AS & A Level Physics - 9702 Paper 4 2021 Summer Zone 2

 An isolated planet is a uniform sphere of radius $3.39 \times 10^6$ m. Its mass of $6.42 \times 10^{23}$ kg may be considered to be a point mass concentrated at its centre. The planet rotates about its axis with a period of 24.6 hours.

For an object resting on the surface of the planet at the equator, calculate, to three significant figures:

(a) The gravitational field strength. [2]

(b) The centripetal acceleration. [2]

(c) The force per unit mass exerted on the object by the surface of the planet. [1]

An ideal gas has a volume of $3.1 \times 10^{-3} \text{m}^3$ at a pressure of $8.5 \times 10^5 \text{Pa}$ and a temperature of $290\text{K}$, as shown in Fig. 2.1.

The gas suddenly expands to a volume of $6.3 \times 10^{-3} \text{m}^3$. During the expansion, no thermal energy is transferred. The final pressure of the gas is $2.7 \times 10^5 \text{Pa}$ at temperature $T_F$, as shown in Fig. 2.1.

(a) Show that the number of gas molecules is $6.6 \times 10^{23}$. [3]

(b) (i) Show that the final temperature $T_F$ of the gas is $190\text{K}$. [1]

(ii) The average translational kinetic energy $E_K$ of a molecule of an ideal gas is given by

$$E_K = \frac{3}{2}kT$$

where $T$ is the thermodynamic temperature and $k$ is the Boltzmann constant.

Calculate the increase in internal energy $\Delta U$ of the gas.   [3]

(c) Use the first law of thermodynamics to explain why the external work $w$ done on the gas during the expansion is equal to the increase in internal energy in (b)(ii).   [2]

A U-shaped tube contains some liquid. The liquid column in each half of the tube has length $L$, as shown in Fig. 3.1.

The liquid columns are displaced vertically. The liquid then oscillates in the tube. The liquid levels are displaced from the equilibrium positions as shown in Fig. 3.2.

The acceleration $a$ of the liquid in the tube is related to the displacement $x$ by the expression

$a = -\left(\frac{g}{L}\right)x$

where $g$ is the acceleration of free fall.

(a) Explain how the expression shows that the liquid in the tube is undergoing simple harmonic motion. [3]

(b) The length $L$ of each liquid column is 18 cm.

Determine the period $T$ of the oscillations.  [3]

(c) The oscillations of the liquid in the tube are damped.

In any one complete cycle of the oscillations, the amplitude decreases by 6.0% of its value at the beginning of the oscillation.   [3]

Determine the ratio  

$\left(\frac{energy  of  oscillations  after  3  cycles}{initial  energy  of  oscillations}\right)$

(a) A sinusoidal carrier wave has a constant amplitude and a frequency of 1.2 MHz. The carrier wave is modulated by a signal wave such that a 1.0 V displacement of the signal wave causes a change in frequency of 25 kHz.

The signal wave has frequency 8.0 kHz and amplitude 2.0 V.

(i) State the name of this type of modulation of the carrier wave.
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(ii) For this modulated carrier wave, determine the variation, if any, in:
1. its amplitude
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2. its frequency.
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(b) An audio signal is transmitted by means of a modulated radio wave.

The variation with frequency of the amplitude of the radio wave is shown in Fig. 4.1.



For this transmission, determine:

(i) the wavelength, in km, of the carrier wave
wavelength = .............................................................. km [2]

(ii) the bandwidth
bandwidth = ......................................................... kHz [1]

(iii) the frequency of the audio signal.
frequency = ......................................................... kHz [1]

(a) An isolated metal sphere of radius $r$ is charged so that the electric potential at its surface is $V_0$. On Fig. 5.1, sketch the variation with distance $x$ from the centre of the sphere of the electric potential. Your graph should extend from $x = 0$ to $x = 3r$.
(b) Photons having wavelength $\lambda$ are incident on a metal surface. The maximum wavelength for which there is emission of electrons is $\lambda_0$. For photons of wavelength $\frac{\lambda_0}{2}$, the maximum kinetic energy of the emitted electrons is $E_{MAX}$. On Fig. 5.2, sketch the variation with wavelength $\lambda$ of the maximum kinetic energy for values of wavelength between $\lambda = \frac{\lambda_0}{3}$ and $\lambda = \lambda_0$.
(c) A pure sample of a radioactive isotope contains $N_0$ nuclei. The half-life of the isotope is $T_{\frac{1}{2}}$. The product of the radioactive decay is stable. The variation with time $t$ of the number $N$ of nuclei of the radioactive isotope is shown in Fig. 5.3.
On Fig. 5.3:

• label, on the time axis, the time $t = 1.0T_{\frac{1}{2}}$ and the time $t = 2.0T_{\frac{1}{2}}$
• sketch the variation with time $t$ of the number of nuclei of the decay product for time $t = 0$ to time $t = T$.

(a) Two flat metal plates are held a small distance apart by means of insulating pads, as shown in Fig. 6.1. Explain how the plates could act as a capacitor. ......................................... [2] (b) The arrangement in Fig. 6.1 has capacitance $C$. The arrangement is connected into the circuit of Fig. 6.2. When the two-way switch is moved to position $A$, the capacitor is charged so that the potential difference across it is $V$. When the switch moves to position $B$, the capacitor fully discharges through the sensitive ammeter. The switch moves repeatedly between $A$ and $B$ so that the capacitor charges and then discharges with frequency $f$. (i) Show that the average current $I$ in the ammeter is given by $I = CVf$. ......................................... [2] (ii) For a potential difference $V$ of 180 V and a frequency $f$ of switching of 50 Hz, the average current $I$ in the ammeter is 2.5 μA. Calculate the capacitance, in pF, of the parallel plates. capacitance = ......................................... pF [2] (c) A second capacitor is connected into the circuit of Fig. 6.2. The two capacitors are connected in parallel. State and explain the change, if any, in the average current in the ammeter. ......................................... [2]

(a) Two properties of an ideal operational amplifier (op-amp) are infinite input impedance and infinite bandwidth.

