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A source \( (S) \) of sound has frequency \( 240 \, \text{Hz} \). When the observer \( (O) \) and the source move towards each other at a speed \( v \) with respect to the ground (as shown in Case \( 1 \) in the figure), the observer measures the frequency of the sound to be \( 288 \, \text{Hz} \). However, when the observer and the source move away from each other at the same speed \( v \) with respect to the ground (as shown in Case \( 2 \) in the figure), the observer measures the frequency of sound to be \( n \) Hz. The value of \( n \) is __________.

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\( S_1 \) and \( S_2 \) are two identical sound sources of frequency \( 656 \, \text{Hz} \). The source \( S_1 \) is located at \( O \) and \( S_2 \) moves anti-clockwise with a uniform speed \( 4\sqrt{2} \, \text{m s}^{-1} \) on a circular path around \( O \), as shown in the figure. There are three points \( P, Q \) and \( R \) on this path such that \( P \) and \( R \) are diametrically opposite while \( Q \) is equidistant from them. A sound detector is placed at point \( P \). The source \( S_1 \) can move along direction \( OP \).
[Given : The speed of sound in air is \( 324 \, \text{m s}^{-1} \) ]

When only \( S_2 \) is emitting sound and it is at \( Q \), the frequency of sound measured by the detector in \( {H_2} \) is_________. 

A string of length \( 1 \, \text{m} \) and mass \( 2 \times 10^{-5} \, \text{kg} \) is under tension \( T \). When the string vibrates, two successive harmonics are found to occur at frequencies \( 750 \, \text{Hz} \) and \( 1000 \, \text{Hz} \). The value of tension \( T \) is _______ Newton.

A point source \( S \) emits unpolarized light uniformly in all directions. At two points \( A \) and \( B \), the ratio \( r = \frac{I_A}{I_B} \) of the intensities of light is \( 2 \). If a set of two polaroids having \( 45^\circ \) angle between their pass-axes is placed just before point \( B \), then the new value of \( r \) will be ________.

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\( S_1 \) and \( S_2 \) are two identical sound sources of frequency \( 656 \, \text{Hz} \). The source \( S_1 \) is located at \( O \) and \( S_2 \) moves anti-clockwise with a uniform speed \( 4\sqrt{2} \, \text{m s}^{-1} \) on a circular path around \( O \), as shown in the figure. There are three points \( P, Q \) and \( R \) on this path such that \( P \) and \( R \) are diametrically opposite while \( Q \) is equidistant from them. A sound detector is placed at point \( P \). The source \( S_1 \) can move along direction \( OP \).
[Given : The speed of sound in air is \( 324 \, \text{m s}^{-1} \) ]

Consider both sources emitting sound. When \( S_2 \) is at \( R \) and \( S_1 \) approaches the detector with a speed \( 4 \, \text{m s}^{-1} \), the beat frequency measured by the detector is ______ \( \text{Hz} \).

A source, approaching with speed \( u \) towards the open end of a stationary pipe of length \( L \), is emitting a sound of frequency \( f_{s} \). The farther end of the pipe is closed. The speed of sound in air is \( v \) and \( f_{0} \) is the fundamental frequency of the pipe. For which of the following combination(s) of \( u \) and \( f_{s} \), will the sound reaching the pipe lead to a resonance?

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?

Consider a system of three connected strings, \( S_1, S_2 \) and \( S_3 \) with uniform linear mass densities \( \mu \, \text{kg} / \text{m} \), \( 4\mu \, \text{kg} / \text{m} \) and \( 16\mu \, \text{kg} / \text{m} \), 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 \( y = y_0 \cos(\omega t - kx) \, \text{cm} \), where \( y_0 \), \( \omega \) and \( k \) are constants of appropriate dimensions.
Which of the following statements is/are correct:

An ideal gas is in thermodynamic equilibrium. The number of degree 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 \( v_{n} \). At a fixed temperature and pressure, which of the following is the correct option?