An aluminium rod having a length $100 \,cm$ is clamped at its middle point and set into longitudinal vibrations. Let the rod vibrate in its fundamental mode. The density of aluminium is $2600 \,kg/m^3$ and its Young's modulus is $7.8 \times 10^{10} \,N/m^2$. The frequency of the sound produced is .............. $Hz$.

$A$ closed pipe of length $10 \, cm$ has its fundamental frequency half that of the second overtone of an open pipe. The length of the open pipe is ......... $cm$.

$A$ closed organ pipe and an open organ pipe have the same length $L$. When they are vibrating simultaneously in their first overtone,they produce $3$ beats per second. If the length of the open pipe is reduced to $\frac{1}{3}$ of its original length and the length of the closed pipe is increased to $3$ times its original length,calculate the number of beats produced per second. (Neglect end correction)

$A$ closed and an open organ pipe have the same lengths. If the ratio of the frequencies of their seventh overtones is $\left(\frac{a-1}{a}\right)$,then the value of $a$ is:

The fundamental frequency of an air column in pipe $A$ closed at one end is in unison with the second overtone of an air column in pipe $B$ open at both ends. The ratio of the length of the air column in pipe $A$ to that of the air column in pipe $B$ is:

The fundamental frequency of a closed pipe of length $L$ is equal to the second overtone of a pipe open at both the ends of length $(XL)$. The value of $X$ is (Neglect end correction).