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An aluminium ring is placed on a coil with the rod of a metal retort stand passing through their centres, as shown in Fig. 1.1.
When an alternating current of frequency $f$ is applied to the coil, the ring rises until it is in equilibrium at a height $h$ above the coil.
It is suggested that the relationship between $h$ and $f$ is
$$h = kf^n$$
where $k$ and $n$ are constants.
Design a laboratory experiment to test the relationship between $h$ and $f$ and determine values for $k$ and $n$. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to
(a) the procedure to be followed,
(b) the measurements to be taken,
(c) the control of variables,
(d) the analysis of the data,
(e) the safety precautions to be taken.
517
593
520
It is suggested that $f$ and $d$ are related by the equation $4(d+k) = \frac{v}{f}$ where $v$ is the speed of sound in air and $k$ is a constant.
(a) A graph is plotted of $d$ on the y-axis against $\frac{1}{f}$ on the x-axis. Determine expressions for the gradient and y-intercept in terms of $k$ and $v$.
gradient = .......................................................
y-intercept = ....................................................
[1]
(b) Values of $f$ and $d$ are given in Fig. 2.2.
$$ \begin{array}{c|c} f / \text{Hz} & d / \text{cm} \\ \hline 320 & 24.5 \pm 0.5 \\ 340 & 23.0 \pm 0.5 \\ 378 & 20.5 \pm 0.5 \\ 428 & 18.0 \pm 0.5 \\ 480 & 16.0 \pm 0.5 \\ 512 & 15.0 \pm 0.5 \end{array} $$
Fig. 2.2
Calculate and record values of $\frac{1}{f} / 10^{-3} \text{s}$ in Fig. 2.2. [2]
(c) (i) Plot a graph of $d/\text{cm}$ against $\frac{1}{f} / 10^{-3} \text{s}$. Include error bars for $d$. [2]
(ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled. [2]
(iii) Determine the gradient of the line of best fit. Include the uncertainty in your answer.
gradient = ........................................................... [2]
(iv) Determine the y-intercept of the line of best fit. Include the uncertainty in your answer.
y-intercept = .......................................................... [2]
(d) Using your answers to c(iii) and c(iv), determine values for $k$ and $v$. Include appropriate units in your answers. Include the absolute uncertainties in $k$ and $v$. [2]
$k$ = ..........................................................
$v$ = ..........................................................
(e) (i) The experiment is repeated with a tuning fork of unknown frequency. The distance $d$ is measured as 31.0 $\pm 0.5 \text{cm}$. Determine the frequency of the tuning fork.
$f$ = ........................................................Hz [1]
(ii) Determine the percentage uncertainty in your value of $f$.
percentage uncertainty = ................................................% [1]
532
550
579
