Draw a labelled diagram to represent a stretched string vibrating at its third harmonic - Leaving Cert Physics - Question iv, v, vi - 2022

Using the formula for the fundamental frequency of a stretched string:

$f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$

Where:

• $f$ is the frequency (440 Hz)
• $L$ is the length of the string (0.341 m)
• $T$ is the tension in the string
• $\mu$ is the mass per unit length, calculated as: $\mu = \frac{m}{L} = \frac{0.21 g}{0.65 m} = 0.000323 kg/m$

Rearranging the formula to find tension:

$T = 4L^2 f^2 \mu$

Now, substituting the values to calculate:

• $L = 0.341 m$
• $f = 440 Hz$
• $\mu = 0.000323 kg/m$

Calculating:

$T = 4(0.341)^2(440)^2(0.000323) = 29.1 N.$

When the tension is reduced by a factor of four:

$T' = \frac{T}{4} = \frac{29.1 N}{4} = 7.275 N$

To find the new frequency, we use the same frequency formula:

$f' = \frac{1}{2L} \sqrt{\frac{T'}{\mu}}$

Substituting $T' = 7.275 N$:

$f' = \frac{1}{2(0.341)} \sqrt{\frac{7.275}{0.000323}}$

Calculating

The new frequency is approximately 220 Hz.