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A 1 m long wire is clamped at both ends...

A 1 m long wire is clamped at both ends. Find out the positions of two bridges that will divide the wire in three parts such that the fundamental frequencies are in the ratio 1 : 2 : 3 .

Text Solution

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Here, ` n_(1) : n_(2) : n_(3) = 1 : 2 : 3 `
As ` n prop (1)/(l)` , we have ` l_(1) : l_(2) : l_(3) = 1 : (1)/(2) : (1)/(2)`
So , ` l_(2) = (l_(1))/(2) and l_(3) = (l_(1))/(3)`
Now , `l_(1) + l_(2) + l_(3) = 1 , so , l_(1) + (l_(1))/(2) + (l_(1))/(3) = 1 `
or , ` (11)/(6)l_(1) = 1 or , l_(1) = (6)/(11)` m
Then, `l_(2) = (l_(1))/(2) = (3)/(11) m and l_(3) = (l_(1))/(3) = (2)/(11) ` m
So, the first bridge is at a distance of ` (6) /(11)` m from one end and the second bridge is at a distance of ` (2)/(11)` m from the other end .
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