State what is meant by:

(i) infinite input impedance
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(ii) infinite bandwidth
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(b) A student uses a negative temperature coefficient thermistor in the circuit of Fig. 7.1 to indicate changes in temperature.



(i) Show that the potential at point X is 0.40 V. [1]

(ii) The thermistor has a resistance of 360 kΩ at a particular temperature.

For this temperature of the thermistor, calculate the magnitude of the reading on the voltmeter.

voltmeter reading = ....................................................... V [3]

(iii) The temperature of the thermistor increases.

State and explain the effect of this change on the magnitude of the reading on the voltmeter.
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(iv) Explain why the amplifier circuit will no longer indicate temperature changes when the magnitude of the gain of the circuit is greater than 12.5.
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(a) Define magnetic flux density.
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(b) Electrons, each of mass $m$ and charge $q$, are accelerated from rest in a vacuum through a potential difference $V$.
Derive an expression, in terms of $m$, $q$ and $V$, for the final speed $v$ of the electrons. Explain your working. [2]

(c) The accelerated electrons in (b) are injected at point $S$ into a region of uniform magnetic field of flux density $B$, as illustrated in Fig. 8.1.

[Image_1: Fig. 8.1]
The electrons move at right angles to the direction of the magnetic field. The path of the electrons is a circle of radius $r$.

(i) Show that the specific charge $\frac{q}{m}$ of the electrons is given by the expression

$$\frac{q}{m} = \frac{2V}{B^2r^2}$$
Explain your working. [2]

(ii) Electrons are accelerated through a potential difference $V$ of 230V. The electrons are injected normally into the magnetic field of flux density 0.38 mT. The radius $r$ of the circular orbit of the electrons is 14 cm.
Use this information to calculate a value for the specific charge of an electron.

specific charge = ................................................ C kg$^{-1}$ [2]

(iii) Suggest why the arrangement outlined in (ii), using the same values of $B$ and $V$, is not practical for the determination of the specific charge of $\alpha$-particles.
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(a) State two situations in which a charged particle in a magnetic field does not experience a force.

1. .......................................................................................................................... ............................................................................................................................
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(b) A loosely coiled metal spring is suspended from a fixed point, as shown in Fig. 9.1.

[Image of spring setup]

Electrical connections are made to the ends of the spring by means of a flexible lead.

The length of the spring is measured before the switch is closed and then again after the switch is closed.

When the switch is closed, a magnetic field is set up around each coil of the spring.

By reference to these magnetic fields, explain why there is a change in length of the spring. State whether the spring extends or contracts.

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(c) With the switch in (b) closed, the small mass on the free end of the spring is now made to oscillate vertically.

Use the principles of electromagnetic induction to explain why small fluctuations in the current in the spring are found to occur.

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(a) By reference to heating effect, explain what is meant by the \textit{root-mean-square (r.m.s.)} value of an alternating current.
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(b) The variations with time \( t \) of two currents \( I_1 \) and \( I_2 \) are shown in Fig. 10.1 and Fig. 10.2.



(i) Use Fig. 10.1 to determine the peak value and the r.m.s. value of the current \( I_1 \).

peak value = .............................................................. A
r.m.s. value = .............................................................. A [1]

(ii) Use Fig. 10.2 to determine the peak value and the r.m.s. value of the current \( I_2 \).

peak value = .............................................................. A
r.m.s. value = .............................................................. A [1]

(c) The variation with time \( t \) of the supply voltage \( V \) to a house is given by the expression
\[ V = 240 \sin kt \]
where \( V \) is in volts, \( t \) is in seconds and \( k \) is a constant with unit rad s\(^{-1}\).

(i) The frequency of the supply voltage is 50 Hz.
Determine \( k \) to two significant figures.

\( k = .............................................................. \) rad s\(^{-1}\) [2]

(ii) The supply voltage is applied to a heater. The mean power of the heater is 3.2 kW.
Calculate the resistance of the heater.

resistance = .............................................................. \( \Omega \) [2]

(a) State the purpose of computed tomography (CT scanning).
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(b) Outline the principles of CT scanning.
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(a) State what is meant by a \textit{photon}.
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(b) A stationary nucleus of samarium-157 $(_{62}^{157}\text{Sm})$ emits a gamma-ray (\(\gamma\)-ray) photon of energy 0.57 MeV.
Determine, for one \(\gamma\)-ray photon:

(i) its wavelength
\(\text{wavelength} = \text{..............................................................}\text{ m} \) [2]

(ii) its momentum.
\(\text{momentum} = \text{..............................................................}\text{ Ns} \) [2]

(c) (i) Using your answer to (b)(ii), determine the speed of the samarium-157 nucleus after emission of the photon.
\(\text{speed} = \text{..............................................................}\text{ ms}^{-1}\) [2]

(ii) By reference to your answer in (c)(i), explain quantitatively why the speed of the samarium-157 nucleus may be assumed to be negligible compared with the speed of the photon.
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