Indian Institute Of Science Education And Research Pune

PHY102: Waves and Matter

Spring 2015 : Mid-sem exam

Date: 26.2.2015 Time: 2 hours. Total Marks : 40.

Instructions

For questions involving drawing sketches, label the axes. Show all the steps clearly in yourcalculations while arriving at an answer. Use the same symbols/notation as given in the question.

1.(a) A block attached to a spring oscillates with an amplitude of 10 cm and time period of 2 sec.At time t = 0, the block is at x = 0. If x(t) = A cos(ωt+ α), then find ω and α.(b) Two harmonic oscillations along a line are described by equations

y1(t) = A cos(8πt), y2(t) = A cos(12πt).

Find the time period of the beats.(c) The displacements of an oscillating particle in x−y plane are x(t) = a sin(ωt) and y(t) = b cos(ωt).Show that the particle traces an ellipse in x− y plane. (3+3+2)

2.(a) In the case of a forced and damped oscillator, what is the physical meaning of steady state ?(b) For a forced, damped oscillator sketch the maximum velocity as a function of driving frequencyω. What is the value of impedance |Z| at resonance.(c) The frequency of a damped simple harmonic oscillator of mass M is given by

ω2

d = ω2

0 −γ2

4M2

where ω0 is the natural frequency of the system and γ is dissipation coefficient. Find the value of Qif ω2

0− ω2

d = 10−6ω20. (2+2+4)

3. (a) Consider a coupled system with 3 beads connected by string.

(i) How many normal modes are there in this system ?(ii) Sketch the normal modes with lowest and highest frequency.(iii) If Ei denotes the energy of i-th normal mode, what is the total energy of the system, assumingno dissipation ?(iv) If inter-bead distance is increased to 2a, how would the normal frequencies change?

(b) A string with N beads executing transverse oscillations has normal mode frequencies,

ω2

j =2T

Ma

(

1− cosjπ

N + 1

)

.

Show that as a → 0, the normal mode frequency is ωj = j(π/L)√

T/ρ, where ρ is the linear densityof the string and L = (N + 1)a. (5+3)

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4. Consider a system of two blocks and springs shown in Figure 1. It oscillates only in the verticaldirection. The mass of each block is M and each spring has spring constant K.(a) Set up the equations of motion.(b) Show that the normal mode frequencies are

ω2 =

(

3±√5

2

)

K

M.

(4+4)

Figure 1 :

5. A platform is executing simple harmonic motion in a vertical direction with restoring force pro-vided by a spring of spring constant K. The amplitude of oscillation is 5 cm and its frequency is10/π vibrations per second. A block is placed on the platform at the lowest point of its path.(a) What is the value of displacement at which the block loses contact with the platform ?(b) How far will the block rise above the highest point reached by the platform ? (4+4)

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