In Melde's experiment,the string vibrates in $4$ loops when a $50 \,g$ weight is placed in the pan of weight $15 \,g$. To make the string vibrate in $6$ loops,the weight that has to be removed from the pan is:

$A$ body is suspended from a string of length $1 \ m$ and mass $2 \ g$. The mass of the body required to produce a fundamental mode of $100 \ Hz$ frequency in the string is (Acceleration due to gravity $= 10 \ m \ s^{-2}$)

Two uniform stretched steel strings $A$ and $B$ are vibrating under the same tension. The first overtone of $A$ is equal to the second overtone of $B$. If the radius of $A$ is twice that of $B$,then the ratio of the lengths of the strings $l_A/l_B$ is:

$A$ wire of density $9 \times 10^{-3} \,kg\, cm^{-3}$ is stretched between two clamps $1 \,m$ apart. The resulting strain in the wire is $4.9 \times 10^{-4}$. The lowest frequency of the transverse vibrations in the wire is......$Hz$ (Young's modulus of wire $Y = 9 \times 10^{10} \,N m^{-2}$),(to the nearest integer).

One insulated conductor from a household extension cord has a mass per unit length of $\mu$. $A$ section of this conductor is held under tension between two clamps. $A$ subsection is located in a magnetic field of magnitude $B$ directed perpendicular to the length of the cord. When the cord carries an $AC$ current of $i$ at a frequency of $f$,it vibrates in resonance in its simplest standing-wave vibration state. Determine the relationship that must be satisfied between the separation $d$ of the clamps and the tension $T$ in the cord.

If the length of a stretched string is reduced by $40 \%$ and the tension is increased by $44 \%$,then the ratio of the final to the initial frequencies of the stretched string is: