AS & A Level Physics - 9702 Paper 4 2022 Summer Zone 2

(a) Starting from the equation for the gravitational potential due to a point mass, show that the gravitational potential energy $E_p$ of a point mass $m$ at a distance $r$ from another point mass $M$ is given by

$$E_p = -\frac{GMm}{r}$$

where $G$ is the gravitational constant. [1]

(b) Fig. 1.1 shows the path of a comet of mass $2.20 \times 10^{14}$ kg as it passes around a star of mass $1.99 \times 10^{30}$ kg.

At point $X$, the comet is $8.44 \times 10^{11}$ m from the centre of the star and is moving at a speed of $34.1$ km s^{-1}.

At point $Y$, the comet passes its point of closest approach to the star. At this point, the comet is a distance of $6.38 \times 10^{10}$ m from the centre of the star.

Both the comet and the star can be considered as point masses at their centres.

(i) Calculate the magnitude of the change in the gravitational potential energy $\Delta E_p$ of the comet as it moves from position $X$ to position $Y$.[2]

(ii) State, with a reason, whether the change in gravitational potential energy in (b)(i) is an increase or a decrease.  [1]

(iii) Use your answer in (b)(i) to determine the speed in $km s^{-1}$ of the comet at point $Y$. [3]

(c) A second comet passes point $X$ with the same speed as the comet in (b) and travelling in the same direction. This comet is gradually losing mass. The mass of this comet when it passes point $X$ is the same as the mass of the comet in (b).

Suggest, with a reason, how the path of the second comet compares with the path shown in Figure 1.1. [1]

(a) State Coulomb’s law.
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(b) Positronium is a system in which an electron and a positron orbit, with the same period, around their common centre of mass, as shown in Fig. 2.1.

The radius $r$ of the orbit of both particles is $1.59 \times 10^{-10}$ m.
(i) Explain how the electric force between the electron and the positron causes the path of the moving particles to be circular.
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(ii) Show that the magnitude of the electric force between the electron and the positron is $2.28 \times 10^{-9}$ N.
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(iii) Use the information in (b)(ii) to determine the period of the circular orbit of the two particles.
period = ............................................................ s

(c) Positronium is highly unstable, and after a very short period of time it becomes gamma radiation.
(i) Describe how gamma radiation is formed from the two particles in positronium.
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(ii) State one medical application of the process described in (c)(i).
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(a) Define specific latent heat of vaporisation.   [2]

(b) The specific latent heat of vaporisation of water at atmospheric pressure of $1.0 \times 10^{5} \text{ Pa} is 2.3 \times 10^{6} \text{ J} \text{ kg}^{-1}$. A mass of $0.37 \text{ kg}$ of liquid water at $100 \degree \text{C}$ is provided with the thermal energy needed to vaporise all of the water at atmospheric pressure.

(i) Calculate the thermal energy $q$ supplied to the water.  [1]

(ii) The mass of $1.0 \text{ mol}$ of water is $18 \text{ g}$. Assume that water vapour can be considered to behave as an ideal gas.

Show that the volume of water vapour produced is $0.64 \text{ m}^{3}$.  [3]

(iii) Assume that the initial volume of the liquid water is negligible compared with the volume of water vapour produced.

Determine the magnitude of the work done by the water in expanding against the atmosphere when it vaporises.  [2]

(iv) Use your answers in (b)(i) and (b)(iii) to determine the increase in internal energy of the water when it vaporises at $100\degree \text{C}$. Explain your reasoning.   [2]

(c) Use the first law of thermodynamics to suggest, with a reason, how the specific latent heat of vaporisation of water at a pressure greater than atmospheric pressure compares with its value at atmospheric pressure.   [2]

(a) State what is meant by resonance.   [2]

(b) Fig. 4.1 shows a heavy pendulum and a light pendulum, both suspended from the same piece of string. This string is secured at each end to fixed points.

Both pendulums have the same natural frequency.

The heavy pendulum is set oscillating perpendicular to the plane of the diagram. As it oscillates, it causes the light pendulum to oscillate.

Fig. 4.2 shows the variation with time $t$ of the displacements of the two pendulums for three oscillations.

The variation with time $t$ of the displacement $x$ of the light pendulum is given by

$$x = 0.25 sin 5.0 pi t$$

where $x$ is in centimetres and $t$ is in seconds.

(i) Calculate the period $T$ of the oscillations.   [2]

(ii) On Fig. 4.2, label both of the axes with the correct scales. Use the space below for any additional working that you need.  [2]

(iii) Determine the magnitude of the phase difference $phi$ between the oscillations of the light and heavy pendulums. Give a unit with your answer.   [2]

(a) Define the capacitance of a parallel plate capacitor. .......................................................... .......................................................... ..........................................................

