A student investigated the variation of the fundamental frequency $f$ of a stretched string with its tension $T$ - Leaving Cert Physics - Question 3 - 2009

To graph the relationship:

1. Plot the fundamental frequency ($f$) on the y-axis and tension ($T$) on the x-axis.
2. Ensure that there are six or more scale values for rac{1}{ ext{sqrt{T}}}.
3. Label both axes clearly—frequency in Hz and tension in N.
4. Plot the six points accurately based on the data provided, ensuring that they form a straight line upon connection.
5. Draw the line of best fit through the data points, indicating a proportional relationship.

The relationship between frequency and tension can be described as:

$f ext{ is proportional to } \sqrt{T}$

This implies that as the tension increases, the fundamental frequency also increases, and the graph should show a straight line through the origin, indicating this proportionality.

From the graph, locate the point where $T = 11 N$. Project down to the frequency axis to estimate the fundamental frequency.

Assuming the estimated frequency reads approximately $f = 3.32$ Hz based on your plotted graph.

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

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

Where:

• $f$ is the frequency,
• $L$ is the length of the string (0.4 m),
• $T$ is the tension (11 N),
• $\\mu$ is the mass per unit length.

Rearranging gives:

$\mu = \frac{T}{(2Lf)^2}$

Substituting in the values gives:

$\mu = \frac{11}{(2 \times 0.4 \times 3.32)^2} = 5.86 \times 10^{-5} \, \text{kg m}^{-1}$