AS & A Level Physics - 9702 Paper 2 2009 Winter Zone 2

A simple pendulum may be used to determine a value for the acceleration of free fall $g$. Measurements are made of the length $L$ of the pendulum and the period $T$ of oscillation.

The values obtained, with their uncertainties, are as shown.

$T = (1.93 \pm 0.03)$ s

$L = (92 \pm 1)$ cm

(a) Calculate the percentage uncertainty in the measurement of

(i) the period $T$, [1]

(ii) the length $L$. [1]

(b) The relationship between $T$, $L$ and $g$ is given by

$$g = \frac{4\pi^2L}{T^2}.$$

Using your answers in (a), calculate the percentage uncertainty in the value of $g$. [1]

(c) The values of $L$ and $T$ are used to calculate a value of $g$ as $9.751 \text{ ms}^{-2}$.

(i) By reference to the measurements of $L$ and $T$, suggest why it would not be correct to quote the value of $g$ as $9.751 \text{ ms}^{-2}$. [1]

(ii) Use your answer in (b) to determine the absolute uncertainty in $g$.

Hence state the value of $g$, with its uncertainty, to an appropriate number of significant figures. [2]

(a) (i) State one similarity between the processes of evaporation and boiling.   [1]

(ii) State two differences between the processes of evaporation and boiling.   [4]

(b) Titanium metal has a density of 4.5 g cm$^{-3}$.

A cube of titanium of mass 48 g contains 6.0 × 10$^{23}$ atoms.

(i) Calculate the volume of the cube.   [1]

(ii) Estimate

1. the volume occupied by each atom in the cube,   [1]
2. the separation of the atoms in the cube.    [1]

A small ball is thrown horizontally with a speed of 4.0 ms$^{-1}$. It falls through a vertical height of 1.96 m before bouncing off a horizontal plate, as illustrated in Fig. 3.1.

Air resistance is negligible.

(a) For the ball, as it hits the horizontal plate,

(i) state the magnitude of the horizontal component of its velocity,

horizontal velocity = ...................................... ms$^{-1}$ [1]

(ii) show that the vertical component of the velocity is 6.2 ms$^{-1}$. [1]

(b) The components of the velocity in (a) are both vectors.

Complete Fig. 3.2 to draw a vector diagram, to scale, to determine the velocity of the ball as it hits the horizontal plate. 

velocity = ...................................... ms$^{-1}$

at ................................. ° to the vertical [3]

(c) After bouncing on the plate, the ball rises to a vertical height of 0.98 m.

(i) Calculate the vertical component of the velocity of the ball as it leaves the plate.

vertical velocity = ...................................... ms$^{-1}$ [2]

(ii) The ball of mass 34 g is in contact with the plate for a time of 0.12 s.

Use your answer in (c)(i) and the data in (a)(ii) to calculate, for the ball as it bounces on the plate,

1. the change in momentum,

change = ...................................... kg ms$^{-1}$ [3]

2. the magnitude of the average force exerted by the plate on the ball due to this momentum change.

force = ...................................... N [2]

(a) Explain what is meant by strain energy (elastic potential energy).   [2]

(b) A spring that obeys Hooke's law has a spring constant $k$.   [3]

Show that the energy $E$ stored in the spring when it has been extended elastically by an amount $x$ is given by 

$$E = \frac{1}{2}kx^2.$$

(c) A light spring of unextended length 14.2 cm is suspended vertically from a fixed point, as illustrated in Fig. 4.1.

A mass of weight 3.8 N is hung from the end of the spring, as shown in Fig. 4.2. The length of the spring is now 16.3 cm.

An additional force $F$ then extends the spring so that its length becomes 17.8 cm, as shown in Fig. 4.3.

The spring obeys Hooke's law and the elastic limit of the spring is not exceeded.

(i) Show that the spring constant of the spring is 1.8 N cm⁻¹. [1]

(ii) For the extension of the spring from a length of 16.3 cm to a length of 17.8 cm,

1. calculate the change in the gravitational potential energy of the mass on the spring,   [2]
2. show that the change in elastic potential energy of the spring is 0.077 J, [1]
3. determine the work done by the force $F$.   [1]

A uniform string is held between a fixed point P and a variable-frequency oscillator, as shown in Fig. 5.1.

The distance between point P and the oscillator is $L$.

The frequency of the oscillator is adjusted so that the stationary wave shown in Fig. 5.1 is formed.

Points X and Y are two points on the string.

Point X is a distance $\frac{1}{8}L$ from the end of the string attached to the oscillator. It vibrates with frequency $f$ and amplitude $A$.

Point Y is a distance $\frac{1}{8}L$ from the end P of the string.

(a) For the vibrations of point Y, state

(i) the frequency (in terms of $f$),   [1]

(ii) the amplitude (in terms of $A$).   [1]

(b) State the phase difference between the vibrations of point X and point Y.   [1]

(c) (i) State, in terms of $f$ and $L$, the speed of the wave on the string.   [1]

(ii) The wave on the string is a stationary wave.

Explain, by reference to the formation of a stationary wave, what is meant by the speed stated in (i).   [3]

(a) Two resistors, each of resistance $R$, are connected first in series and then in parallel.
Show that the ratio
$$\frac{\text{combined resistance of resistors connected in series}}{\text{combined resistance of resistors connected in parallel}}$$ is equal to 4.

(b) The variation with potential difference $V$ of the current $I$ in a lamp is shown in Fig. 6.1.
Calculate the resistance of the lamp for a potential difference across the lamp of 1.5V.
resistance = ........................................ $\Omega$

(c) Two lamps, each having the $I-V$ characteristic shown in Fig. 6.1, are connected first in series and then in parallel with a battery of e.m.f. 3.0V and negligible internal resistance.
Complete the table of Fig. 6.2 for the lamps connected to the battery.
[Table of Fig. 6.2]
\begin{tabular}{|c|c|c|c|} \hline & \text{p.d. across each lamp/V} & \text{resistance of each lamp/} \Omega & \text{combined resistance of lamps/} \Omega \\ \hline \text{lamps connected in series} & \text{.........................} & \text{.........................} & \text{.........................} \\ \text{lamps connected in parallel} & \text{.........................} & \text{.........................} & \text{.........................} \\ \hline \end{tabular}

(d) (i) Use data from the completed Fig. 6.2 to calculate the ratio
$$\frac{\text{combined resistance of lamps connected in series}}{\text{combined resistance of lamps connected in parallel}}$$
ratio = ........................................

(d) (ii) The ratios in (a) and (d)(i) are not equal.
By reference to Fig. 6.1, state and explain qualitatively the change in the resistance of a lamp as the potential difference is changed.
....................................................................................................................................................
....................................................................................................................................................
....................................................................................................................................................

Tungsten-184 \( \left( ^{184}_{74}\text{W} \right) \) and tungsten-185 \( \left( ^{185}_{74}\text{W} \right) \) are two isotopes of tungsten.
Tungsten-184 is stable but tungsten-185 undergoes \( \beta \)-decay to form rhenium (Re).
(a) Explain what is meant by \textit{isotopes}.
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
................................................................................................................................................................. [2]
(b) The \( \beta \)-decay of nuclei of tungsten-185 is spontaneous and random.
State what is meant by
(i) \textit{spontaneous decay},
.................................................................................................................................................................
................................................................................................................................................................. [1]
(ii) \textit{random decay}.
.................................................................................................................................................................
................................................................................................................................................................. [1]
(c) Complete the nuclear equation for the \( \beta \)-decay of a tungsten-185 nucleus. \( ^{185}_{74}\text{W} \rightarrow \text{................} + \text{................} \) [2]