A sonometer wire is stretched by hanging a 10 cm high brass cylinder vertically from one of its ends. The wire resonates with a tuning fork of frequency256 Hz . Now the cylinder is partially immersed in water. If the wire and the tuningfork are vibrated simultaneously, 4 beats are heard per second. Calculate the length of the portion of the cylinder that was immersed in water.Given , density of brass= 8 . 5 g * cm^(-3) .

A sonometer wire is stretched by hanging a 10 cm high brass cylinder v

1.	A sonometer wire is stretched by hanging a 10 cm high brass cylinder vertically from one of its ends. The wire resonates with a tuning fork of frequency256 Hz . Now the cylinder is partially immersed in water. If the wire and the tuningfork are vibrated simultaneously, 4 beats are heard per second. Calculate the length of the portion of the cylinder that was immersed in water.Given , density of brass= 8 . 5 g * cm^(-3) .
Answer» Solution :The apparent loss of weight of the partially immersed cylinder reduces the tension in the wire. So , the frequency of the vibrating wire decreases. When the cylinder is partially immersed I water, the FUNDAMENTAL frequency of thewire is ` n_(2) = n_(1) - 4 = 256 - 4 = 252 ` HZ Now ,` (n_(1))/(n_(2))= sqrt((T_(1))/(T_(2)))`or ,`(256)/(252) = sqrt((T_(1))/(T_(2)))` Here, `T_(1)` = weight of the cylinder = ` 10 alpha rho g =10 alpha XX 8 . 5 xx g ` ` T_(2)`= weight of the partially immersedcylinder = weight of the cylinder in air - weight of the water displaced ` = 1= alpha xx 8 . 5 xx g - lalpha xx 1 xx g ` where `alpha = `area of cross section of the cylinder l = height of the cylinder immersed in water `:. (256)/(252) = sqrt((10 alpha xx 8 . 5 xx g )/(10 alpha xx 8.5 xx g - l alpha xx 1 xx g )) = sqrt((85)/(85 - l ))` or ,`(85)/(85 - l) = ((256)/(252))^(2) or, 85 - l = 85 xx ((252)/(256))^(2)` or,`l = 85 xx [ 1 - ((252)/(256))^(2)] = 2 . 64 ` cm .