A student is investigating stationary waves in the air column inside a tube, using the apparatus shown in Fig - OCR - A-Level Physics A - Question 5 - 2021 - Paper 1

To show that the line of best fit has a gradient equal to $\frac{v}{4}$ and a $y$-intercept of $-k$, we start from the equation: $4(l + k) = \frac{v}{f}$ Rearranging gives: $f = \frac{v}{4(l + k)}$

If we plot $\frac{1}{f}$ against $\frac{1}{\lambda}$ and recognize that: $\lambda = \frac{v}{f}$ The equation becomes: $\frac{1}{f} = \frac{4(l+k)}{v}$ Thus, we can see that the slope (gradient) of the line is $\frac{v}{4}$ and upon extrapolating, the $y$-intercept equals $-k$.

To calculate $v$ from the gradient of the line, substitute the gradient value $m$ from the graph into the relationship: $m = \frac{v}{4}$ Rearranging gives: $v = 4 \cdot m$

After determining the gradient from the graph line of best fit, multiply it by 4 to obtain the speed of sound $v$.

Using the line of best fit from the graph, locate the corresponding $\frac{1}{f}$ value for the given $\frac{1}{\lambda}$ value of the tuning fork experiment to estimate the frequency $F$. This will provide an approximate frequency of the unlabelled tuning fork based on the graph.

The percentage uncertainty in the value of $F$ can be expressed as: $\text{Percentage Uncertainty} = 100 \frac{\Delta F}{F}$ To express $\Delta F$ in terms of other uncertainties, apply the propagation of uncertainty rules:

1. Since $F$ is dependent on $v$, $l$, and $k$, we can write the total uncertainty as:

$\Delta F = F \left( \frac{\Delta v}{v} + \frac{\Delta l}{l} + \frac{\Delta k}{k} \right)$

2. Combine the expressions to find: $\text{Percentage Uncertainty} = 100 \left( \frac{\Delta v}{v} + \frac{\Delta l}{l} + \frac{\Delta k}{k} \right).$