Question

If the tension in sonometer wire is increased by 21%, compare the initial frequency with the later.

Answer 1

The fundamental frequency (f) of a sonometer wire is given by the formula:

f = `1/(2 L) sqrt(T/mu)`

Since length and mass density remain constant, frequency is directly proportional to the square root of the tension:

f ∝ `sqrt T`

Let the initial tension be T₁, and the initial frequency be f₁. If the tension is increased by 21%, the new tension T₂ is:

T₂ = T₁ + 0.21 T₁

= 1.21 T₁​

To find the new frequency f₂, we use the ratio of the two states:

`f_2/f_1 = sqrt (T_2/T_1`

`f_2/f_1 = sqrt((1.21  T_1)/T_1`

`f_2/f_1 = sqrt 1.21`

`f_2/f_1` = 1.1

f₂ = 1.1 f₁

To find the percentage increase:

Percentage increase = `((f_2 - f_1)/f_1) xx 100`

= (1.1 − 1) × 100

= 0.1 × 100

= 10%

So, the frequency increases to 1.1 times, i.e., by 10%.