$A$ massless rod of length $L$ 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 the $1^{st}$ harmonic in $AB$ is equal to the $2^{nd}$ harmonic frequency in $CD$. The value of $x$ is

$A$ man standing in front of a mountain beats a drum at regular intervals. The rate of drumming is gradually increased and he finds that the echo is not heard distinctly when the rate becomes $40$ per minute. He then moves nearer to the mountain by $90 \ m$ and finds that the echo is again not heard when the drumming rate becomes $60$ per minute. The distance between the mountain and the initial position of the man is .... $m$

At a certain moment, the photograph of a string on which a harmonic wave is travelling to the right is shown. Then, which of the following is true regarding the velocities of the points $P$, $Q$ and $R$ on the string?

$A$ wire of linear mass density $\mu = 9.8 \times 10^{-3} \, kg \, m^{-1}$ passes over a frictionless light pulley fixed on the top of a frictionless inclined plane which makes an angle of $30^{\circ}$ with the horizontal. Masses $m$ and $M$ are tied at the two ends of the wire such that $m$ rests on the plane and $M$ hangs freely vertically downwards. The entire system is in equilibrium and a transverse wave propagates along the wire with a velocity of $100 \, m \, s^{-1}$. Choose the correct option for $m$ in $kg$.

In a large room,a person receives direct sound waves from a source $120 \ m$ away from him. He also receives waves from the same source which reach him after being reflected from the $25 \ m$ high ceiling at a point halfway between them. The two waves interfere constructively for wavelengths of

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