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A student is investigating the acceleration of a trolley moving up an inclined plane as shown in Fig. 1.1.
The student is investigating the relationship between the acceleration $a$ of the trolley and the angle $\theta$ of the inclined plane when a force $F$ is applied to the trolley.
It is suggested that the relationship is
$ma = F - (mg \sin \theta + k)$
where $g$ is the acceleration of free fall, $m$ is the mass of the trolley and $k$ is a constant.
Design a laboratory experiment to test the relationship between $a$ and $\theta$. Explain how your results could be used to determine a value for $k$. You should draw a diagram, on page 3, showing the arrangement of your equipment. In your account you should pay particular attention to
• the procedure to be followed,
• the measurements to be taken,
• the control of variables,
• the analysis of the data,
• any safety precautions to be taken.
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A student is investigating how the resistance of a wire depends on the diameter of the wire.
The circuit is set up as shown in Fig. 2.1.
The resistance $R$ of the wire is measured using an ohmmeter.
The experiment is repeated for wires of the same material and same length $L$ but different diameter $d$.
It is suggested that $R$ and $d$ are related by the equation
$$R = \frac{4 \rho L}{\pi d^{2}}$$
where $\rho$ is a constant.
(a) A graph is plotted of $R$ on the $y$-axis against $\frac{1}{d^{2}}$ on the $x$-axis.
Determine an expression for the gradient.
gradient = .....................................................[1]
(b) Values of $d$ and $R$ are given in Fig. 2.2.
[Table_1]
Calculate and record values of $\frac{1}{d^{2}} / 10^{6} \text{m}^{-2}$ in Fig. 2.2.
Include the absolute uncertainties in $\frac{1}{d^{2}}$ . [3]
(c) (i) Plot a graph of $R/\Omega$ against $\frac{1}{d^{2}} / 10^{6} \text{m}^{-2}$.
Include error bars for $\frac{1}{d^{2}}$. [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 absolute uncertainty in your answer.
gradient = .....................................................[2]
(d) (i) Using your answers to (a) and (c)(iii), determine the value of $\rho$. Include an appropriate unit.
Data: $L = 1.00 \pm 0.01 \text{ m}$.
$\rho =$ .....................................................[2]
(ii) Determine the percentage uncertainty in $\rho$.
percentage uncertainty in $\rho =$ ..................................................... % [1]
(e) The experiment is repeated with a thinner wire of diameter $0.23 \pm 0.01$ mm. The wire is of the same material and length.
Determine the resistance $R$ of the wire. Include the absolute uncertainty in your answer.
$R =$ .....................................................\Omega [2]
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