A stretched string of length 1 m and mass 40 g is connected to a...

Given:
Length of the string, $L = 1 \text{ m}$
Mass of the string, $m = 40 \text{ g} = 0.04 \text{ kg}$
Linear mass density, $\mu = \dfrac{m}{L} = 0.04 \text{ kg/m}$
Frequency of the tuning fork, $f = 128 \text{ Hz}$
Number of loops, $p = 4$

Since the tuning fork vibrates perpendicular to the string in Melde's experiment, the frequency of the string is equal to the frequency of the tuning fork.

The frequency of a stretched string vibrating in $p$ loops is given by:
$f = \dfrac{p}{2L} \sqrt{\dfrac{T}{\mu}}$

Substituting the given values:
$128 = \dfrac{4}{2 \times 1} \sqrt{\dfrac{T}{0.04}}$

$128 = 2 \sqrt{\dfrac{T}{0.04}}$

$64 = \sqrt{\dfrac{T}{0.04}}$

Squaring both sides:
$4096 = \dfrac{T}{0.04}$

$T = 4096 \times 0.04 = 163.84 \text{ N}$

Rounding to the nearest integer, the required tension is $164 \text{ N}$.