$A$ string is divided into three segments such that the segments possess fundamental frequencies in the ratio $1: 2: 3$. Then,the lengths of the segments are in the ratio:

The frequency of a sonometer wire is $100 \ Hz$. When the weights producing the tension are completely immersed in water,the frequency becomes $80 \ Hz$,and on immersing the weights in a certain liquid,the frequency becomes $60 \ Hz$. The specific gravity of the liquid is:

$A$ wire of length $0.4 \,m$ stretched at both ends vibrates $250$ times per second. If the length of the wire is increased by $0.1 \,m$ and the stretching force is reduced to $1/4$ of its original value, then the new frequency is (in $\,Hz$)

Two wires of the same material with radii $r$ and $2r$ respectively are welded together end to end. The combination is then used as a sonometer wire under tension $T$. The joint is kept midway between the two bridges. The ratio of the number of loops in the wires,given that the joint is a node,is:

Length of a sonometer wire is either $95 \ cm$ or $100 \ cm$. In both the cases,a tuning fork produces $5 \ beats/sec$ with the wire. The frequency of the tuning fork is- (in $Hz$)

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: