The frequency v of the stretched string may depend on (i) the length of the vibrating segment 1 (ii) the tension in the stringand (iii) the mass per unit length m. Show that v prop 1/l sqrt(F/m).

The frequency v of the stretched string may depend on (i) the length o

1.	The frequency v of the stretched string may depend on (i) the length of the vibrating segment 1 (ii) the tension in the stringand (iii) the mass per unit length m. Show that v prop 1/l sqrt(F/m).
Answer» Solution :Tension is a force and its dimensions are `MLT^(-2)` Dimensions of mass PER unit length are `ML^(-1)` According to the given DATA `v prop l^(x)F^(y)m^(z)` `v=kl^(x)F^(y)m^(z)`. K is a constnat WITHOUT any dimension. Taking dimensions of terms on both sides `T^(-1)=L^(x)(MLT^(-2))^(y)(ML^(-1))^(z)=L^(x)M^(y)T^(-2y)M^(z)L^((-z)` `M^(0)L^(0)T^(-1)=L^(x+y-z)M^(y+z)T^(-2y)` Equating the dimensions of T on both sides `-1=-2y,y=1/2` Equating the dimensions of M `z+y=0, z=-y=(-1)/2` Equating the dimensions of L `0=xy-z+x+1/2+1/2, x=-1` `v=kl^(-1)F^(1//2)m^(-1//2),v=1/lsqrt(F/m)`