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(a) Assemble the apparatus as shown in Fig. 1.1, with the rods of both clamps at an equal height of approximately 50 cm above the bench.
The angle between the two lengths of string is $\theta$, as shown in Fig. 1.1. Adjust the positions of the stands so that $\theta$ is approximately $165^\circ$.
Use the two G-clamps to secure the stands to the bench. The stands should remain in these positions for the rest of the experiment.
(ii) Add 200 g to the mass hanger.
(iii) Record the total mass $M$ of the mass hanger and added mass.
$M$ = ............................................
(iv) Measure and record the height $h$ of the bottom of the mass hanger above the bench.
$h$ = ........................................... cm
(v) Pull down the mass hanger through a distance of approximately 2 cm. Release the mass hanger so that it oscillates vertically. Determine the period $T$ of the vertical oscillations.
$T$ = ............................................ s [2]
(b) Change $M$ and repeat (a)(iii), (a)(iv) and (a)(v) until you have six sets of values of $M$, $h$ and $T$. Record your results in a table. Include values of $T^3$ in your table. [10]
(c) (i) Plot a graph of $T^3$ on the $y$-axis against $h$ 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 = ...........................................
$y$-intercept = ........................................ [2]
(d) It is suggested that the quantities $T$ and $h$ are related by the equation:
$$T^3 = ah + b$$
where $a$ and $b$ are constants.
Use your answers in (c)(iii) to determine the values of $a$ and $b$. Give appropriate units.
$a$ = ................................................
$b$ = ................................................ [2]
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(a) Assemble the apparatus as shown in Fig. 2.1. The nail should pass through the hole in the wooden strip and be held in the boss. The bottom of the plastic pipe should be approximately 2cm above the bottom of the container.
Position the string loop with paper clips so that the wooden strip is parallel to the bench.
The distance from the nail to the string loop holding the paper clips is $x_0$, as shown in Fig. 2.1. Measure and record $x_0$.
$x_0 = \text{............................................. cm}$ [1]
(b) (i) Move the string loop holding the paper clips approximately 4 cm further from the nail. Let the wooden strip settle at an angle, as shown in Fig. 2.2.
(ii) Measure and record the new distance $x$ from the nail to the string loop holding the paper clips, as shown in Fig. 2.2.
$x = \text{.......................................... cm}$ [1]
(iii) Measure and record the larger angle $\phi$ between the wooden strip and the plumb-line, as shown in Fig. 2.2.
$\phi = \text{.......................................... °}$ [1]
(iv) Calculate $(\phi - 90°)$.
$(\phi - 90°) = \text{............................................ °}$
(v) Estimate the percentage uncertainty in your value of $(\phi - 90°)$.
percentage uncertainty = $\text{.........................................}$ [1]
(c) (i) Move the string loop holding the paper clips approximately 3 cm further from the nail. Let the wooden strip settle at a new angle.
(ii) Repeat (b)(ii), (b)(iii) and (b)(iv).
$x = \text{............................................. cm}$
$\phi = \text{............................................. °}$
$(\phi - 90°) = \text{............................................. °}$ [3]
(d) It is suggested that the relationship between $\phi$, $x_0$ and $x$ is
$\sin(\phi - 90°) = k(x - x_0)$
where $k$ is a constant.
(i) Using your data, calculate two values of $k$.
first value of $k = \text{............................................}$
second value of $k = \text{..............................................}$ [1]
(ii) Justify the number of significant figures you have given for your values of $k$.
.......................................................................................................
.......................................................................................................
[1]
(iii) Explain whether your results in (d)(i) support the suggested relationship.
.......................................................................................................
.......................................................................................................
[1]
(e) (i) Remove the plastic pipe from the water.
(ii) Measure and record the outside diameter $D$ and the inside diameter $d$ of the pipe, as shown in Fig. 2.3.
$D = \text{.............................................. cm}$
$d = \text{.............................................. cm}$ [1]
(iii) The mass $m$ of the paper clips and string loop is written on the card. Record $m$.
$m = \text{.......................................... g}$
(iv) Using your second value of $k$, calculate the density $\rho$ of the water using the relationship
$\rho = \frac{m}{Ck(D^2 - d^2)}$
where $C = 201 \text{ cm}^2$.
$\rho = \text{............................................. g cm}^{-3}$ [1]
(f) (i) Describe four sources of uncertainty or limitations of the procedure for this experiment.
1. ..................................................................................................
2. ..................................................................................................
3. ..................................................................................................
4. .......................................................................................................... [4]
(ii) Describe four improvements that could be made to this experiment. You may suggest the use of other apparatus or different procedures.
1. ..................................................................................................
2. ..................................................................................................
3. ..................................................................................................
4. .......................................................................................................... [4]
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