Question

Consider the situation shown in figure.The wire which has a mass of 4.00 g oscillates in its second harmonic and sets the air column in the tube into vibrations in its fundamental mode. Assuming that the speed of sound in air is 340 m s−1, find the tension in the wire.

Answer 1

Thus the tension in the wire is T = 11.6 Newton

Explanation:

• Speed of sound in air v = 340 ms−1
• Length of the wire l = 40 cm = 0.4 m
• Mass of the wire M = 4 g

Mass per unit length of wire (m)m is given by:

• m = Mass Unit length = 10^−2 kg/m
• n0  = frequency of the tuning fork
• T = tension of the string

Fundamental frequency:

n0 = 1 / 2 L √ T / m

For second harmonic, n1 = 2n0

n1 = 2 / 2L  √ T/ m .....(i)

n1 = 2n0 = 340 / 4 × 1 = 85 Hz

Now put the values

85 = 2 / 2 × 0.4 √T / 10−2

T = (85)^2 × (0.4)^2 × 10^−2

T = 11.6 Newton  

Thus the tension in the wire is T = 11.6 Newton

Answer 2

Given that,

Mass of wire = 4.00 g

Speed of sound = 340 m/s

Suppose the length of the wire is 40 cm and length of air column is 1 m.

We need to calculate the mass per unit length of the wire

Using formula of mass per unit length

$m=\dfrac{M}{L}$

Put the value into the formula

$m=\dfrac{4.00}{40\times10^{-2}}$

$m=10^{-2}\ kg/m$

We need to calculate the second harmonic frequency of air column

Using formula of second harmonic frequency

$f=\dfrac{v}{4l}$

Where, v = speed of sound

L = length of wire

Put the value into the formula

$f=\dfrac{340}{4\times1}$

$f=85\ Hz$

We know that,

The fundamental frequency is

$f'=\dfrac{1}{2L}\sqrt{\dfrac{T}{m}}$

We need to calculate the tension in the wire

Using relation between fundamental frequency and second harmonic frequency

$f=2f'$

Put the value of f'

$f=\dfrac{2}{2L}\sqrt{\dfrac{T}{m}}$

Put the value into the formula

$85=\dfrac{2}{2\times40\times10^{-2}}\sqrt{\dfrac{T}{10^{-2}}}$

$T=(85\times40\times10^{-2})^2\times10^{-2}$

$T=11.56\ N$

Hence, The tension in the wire is 11.56 N.