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The lengths of two wires made of the sam...

The lengths of two wires made of the same material are in the ratio 2 : 3 . Their diameters are equal and the fundamental of the shorter wire is one octave higher than of the longer wire. Find the ratio between the tensions in the two wires.

Text Solution

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The two wires are of the same material and the diameters are equal. So, the mass per unit length m is the same.
Here, the ratio 2 : 3 implies that the 1st wire is shorter :
` :. N_(1) = 2 n_(2) ` or, `(n_(1))/(n_(2)) = (2)/(1)*`
The fundamental frequency, n ` =(1)/(2l) sqrt((T)/(m))*`
So, for the two wires,
` (n_(1))/(n_(2)) = (l_(2))/(l_(1)) sqrt((T_(1))/(T_(2)))`
or, ` (T_(1))/(T_(2)) = ((n_(1))/(n_(2)))^(2)*((l_(1))/(l_(2)))^(2) = ((2)/(1))^(2) * ((2)/(3))^(2) = (16)/(9)`
i.e., the ratio between the tensions is 16 : 9 .
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