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A student is investigating the stability of a wooden block resting on a bench.
A strip is attached by a nail to the centre of the top of the block and is able to rotate, as shown in Fig. 1.1 and Fig. 1.2.
A load of mass $m$ is attached to the free end of the strip at point $P$. The student is investigating the position of the strip indicated by angle $\theta$, as shown in Fig. 1.2, at which the block just topples.
It is suggested that the relationship between $m$ and $\theta$ is
$$\alpha Vw = 2mL\cos \theta - mw$$
where $\alpha$ is a constant, $V$ is the volume of the block, $w$ is the width of the block and $L$ is the distance between the centre of the nail and the centre of the load.
Design a laboratory experiment to test the relationship between $m$ and $\theta$. Explain how your results could be used to determine a value for $\alpha$.
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.
[Diagram on page 3]
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A student is investigating the oscillations of a mass attached to an arrangement of springs. Fig. 2.1 shows a mass attached to two springs connected in series.
The student determines the spring constant $k$ for the arrangement of the springs. A stopwatch is used to measure the time $t$ for 20 oscillations. The measurement of $t$ is repeated and the average period $T$ is determined.
The experiment is repeated for different arrangements and different numbers of springs.
It is suggested that $T$ and $k$ are related by the equation
$$ T = 2\pi \sqrt{\frac{M}{k}} $$ where $M$ is the mass.
(a) A graph is plotted of $T^2$ on the $y$-axis against $\frac{1}{k}$ on the $x$-axis.
Determine an expression for the gradient.
gradient = .............................. [1]
(b) Values of $k$, $\frac{1}{k}$ and the measurements of $t$ are given in Fig. 2.2.
[Table_1]
Calculate and record values of $T/s$ and $T^2/s^2$ in Fig. 2.2.
Include the absolute uncertainties in $T$ and $T^2$. [4]
(c)
(i) Plot a graph of $T^2/s^2$ against $\frac{1}{k}/\text{mN}^{-1}$.
Include error bars for $T^2$. [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 $M$. Include an appropriate unit.
$M = ........................................... $ [1]
(ii) Determine the percentage uncertainty in $M$.
percentage uncertainty = .......................... % [1]
(e) Determine the spring constant $k$ for an arrangement of springs using the same mass that would have a period of 2.50 ± 0.01 s. Include the absolute uncertainty in your answer.
$k = ...............................................\text{ Nm}^{-1}$ [2]
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