Figure 4 shows the structure of a violin and Figure 5 shows a close-up image of the tuning pegs - AQA - A-Level Physics - Question 3 - 2018 - Paper 1

To determine the mass of the 1.0 m length of the string, we can use the relationship between tension, frequency, and the mass per unit length.

Starting with the formula for the frequency of a vibrating string: $f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$ where:

• $f$ = frequency (370 Hz)
• $L$ = vibrating length (0.33 m)
• $T$ = tension (25 N)
• $\mu$ = mass per unit length (mass/length).

Rearranging this yields: $\mu = \frac{T}{(2Lf)^2}$ We can now plug in the values: $\mu = \frac{25 N}{(2 \times 0.33 m \times 370 Hz)^2} = 4 \times 10^{-4} \text{ kg/m}$.

The wave speed $v$ on a string can be calculated using the formula: $v = f \cdot \lambda$ where:

• $f$ = frequency (370 Hz)
• $\lambda$ = wavelength (for the first harmonic, $\lambda = 2L$, where $L$ is the vibrating length).

Calculating the wavelength: $\lambda = 2 \times 0.33 m = 0.66 m$

Now calculating wave speed: $v = 370 Hz \cdot 0.66 m = 244.2 m/s$.

To determine the higher frequency produced after stretching the string, we first need to calculate the new tension after the angle of rotation affects the effective length. The extension due to turning can be estimated based on the diameter and rotation:

The circumference of the circular peg is calculated as: $C = \pi d = \pi (0.00702 m) \approx 0.022 mm$

The effective extension of the string when tightened can be calculated as: $\text{extra extension} = 22 mm \times \frac{75°}{360°} \approx 3.75 mm$

Adding this to the original length: $\text{total length} = 1.0 m + 0.00375 m = 1.00375 m$

Now, the new tension can then be recalculated using: $T' = \text{original } T + \text{increase in tension from extension}$

Using the new length and the relation for frequency: $v' = \sqrt{\frac{T'}{\mu}} ext{ and substitute it back into frequency calculations }$

After completing the calculations, the higher produced frequency will be determined, potentially ending up between 448 Hz and 455 Hz.