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(a) (i) • Assemble the apparatus as shown in Fig. 1.1.
The top end of the track is a distance $d$ above the bench. The initial value of $d$ should be approximately 5 cm.
• Ensure that the distance between the bottom of the track and the target wooden strip is 50.0 cm.
(ii) • Place the sphere on the track and hold it gently against the tape, as shown in Fig. 1.2.
• Measure and record the height $h$ of the top of the sphere above the bench.
$h = \text{..................................................}$ [1]
• Release the sphere.
• Measure and record the time $t$ from release for the sphere to reach the target.
$t = \text{..................................................}$ [2]
(b) Change $d$ and repeat (a) until you have six sets of values of $h$ and $t$. Do not use values of $d$ greater than 10 cm. Ensure that the distance from the bottom of the track to the target is always 50.0 cm.
Record your results in a table. Include values of $\frac{1}{t^2}$ in your table. [9]
(c) (i) Plot a graph of $\frac{1}{t^2}$ 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
\[ \frac{1}{t^2} = ah + b \]
where $a$ and $b$ are constants.
Use your answers in (c)(iii) to determine the values of $a$ and $b$. Give appropriate units.
$a = \text{..................................................}$
$b = \text{..................................................}$ [2]
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(a) You are provided with two pieces of string of different diameters. Each piece of string has a 10 g mass attached to one end. Using the thinner string, assemble the apparatus as shown in Fig. 2.1. The length of string between the hacksaw blade and the 10 g mass is $L$, as shown in Fig. 2.1. Measure and record $L$.
$L = \text{.................................}$ [1]
(b) The length of hacksaw blade outside the blocks is $x$, as shown in Fig. 2.2. When the end of the hacksaw blade is moved a small distance to one side and released, the string vibrates. For certain values of $x$, the string vibrates in stationary wave patterns. Examples of these patterns are shown in Fig. 2.3. Press down on the unclamped wooden block. Move the end of the hacksaw blade a small distance to one side and release it. Change $x$ in small steps. Keep changing $x$, testing for a pattern at each step, until the pattern B with three loops is clearly produced. Measure and record $x$.
$x = \text{...........................}$ [2]
(c) Estimate the percentage uncertainty in your value of $x$.
percentage uncertainty = ...................................................... [1]
(d) Calculate the value of $\lambda$ in metres, using $\lambda = \frac{2L}{3}$.
$\lambda = \text{....................... m}$ [1]
(e) Justify the number of significant figures you have given for your value of $\lambda$.
...................................................................................................................... [1]
(f) You are provided with a card stating the mass per unit length $\mu$ of the string. Record the value of $\mu$ from the card.
$\mu = \text{...................... kg m}^{-1}$ Calculate the frequency $f$ of the vibrations, using $f = \frac{1}{\lambda} \sqrt{\frac{mg}{\mu}}$ where $m = 0.010 \text{ kg}$ and $g = 9.81 \text{ m s}^{-2}$.
$f = \text{...................... Hz}$ [1]
(g) Repeat (a), (b), (d) and (f) using the thicker string.
$L = \text{...........................}$
$x = \text{...........................}$
$\lambda = \text{....................... m}$
$\mu = \text{...................... kg m}^{-1}$
$f = \text{........................ Hz}$ [3]
(h) It is suggested that the relationship between $f$ and $x$ is $f = \frac{k}{x^2}$ where $k$ is a constant.
(i) Using your data, calculate two values of $k$.
first value of $k = .........................$
second value of $k = .........................$ [1]
(ii) Explain whether your results support the suggested relationship.
...................................................................................................................... [1]
(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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