The length of a sonometer wire between two fixed ends is 1.10 m. Where should the two bridges be placed to provide the wire into three segments whose fundamental frequencies are in the ratio of 1 : 2 : 3?

The length of a sonometer wire between two fixed ends is 1.10 m. Where

1.	The length of a sonometer wire between two fixed ends is 1.10 m. Where should the two bridges be placed to provide the wire into three segments whose fundamental frequencies are in the ratio of 1 : 2 : 3?
Answer» Solution :Let `l_(1) , l_(2) , l_(3)` be the lengths of three segments. Given ` l_(1) + l_(2) + l_(3) = 1.10`(i) From relation` n = (1)/( 2l) sqrt ((T)/(mu))` If tension `T` and mass per UNIT length ` mu` are fixed , then ` n prop 1//l` So ` nl = constant` ` n_(1) l_(1) = n_(2) l_(2) = n_(3) l_(3)` `:. l_(1) : l_(2) : l_(3) = (1)/( n_(1)) : (1)/( n_(2)) : (1)/( n_(3)) = (1)/(1) : (1)/(2) : (1)/(3) = ( 6 : 3 : 2)/( 6) = 6 : 3 :2` `:. l_(1) = 6 K , l_(2) = 3k , l_(3) = 2k , k` being a constant . From EQ. (i) `6 k + 3k + 2k = 1.10 or 11k = 1.10` `:.k = 1.10 //11 = 0.1` `:. l_(1) = 6 XX 0.1 = 0.6 m` `l_(2) = 3 xx 0.1 = 0.3 m` `l_(3) = 2 xx 0.1 = 0.2 m` Therefore the bridges must be placed at distance `0.6 m and ( 0.6 + 0.3 ) = 0.9 m`