(b) Two capacitors, of capacitances $C_1$ and $C_2$, are connected in parallel to a power supply of electromotive force (e.m.f.) $E$, as shown in Fig. 5.1. Show that the combined capacitance $C_T$ of the two capacitors is given by $C_T = C_1 + C_2$. Explain your reasoning. You may draw on Fig. 5.1 if you wish.

(c) Two capacitors of capacitances 22 µF and 47 µF, and a resistor of resistance 2.7 MΩ, are connected into the circuit of Fig. 5.2. The battery has an e.m.f. of 12 V. (i) Show that the combined capacitance of the two capacitors is 15 µF. [1] (ii) The two-way switch $S$ is initially at position $X$, so that the capacitors are fully charged. Use the information in (c)(i) to calculate the total energy stored in the two capacitors. total energy = .................................................. J [2] (iii) The two-way switch is now moved to position $Y$. Determine the time taken for the potential difference (p.d.) across the 22 µF capacitor to become 6.0 V. time = .................................................. s [3]

(a) State the two conditions that must be satisfied for a copper wire, placed in a magnetic field, to experience a magnetic force.
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(b) A long air-cored solenoid is connected to a power supply, so that the solenoid creates a magnetic field. Fig. 6.1 shows a cross-section through the middle of the solenoid.

Fig. 6.1
The direction of the magnetic field at point W is indicated by the arrow. Three other points are labelled X, Y and Z.
(i) On Fig. 6.1, draw arrows to indicate the direction of the magnetic field at each of the points X, Y and Z. [3]
(ii) Compare the magnitude of the flux density of the magnetic field:
• at X and at W ..........................................................................................
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• at Y and at Z. .........................................................................................
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(c) Two long parallel current-carrying wires are placed near to each other in a vacuum.
Explain why these wires exert a magnetic force on each other. You may draw a labelled diagram if you wish.
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(a) State Faraday’s law of electromagnetic induction.
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(b) Two coils are wound on an iron bar, as shown in Fig. 7.1.

!(https://dummyimage.com/fig1)

Coil 1 is connected to a potential difference (p.d.) $V_1$ that gives rise to a magnetic field in the iron bar.
Fig. 7.2 shows the variation with time $t$ of the magnetic flux density $B$ in the iron bar.

!(https://dummyimage.com/fig2)

A voltmeter measures the electromotive force (e.m.f.) $V_2$ that is induced across coil 2.

On Fig. 7.3, sketch the variation with $t$ of $V_2$ between $t = 0$ and $t = 0.40$ s.

!(https://dummyimage.com/fig3)

[4]

(c) Coil 2 in (b) is now replaced with a copper ring that rests loosely on top of coil 1. The supply to coil 1 is replaced with a cell and a switch that is initially open, as shown in Fig. 7.4.

!(https://dummyimage.com/fig4)

(i) The switch is now closed. As it is closed, the copper ring is observed to jump upwards.
Explain why this happens.
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(ii) Suggest, with a reason, what would be the effect of repeating the procedure in (c)(i) with the terminals of the cell reversed.
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[Total: 10]

(a) State one piece of experimental evidence for:

(i) the particulate nature of electromagnetic radiation
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(ii) the wave nature of matter.
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(b) (i) Calculate the de Broglie wavelength $\lambda$ of an alpha-particle moving at a speed of 6.2 \times 10^7 \text{ms}^{-1}$.

$$\lambda = \text{........................................................... m}$$ [3]

(ii) The speed $v$ of the alpha-particle in (b)(i) is gradually reduced to zero.
On Fig. 8.1, sketch the variation with $v$ of $\lambda$.



[2]

(c) Suggest an explanation for why people are not observed to diffract when they walk through a doorway.

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(a) (i) State Hubble’s law.
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(ii) Explain how cosmologists use observations of emission spectra from stars in distant galaxies to determine that the Universe is expanding.
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(b) Explain how Hubble’s law and the idea of the expanding Universe lead to the Big Bang theory of the origin of the Universe.
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(a) State what is meant by radioactive decay.
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(b) A radioactive sample consists of an isotope X of half-life $T$ that decays to form a stable product. Only X and the stable product are present in the sample.
At time $t = 0$, the sample has an activity of $A_0$ and contains $N_0$ nuclei of X.
(i) On Fig. 10.1, sketch the variation with $t$ of the number $N$ of nuclei of X present in the sample. Your line should extend from time $t = 0$ to time $t = 3T$.
[3]

(ii) On Fig. 10.2, sketch the variation with $N$ of the activity $A$ of the sample for values of $N$ between $N = 0$ and $N = N_0$.
[2]

(c) State the name of the quantity represented by the gradient of your line in:
(i) Fig. 10.1
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(ii) Fig. 10.2.
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(d) For the sample in (b), calculate the fraction $\frac{N}{N_0}$ at time $t = 1.70T$.
\[ \frac{N}{N_0} = \text{..........................................................} \] [2]
[Total: 11]