Question

A sonometer wire is in unison with a tuning fork, when it is stretched by weight w and the corresponding resonating length is L₁· If the weight is reduced to `("w"/4)`, the corresponding resonating length becomes L₂. The ratio `("L"_1/"L"_2)` is ______.

Options

• 4 : 1

• 1 : 4

• 1 : 2

• 2 : 1

Answer 1

A sonometer wire is in unison with a tuning fork, when it is stretched by weight w and the corresponding resonating length is L₁· If the weight is reduced to `("w"/4)`, the corresponding resonating length becomes L₂. The ratio `("L"_1/"L"_2)` is 2 : 1.

Explanation:

When sonometer is stretched by weight w (tension), then frequency of vibration In the string

v = `1/(2l) sqrt("T"/"m")`

v = `1/(2"L"_1) sqrt("w"/"m")`    ....(i)

where, L₁ is the resonating length and mis mass per unit length of the string.

Similarly, when weight reduces to `"w"/4` and resonating length is L₂, then

v = `1/(2"L"_2) sqrt(("w"/4)/"m")`

`=> "v" = 1/(4"L"_2)sqrt("w"/"m")`    ...(ii)

From Eqs. (i) and (ii), we have

`1/(2"L"_1)sqrt("w"/"m") = 1/(4"L"_2)sqrt("w"/"m")`

`=> "L"_1 = 2"L"_2`

`therefore "L"_1/"L"_2 = 2`

∴ L₁ : L₂ = 2 : 1