A student was studying musical instruments: (a) The student set up a standing wave on a string using the apparatus shown - Edexcel - A-Level Physics - Question 12 - 2023 - Paper 2

When the guitar string is plucked, it vibrates back and forth. These vibrations disturb the surrounding air molecules, causing them to oscillate. As the string moves away from its equilibrium position, it compresses the air molecules, creating areas of high pressure (compressions). Conversely, as it returns, it creates areas of low pressure (rarefactions). This cycle of compressions and rarefactions propagates through the air as a sound wave.

The fundamental frequency of a vibrating string can be determined using the formula:

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

where:

• $f$ is the frequency
• $L$ is the length of the string
• $T$ is the tension in the string (56 N)
• $\mu$ is the mass per unit length of the string ($5.0 \times 10^{-1} kg m^{-1}$).

To determine the range of frequencies, we check the frequency for the maximum length (63 cm = 0.63 m) and minimum length (21 cm = 0.21 m):

1. For $L = 0.63 m$: $f_{max} = \frac{1}{2(0.63)} \sqrt{\frac{56}{5.0 \times 10^{-1}}} = \frac{1}{1.26} \sqrt{112} \approx 2.87 Hz$

2. For $L = 0.21 m$: $f_{min} = \frac{1}{2(0.21)} \sqrt{\frac{56}{5.0 \times 10^{-1}}} = \frac{1}{0.42} \sqrt{112} \approx 6.73 Hz$

Thus, the fundamental frequency lies between 2.87 Hz and 6.73 Hz. Since the frequency of 196 Hz exceeds this range, it cannot be played on the string.