A string is stretched between fixed points separated by \(75.0~\text{cm}.\) It is observed to have resonant frequencies of \(420~\text{Hz}\) and \(315~\text{Hz}\). There are no other resonant frequencies between these two. The lowest resonant frequency for these strings is:

1.	\(155~\text{Hz}\)	2.	\(205~\text{Hz}\)
3.	\(10.5~\text{Hz}\)	4.	\(105~\text{Hz}\)

A wave travelling in the positive x-direction having maximum displacement along y-direction as 1m, wavelength 2π m and frequency of 1/π Hz is represented by

1. y=sin(x-2t)
2. y=sin(2πx-2πt)
3. y=sin(10πx-20πt)
4. y=sin(2πx+2πt)

When a string is divided into three segments of lengths $l_1, l_2 and l_3,$ the fundamental frequencies of these three segments are $v_1, v_2 and v_3$ respectively. The original fundamental frequency (v) of the string is 

1. $\sqrt{v}=\sqrt{v_1}+\sqrt{v_2}+\sqrt{v_3}$

2. $v=v_1+v_2+v_3$

3. $\frac{1}{v}=\frac{1}{v_1}+\frac{{\displaystyle 1}}{{\displaystyle v_2}}+\frac{{\displaystyle 1}}{{\displaystyle v_3}}$

4. $\frac{1}{\sqrt{v}}=\frac{1}{\sqrt{v_1}}+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{v_2}}}+\frac{{\displaystyle 1}}{{\displaystyle \sqrt{v_3}}}$

The equation of a simple harmonic wave is given by 

          $\mathrm{y}=3 \sin \frac{\mathrm{\pi }}{2}(50\mathrm{t}-\mathrm{x})$

where x and y are in meters and t is in seconds. The ratio of maximum particle velocity to the wave velocity is

1. $2\mathrm{\pi }$

2. $\frac{3}{2}\mathrm{\pi }$

3. $3\mathrm{\pi }$

4. $\frac{2}{3}\mathrm{\pi }$

Two waves are represented by the equations ${\mathrm{y}}_1=\mathrm{a} \sin (\mathrm{\omega t}+\mathrm{kx}+0.57)\mathrm{m}$ and ${\mathrm{y}}_2=\mathrm{a} \cos (\mathrm{\omega t}+\mathrm{kx})\mathrm{m}$, where x is in metre and t in second. The phase difference between them is?

1. 1.25 rad

2. 1.57 rad

3. 0.57 rad

4. 1.0 rad

Sound waves travel at 350 m/s through a warm air and at 3500 m/s through brass. The wavelength of a 700 Hz acoustic wave as it enters brass from warm air :

1. increases by factor 20

2. increases by factor 10

3. decreases by factor 20

4. decreases by factor 10

Each of the two strings of length 51.6 cm and 49.1 cm are tensioned separately by 20N force. Mass per unit length of both the strings is same and equal to 1 $g{m}^{-1}$When both the strings vibrate simultaneously the number of beats is :

1. 5                                                   

2. 7

3. 8                                                   

4. 3

Two waves are represented by the equations $y_1=a\sin \omega t$ and $y_2=a\cos \omega t.$ The first wave 

(1) Leads the second by $\pi$

(2) Lags the second by $2\pi$

(3) Leads the second by $\frac{\pi }{2}$

(4) Lags the second by $\frac{\pi }{2}$

The distance between two consecutive crests in a wave train produced in a string is 5 cm. If 2 complete waves pass through any point per second, the velocity of the wave is :

1. 10 cm/sec

2. 2.5 cm/sec

3. 5 cm/sec

4. 15 cm/sec