Question

A stretched wire of length 260 cm is set into vibrations. It is divided into three segments whose frequencies are in the ratio 2 : 3 : 4. Their lengths must be ______.

Options

• 80 cm, 60 cm, 120 cm

• 120 cm, 80 cm, 60 cm

• 60 cm, 80 cm, 120 cm

• 120 cm, 60 cm, 80 cm

Answer 1

A stretched wire of length 260 cm is set into vibrations. It is divided into three segments whose frequencies are in the ratio 2 : 3 : 4. Their lengths must be 120 cm, 80 cm, 60 cm.

Explanation:

The frequency produced by a stretched wire Is given by

f = `"p"/(2"l") sqrt("T"/"m")`     ....(i)

where, p = number of loops formed in vibration,

I = length of wire,

T =tension in wire,

and m = mass per unit length of wire

So, from Eq (i), we get

`"f" prop 1/"l"`

Here, the ratio of frequencies of three segment is

f₁ : f₂ : f₃ = 2 : 3 : 4

⇒ l₁ : l₂ : l₃ = `1/2 : 1/3 : 1/4` = 6 : 4 : 3

The length of wire, L = 260 cm, so

`"l"_1 = 6/13 xx 260` = 120 cm

`"l"_2 = 4/13 xx 260` = 80 cm

and `"l"_3 = 3/13 xx 260` = 60 cm