A student investigated how the fundamental frequency $f$ of a stretched string varied with its tension $T$ - Leaving Cert Physics - Question 3 - 2021

The student used the apparatus in the following manner:

1. Setting up the Tuning Fork: The vibrating tuning fork was held close to the string to excite it, allowing the string to resonate at its fundamental frequency.
2. Adjusting the Tension: The student altered the tension in the string by adding or removing weights from the newton meter, ensuring that the changes in frequency could be recorded.
3. Observing Vibrations: The string's sound was heard, with the paper rider falling off when the sound was loudest, indicating the fundamental frequency.

To create an appropriate graph:

1. Values for f or rac{1}{T}: Use the frequency as the dependent variable (y-axis) and the tension as the independent variable (x-axis).
2. Labelled Axes: Label the y-axis as 'Frequency (Hz)' and the x-axis as 'Tension (N)'.
3. Correct Points Plotted: Plot the given data points accurately on the graph based on the recorded values of f and T.
4. Line of Best Fit: Draw a smooth line (or curve) that best represents the data points on the graph.

To calculate the mass per unit length from the graph:

1. Slope Formula: The slope of the line on the graph is used, calculated as:

   $Slope = \frac{\Delta f}{\Delta T}$

2. Value of Slope: For example, if the slope is calculated as approximately 64.8 Hz/N, it can be noted here.

3. Formula for Linear Density: The linear density $\mu$ can be calculated using:

   $\mu = \frac{g}{(4L^2 f^2)}$ where $g$ is the gravitational acceleration and $L$ is the length of the string.

4. Final Calculation: Substituting the appropriate values yields:

   $\mu = 1.4 \times 10^{-4} \text{ kg m}^{-1}$