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A hot air balloon is tied to the ground using a rope. As the wind blows with speed $v$, the rope makes an angle $\theta$ to the horizontal, as shown in Fig 1.1
It is suggested that $\tan \theta$ is inversely proportional to $v^2$.
To model the hot air balloon in the laboratory, a balloon filled with helium is used. Design a laboratory experiment using a small helium-filled balloon to test the relationship between $\theta$ and $v$. 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.
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A student investigates how the resonant length $L$ of a loaded wire varies with frequency $f$.
For six different frequencies, the student records the length $L$.
It is suggested that $f$ and $L$ are related by the equation $$f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$$ where $T$ is the tension in the wire and $\mu$ is a constant.
(a) A graph is plotted of $f$ on the $y$-axis against $1/L$ on the $x$-axis. Determine an expression for the gradient in terms of $T$ and $\mu$. [1]
(b) Values of $f$ and $L$ are given in Fig. 2.2.
Calculate and record values of $(1/L)/\text{m}^{-1}$ in Fig. 2.2. Include the absolute uncertainties in $1/L$. [3]
(c) (i) Plot a graph of $f/\text{Hz}$ against $(1/L)/\text{m}^{-1}$. Include error bars for $1/L$. [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) (i) The tension $T$ in the wire is $30 \pm 3$ N. Using your answer to (c)(iii), determine the value of $\mu$. Include an appropriate unit in your answer. [2]
(ii) Determine the percentage uncertainty in $\mu$. [1]
(e) An expression for $\mu$ is $$\mu = \rho \pi r^2$$ where the density $\rho$ of the wire is $8800$ kg m$^{-3}$ and $r$ is the radius of the wire.
(i) Using your answer to (d)(i), determine a value for $r$. [1]
(ii) Determine the percentage uncertainty in your value of $r$. [1]
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