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A hammer is often used to force a nail into wood. The faster the hammer moves, the deeper the nail moves into the wood.
This can be represented in a laboratory by a mass falling vertically onto a nail.
It is suggested that the depth $d$ of the nail in the wood (see Fig. 1.1) is related to the velocity $v$ of the mass at the instant it hits the nail by the equation $d = kv^n$
where $k$ and $n$ are constants.
Design a laboratory experiment to investigate the relationship between $v$ and $d$ so as to determine a value for $n$. You should draw a diagram 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.
572
594
556
The reactance $X_C$ of a capacitor is defined as
$$X_C = \frac{V_0}{I_0}$$
where $V_0$ is the peak voltage across the capacitor and $I_0$ is the peak current through the capacitor.
An experiment is carried out to investigate how the reactance of a capacitor varies with the frequency $f$ of the a.c. supply to the capacitor.
The equipment is set up as shown in Fig. 2.1.
The dual-beam oscilloscope is used to determine values of $V_0$ and $I_0$.
It is suggested that $X_C$ and $f$ are related by the equation
$$X_C = \frac{1}{2\pi fC}$$
where $C$ is the capacitance of the capacitor.
(a) A graph is plotted with $X_C$ on the y-axis and $\frac{1}{f}$ on the x-axis. Express the gradient in terms of $C$. [1]
(b) Values of $f$, $V_0$ and $I_0$ are given in Fig. 2.2.
Calculate and record values of $\frac{1}{f}$ and $X_C$ in Fig. 2.2. Include the absolute uncertainties in $X_C$. [3]
(c) (i) Plot a graph of $X_C/\Omega$ against $\frac{1}{f}/10^{-3}$ s. Include error bars for $X_C$. [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. [2]
(d) Using your answer to (c)(iii), determine the value of $C$. Include the absolute uncertainty in your value and an appropriate unit. [3]
(e) The time constant $\tau$ is defined as $\tau = CR$ where $R$ is the total resistance of the circuit.
(i) $C$ is placed in a circuit with total resistance 220 k$\Omega$. Determine the value of $\tau$. [1]
(ii) The percentage uncertainty in the total resistance of the circuit is $\pm$10%. Determine the percentage uncertainty in $\tau$. [1]
514
554
593
