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1.

A wave travelling along a string is described by, \(y(x,~t)=0.005 ~\sin(80.0x-3.0t),\) in which the numerical constants are in SI units. The displacement \(y\) of the wave at a distance \(x = 30.0~\text {cm}\) and time \(t=20~\text{s}\) is:
1. \(0.5~\text{mm}\)
2. \(5~\text{mm}\)
3. \(5~\text{m}\)
4. \(5~\text{cm}\)

2.

A steel wire \(0.72~\text{m}\) long has a mass of \(5\times10^{-3}~\text{kg}\). If the wire is under tension of \(60~\text{N}\), the speed of transverse waves on the wire will be:
1. \(85~\text{m/s}\)
2. \(83~\text{m/s}\)
3. \(93~\text{m/s}\)
4. \(100~\text{m/s}\)

3.

Two sitar strings A and B playing the note ‘Dha’ are slightly out of tune and produce beats of frequency \(5\) Hz. The tension of the string B is slightly increased and the beat frequency is found to decrease to \(3\) Hz. What is the original frequency of B if the frequency of A is \(427\) Hz?
1. \(432\) Hz
2. \(424\) Hz
3. \(430\) Hz
4. \(422\) Hz

4.

A stone dropped from the top of a tower of height \(300~\text m\) splashes into the water of a pond near the base of the tower. When is the splash heard at the top?
(Given that the speed of sound in air is \(340~\text{m/s}\) and \(g=9.8~\text{m/s}^{2}\) )
1. \(7.7~\text s\)
2. \(8.7~\text s\)
3. \(6.7~\text s\)
4. \(7.8~\text s\)

5. A transverse harmonic wave on a string is described by, \(y(x,t) = 3.0 ~\sin \left(36t + 0.018x + {\dfrac {\pi} 4}\right)\) where \(x\) and \(y\) are in cm and \(t\) in sec. The positive direction of \(x\) is from left to right. What is the shortest distance between two successive crests in the wave?

1.	\(1.3\) m	2.	\(3.0\) m
3.	\(2.5\) m	4.	\(3.5\) m

6. The transverse displacement of a string (clamped at both ends) is given by;
\(y(x,t)=0.06\sin\Big(\dfrac{2\pi}{3}x\Big)\cos(120\pi t)\)
where \(x\) and \(y\) are in meter and \(t\) in second. The length of the string is \(1.5~\text{m}\) and its mass is \(3\times10^{-2}~\text{kg}.\) The tension in the string is:
1. \(540~\text{N}\) 
2. \(648~\text{N}\) 
3. \(200~\text{N}\) 
4. \(425~\text{N}\)

7.

A bat emits an ultrasonic sound of frequency \(1000\) kHz in the air. If the sound meets a water surface, what is the wavelength of the reflected sound? (The speed of sound in air is \(340\) m/sec and in water is \(1486\) m/sec)
1. \(3.4 \times 10^{-4}~\text{m}\)
2. \(1 . 49 \times 10^{- 3}  ~ \text{m}\)
3. \(2 . 34 \times 10^{- 2}   ~\text{m}\)
4. \(1 . 73 \times10^{- 3}   ~\text{m}\)

8. A \(1~\text m,\) long tube open at one end with a movable piston at the other end shows resonance with a fixed frequency source (a tuning fork of frequency \(340~\text{Hz}\)) when the minimum tube length is \(25.5~\text{cm}.\) The speed of sound in air at the temperature of the experiment is:
(The edge effects may be neglected.)
1. \(324.16~\text{m/s}\)
2. \(320~\text{m/s}\) 
3. \(345~\text{m/s}\)
4. \(346.8~\text{m/s}\)

9.

A steel rod \(100~\text{cm}\) long is clamped at its middle. The fundamental frequency of the longitudinal vibrations of the rod is given to be \(2.53~\text{kHz}.\) What is the speed of sound in steel?
1. \(5.06~\text{km/s}\) 
2. \(5.12~\text{km/s}\) 
3. \(4.29~\text{km/s}\) 
4. \(4.01~\text{km/s}\)

10.

Two sitar strings, \(A\) and \(B,\) playing the note \(\mathrm{Ga},\) are slightly out of tune and produce \(6~\text{Hz}\) beats. The tension in the string \(A\) is slightly reduced, and the beat frequency is found to be reduced to \(3~\text{Hz}.\) If the original frequency of \(A\) is \(324~\text{Hz},\) what is the frequency of \(B?\)

1.	\(316~\text{Hz}\)	2.	\(318~\text{Hz}\)
3.	\(319~\text{Hz}\)	4.	\(314~\text{Hz}\)