sin

t

x t

where 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) are

superposed. Calculate the (i) maximum and minimum amplitudes, and (ii) the frequency of

amplitude modulation. (5+5)

3. For a damped harmonic oscillation, the equation of motion is

0, 2

2

+ g + kx =

dt

dx

dt

d x

m

with 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 initial

value, (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 the

lower end of the spring. After equilibrium has been reached, the upper end of the spring is

moved 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 as

y (t) = a sin (wt ± f)

whereas the equation of a wave is written as

y (x, t) = a sin (wt − kx)

Highlight the differences between the two. (5)

4

b) The equation of transverse wave on a rope is

y (x, t) = 5 sin (4.0t −0.02x)

where y and x are measured in cm and t is expressed in second. Calculate the maximum

speed 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 as

follows:

y1 = 0.2 sin 3pt

and y2 = 0.2 sin (3pt +

8

p

)

Calculate the wavelength and speed of the associated wave. (5+5)

7. Two waves, travelling along the same direction, are given by

y1 = 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 an

expression for the resultant wave obtained by their superposition, and (ii) explain the

formation of wave packet. (5+5)

8. a) A stretched string is observed to vibrate with frequency 30 Hz in its fundamental mode

when the supports are 60 cm apart. The amplitude at the antinode is 3 cm. The string has

a mass of 30g. Calculate the speed of propagation of the wave and the tension in the

string. (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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