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In this experiment, you will investigate the equilibrium of a rod supported by a spring.
(a) The apparatus has been assembled for you as shown in Fig. 1.1.
(i) Lift the end of the rod attached to the spring so that there is no tension in the spring. Measure and record the unstretched length \( l_0 \) of the spring, as shown in Fig. 1.2.
\( l_0 = \text{...................................} \text{ cm} \) [1]
(ii) Release the end of the rod. Suspend the mass \( M \) from the string loop.
(b) (i) Move the string in the clamp to make the rod horizontal.
(ii) Use the plumb-line to ensure that nail B is vertically below nail A, as shown in Fig. 1.3.
(iii) Measure and record the distance \( h \) between the two nails, and the length \( l \) of the spring, as shown in Fig. 1.3.
\( h = \text{...................................} \text{ cm} \)
\( l = \text{...................................} \text{ cm} \) [1]
(c) Raise nail B and repeat (b) until you have six sets of values of \( h \) and \( l \). Include values for \( \frac{1}{h^2} \) and \( (l - l_0)^2 \) in your table.
(d) (i) Plot a graph of \( (l - l_0)^2 \) on the y-axis against \( \frac{1}{h^2} \) on the x-axis. [3]
(ii) Draw the straight line of best fit. [1]
(iii) Determine the gradient and \( y \)-intercept of this line.
gradient = \text{...................................}
\( y \)-intercept = \text{...................................} [2]
(e) The quantities \( l \) and \( h \) are related by the equation
\( (l - l_0)^2 = \frac{p}{h^2} + q \)
where \( p \) and \( q \) are constants.
Using your answers from (d)(iii), determine the values of \( p \) and \( q \). Give appropriate units.
\( p = \text{...................................} \)
\( q = \text{...................................} \) [2]
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(a) You are provided with a spring.
(i) Measure and record the diameter $d$ of the coiled section of the spring as shown in Fig. 2.1. Record the number $n$ of turns in the coiled section.
$$d = \text{...............................} \text{cm}$$
$$n = \text{...............................}$$
[1]
(ii) Calculate the length $l$ of wire used to make the coiled section of the spring using the relationship $l = \pi nd$.
$$l = \text{...............................} \text{cm}$$
[1]
(b) (i) Set up the apparatus as shown in Fig. 2.2 with mass 1 suspended from the spring and secured with Blu-Tack.
(ii) Pull the mass down approximately 2 cm and release it. One vertical oscillation is shown in Fig. 2.2. Measure and record the time $t$ for the mass to make 10 vertical oscillations.
$$t = \text{...............................}$$
[2]
(c) (i) Lower the bolt until the bottom of the stationary mass is approximately 6 cm above the bench, as shown in Fig. 2.3.
(ii) Pull the mass down until it touches the bench. Release the mass and watch the loop on the bolt, looking to see if the loop rises above the bolt producing a gap as shown in Fig. 2.4.
(iii) Keep raising the bolt and repeating (ii) until the loop just rises above the bolt at the top of the first oscillation. With the mass stationary, measure and record the distance $A$ from the bottom of the mass to the bench.
$$A = \text{...............................}$$
[1]
(d) Estimate the percentage uncertainty in your value of $A$.
$$\text{percentage uncertainty} = \text{...............................}$$
[1]
(e) Detach the mass from the spring. Repeat (b) and (c) using mass 2.
$$t = \text{...............................}$$
$$A = \text{...............................}$$
[2]
(f) It is suggested that the relationship between $A$, $t$ and $l$ is $$A = kt^2 l$$ where $k$ is a constant.
(i) Using your data, calculate two values of $k$.
$$\text{first value of } k = \text{...............................}$$
$$\text{second value of } k = \text{...............................}$$
[2]
(ii) Justify the number of significant figures that you have given for your values of $k$.
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[1]
(iii) Explain whether your results in (f)(i) support the suggested relationship.
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[1]
(g) (i) Describe four sources of uncertainty or limitations of the procedure for this experiment.
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[4]
(ii) Describe four improvements that could be made to this experiment. You may suggest the use of other apparatus or different procedures.
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[4]
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