Two uniform strings of mass per unit length $\mu$ and $4 \mu$,and length $L$ and $2 L$,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 $v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$,which of the following statement$(s)$ is(are) correct?
$(A)$ With a node at $O$,the minimum frequency of vibration of the composite string is $v_0$
$(B)$ With an antinode at $O$,the minimum frequency of vibration of the composite string is $2 v_0$
$(C)$ When the composite string vibrates at the minimum frequency with a node at $O$,it has $6$ nodes,including the end nodes
$(D)$ No vibrational mode with an antinode at $O$ is possible for the composite string

If a microwave and an ultrasonic sound wave have the same wavelength, the ratio of their frequencies is approximately:

$A$ parallel beam of light is incident on a tank filled with water up to a height of $61.5 \,mm$ as shown in the figure below. Ultrasonic waves of frequency $0.5 \,MHz$ are sent along the length of the water column using a transducer placed at the top and they form longitudinal standing waves in the water. Which of the schematic plots below best describes the intensity distribution of the light as seen on the screen? (Take the speed of sound in water to be $1500 \,m/s$)

State whether the following statements are True or False:
$(i)$ Compared to moist air,the speed of sound is greater in dry air.
$(ii)$ When the prongs of a tuning fork are rubbed,its frequency decreases.
$(iii)$ In the case of a stationary wave,the amplitudes of particles in any one loop are the same.
$(iv)$ In the case of a stationary wave,the amplitude of a particle decreases from node to antinode.

The wavelength of light in the visible part $(\lambda_V)$ and for sound $(\lambda_S)$ are related as:

The persistence of sound in a room after the source of sound is turned off is called reverberation. The measure of reverberation time is the time required for sound intensity to decrease by $60 \,dB$. It is given that the intensity of sound falls off as $I = I_0 \exp(-c_1 \alpha)$,where $I_0$ is the initial intensity,$c_1$ is a dimensionless constant with value $1/4$. Here,$\alpha$ is a positive constant which depends on the speed of sound $v_s$,volume of the room $V$,reverberation time $t$,and the effective absorbing area $A_e$. The value of $A_e$ is the product of the absorbing coefficient and the area of the room. For a concert hall of volume $V = 600 \,m^3$,the value of $A_e$ (in $m^2$) required to give a reverberation time of $t = 1 \,s$ is closest to (speed of sound in air $v_s = 340 \,m/s$):