Speed of waves

Consider a system of three connected strings, $S_1$, $S_2$ and $S_3$ with uniform linear mass densities $\mu$ kg/m, $4\mu$ kg/m and $16\mu$ kg/m, respectively, as shown in the figure. $S_1$ and $S_2$ are connected at the point $P$, whereas $S_2$ and $S_3$ are connected at the point $Q$, and the other end of $S_3$ is connected to a wall. A wave generator O is connected to the free end of $S_1$. The wave from the generator is represented by $y = y_0\cos(\omega t - kx)$ cm, where $y_0$, $\omega$ and $k$ are constants of appropriate dimensions. Which of the following statements is/are correct:

Two uniform strings of mass per unit length $\mu$ and $4\mu$, and length $L$ and $2L$, respectively, are joined at point O, and tied at two fixed ends P and Q, as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $\nu_0 = \dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$, which of the following statement(s) is(are) correct?

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $\mathrm v_n$. At a fixed temperature and pressure, which of the following is the correct option?

A string of length 1 m and mass $2\times10^{-5}$ kg is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies 750 Hz and 1000 Hz. The value of tension $T$ is _____ Newton.

If a wave gets refracted into a denser medium, then which of the following is true?

Sound travels in a mixture of two moles of helium and n moles of hydrogen. If rms speed of gas molecules in the mixture is $\sqrt2$ times the speed of sound, then the value of n will be :

A wire of length 30 cm, stretched between rigid supports, has it's n$^{\text{th}}$ and (n+1)$^{\text{th}}$ harmonics at 400 Hz and 450 Hz, respectively. If tension in the string is 2700 N, it's linear mass density is ______ kg/m.

The speed of a transverse wave passing through a string of length 50 cm and mass 10 g is 60 m s$^{-1}$. The area of cross-section of the wire is 2.0 mm$^2$ and its Young's modulus is $1.2\times10^{11}$ N m$^{-2}$. The extension of the wire over its natural length due to its tension will be $x\times10^{-5}$ m. The value of $x$ is _______.

If the tension in each string is $T_0$, the correct match for the highest fundamental frequency in $f_0$ units will be,

The length of the strings 1, 2, 3 and 4 are kept fixed at $L_0$, $\dfrac{3L_0}2$, $\dfrac{5L_0}4$, and $\dfrac{7L_0}4$, respectively. Strings 1, 2, 3, and 4 are vibrated at their 1st, 3rd, 5th, and 14th harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of $T_0$ will be,

A block $M$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at $O$. A transverse wave pulse (Pulse 1) of wavelength $\lambda_0$ is produced at point $O$ on the rope. The pulse takes time $T_{OA}$ to reach point $A$. If the wave pulse of wavelength $\lambda_0$ is produced at point $A$ (Pulse 2) without disturbing the position of $M$ it takes time $T_{AO}$ to reach point $O$. Which of the following options is/are correct?

Which one of the following options correctly represents a thermodynamic process that is used as a correction in the determination of the speed of sound in an ideal gas?

Two wires $W_1$ and $W_2$ have the same radius r and respective densities $\rho_1$ and $\rho_2$ such that $\rho_2=4\rho_1$. They are joined together at the point O, as shown in the figure. The combination is used as a sonometer wire and kept under tension T. The point O is midway between the two bridges. When a stationary wave is set up in the composite wire, the joint is found to be a node. The ratio of the number of antinodes formed in $W_1$ to $W_2$ is :

A uniform string of length 20 m is suspended from a rigid support. A short wave pulse is introduced at its lowest end. It starts moving up the string. The time taken to reach the support is :

(take g = 10 ms$^{-2}$)

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature $T$. Assuming the gases are ideal, the correct statement(s) is(are)

A student is performing an experiment using a resonance column and a tuning fork of frequency $244\ \mathrm s^{-1}$. He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is $(0.350\pm0.005)$ m, the gas in the tube is

(Useful information: $\sqrt{167RT}=640\ \mathrm{J^{1/2}\,mole^{-1/2}}$; $\sqrt{140RT}=590\ \mathrm{J^{1/2}\,mole^{-1/2}}$. The molar masses $M$ in grams are given in the options. Take the values of $\sqrt{10/M}$ for each gas as given there.)

The total length of a sonometer wire between fixed ends is 110 cm. Two bridges are placed to divide the length of wire in ratio 6 : 3 : 2. The tension in the wire is 400 N and the mass per unit length is 0.01 kg/m. What is the minimum common frequency with which three parts can vibrate ?

A 20 cm long string, having a mass of 1.0 g, is fixed at both the ends. The tension in the string is 0.5 N. The string is set into vibrations using an external vibrator of frequency 100 Hz. Find the separation (in cm) between the successive nodes on the string.

The speed of sound of the whistle is