Two uniform strings of mass per unit length $\mu$ and $4 \mu$, and length $L$ and $2 L$, respectively, are joined at point $\mathrm{O}$, and tied at two fixed ends $\mathrm{P}$ and $\mathrm{Q}$, as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$, which of the following statement(s) is(are) correct?

Two vehicles, each moving with speed u on the same horizontal straight road, are approaching each other. Wind blows along the road with velocity w. One of these vehicles blows a whistle of frequency f₁. An observer in the other vehicle hears the frequency of the whistle to be f₂. The speed of sound in still air is V. The correct statement(s) is(are)

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation

y(x, t) = (0.01 m) sin[(62.8 m^$-$1)x] cos[(628 s^$-$1)t]

Assuming $\pi$ = 3.14, the correct statement(s) is(are)

Column I shows four systems, each of the same length L, for producing standing waves. The lowest possible natural frequency of a system is called its fundamental frequency, whose wavelength is denoted as $\lambda$_f. Match each system with statements given in Column II describing the nature and wavelength of the standing waves :

Under the influence of the Coulomb field of charge +Q, a charge $-$q is moving around it in an elliptical orbit. Find out the correct statement(s):

A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of 10 cm/s. The wavelength of the waves is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is :

A vibrating string of certain length 1 under a tension T resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length 75 cm inside a tube closed at one end. The string also generates 4 beats per second when excited along with a tuning fork of frequency n. Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be 340 m/s, the frequency n of the tuning fork in Hz is:

In the experiment to determine the speed of sound using a resonance column,

The speed of sound of the whistle is

The distribution of the sound intensity of the whistle as observed by the passengers in train $\mathrm{A}$ is best represented by

The spread of frequency as observed by the passengers in train B is

Column I describe some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situation in Column I with the characteristics in Column II and indicate your answer by darkening appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

A massless rod is suspended by two identical strings AB and CD of equal length. A block of mass $m$ is suspended from point $O$ such that BO is equal to $x$. Further, it is observed that the frequency of 1st harmonic (fundamental frequency) in AB is equal to 2 nd harmonic frequency in CD. Then, length of BO is

Find the number of times the intensity is maximum in the time interval of 1 sec.

Find the wave velocity of louder sound.

Find the number of times $y_1+y_2=0$ at $x=0$ in $1 s$.

A whistling train approaches a junction. An observer standing at the junction observes the frequency to be 2.2 kHz and 1.8 kHz of the approaching and the receding train. Find the speed of the train (speed of sound = 300 m/s).

A transverse harmonic disturbance is produced in a string. The maximum transverse velocity is 3 m/s and the maximum transverse acceleration is 90 m/s$^2$ . If the wave velocity is 20 m/s, then find the waveform.