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A student is investigating the angle at which a glass cylinder containing oil topples, as shown in Fig. 1.1.
A cylinder containing a mass $m$ of oil can be tilted through a maximum angle $\phi$ from the vertical before it topples.
It is suggested that the relationship between $m$ and $\phi$ is
$$\frac{1}{\tan \phi} = \frac{am}{\rho d^3} + b$$
where $d$ is the diameter of the cylinder, $\rho$ is the density of the oil and $a$ and $b$ are constants.
Design a laboratory experiment to test the relationship between $\phi$ and $m$. Explain how your results could be used to determine values for $a$ and $b$. 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.
595
539
502
A student is investigating an electrical circuit containing a length of nichrome wire.
The circuit is set up as shown in Fig. 2.1.
The length $L$ of the wire in the circuit is varied and the current $I$ is measured.
It is suggested that $I$ and $L$ are related by the equation
$$ \frac{1}{I} = \frac{4 \rho L}{\pi E d^2} + \frac{r}{E} $$
where $E$ is the e.m.f. of the battery, $d$ is the diameter of the wire and $\rho$ and $r$ are constants.
(a) A graph is plotted of $\frac{1}{I}$ on the $y$-axis against $L$ on the $x$-axis.
Determine expressions for the gradient and $y$-intercept.
gradient = ..................................................
$y$-intercept = ..................................................
(b) Values of $L$ and $I$ are given in Fig. 2.2.
[Table_1]
Calculate and record values of $\frac{1}{I} \text{A}^{-1}$ in Fig. 2.2.
Include the absolute uncertainties in $\frac{1}{I} \text{A}^{-1}$.
(c) (i) Plot a graph of $\frac{1}{I} \text{A}^{-1}$ against $L/10^{-2} \text{m}$.
Include error bars for $\frac{1}{I} \text{A}^{-1}$.
(ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Both lines should be clearly labelled.
(iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer.
gradient = ..................................................
(iv) Determine the $y$-intercept of the line of best fit. Include the absolute uncertainty in your answer.
$y$-intercept = ..................................................
(d) (i) Using your answers to (a), (c)(iii) and (c)(iv), determine the values of $\rho$ and $r$. Include appropriate units.
Data: $E = 3.2 \pm 0.1 \text{V}$ and $d = 0.31 \pm 0.01 \text{mm}$.
$\rho$ = ..................................................
$r$ = ..................................................
(ii) Determine the percentage uncertainty in $\rho$.
percentage uncertainty in $\rho$ = .................................................. %
531
557
522
