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(a) Define the radian. [1]
(b) A circular metal disc spins horizontally about a vertical axis, as shown in Fig. 1.1.
A piece of modelling clay is attached to the disc.
For the instant when the piece of modelling clay is in the position shown, draw on Fig. 1.1:
(i) an arrow, labelled V, showing the direction of the velocity of the modelling clay [1]
(ii) an arrow, labelled A, showing the direction of the acceleration of the modelling clay. [1]
(c) The metal disc in Fig. 1.1 has a radius of 9.3 cm.
The centre of gravity of the modelling clay is 1.2 cm from the rim of the disc and moves with a speed of $0.68 m s^{-1}$.
(i) Calculate the angular speed \( \omega \) of the disc. [2]
(ii) Calculate the acceleration \( a \) of the center of gravity of the modelling clay. [2]
(d) A second piece of modelling clay is attached to the disc in the position shown in Fig. 1.2.
The second piece of modelling clay has a larger mass than the first piece.
By placing one tick (✓) in each row, complete Table 1.1 to show how the quantities indicated compare for the two pieces of modelling clay. [3]
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(a) With reference to thermal energy, state what is meant by two objects being in thermal equilibrium.
(b) Two cylinders X and Y each contain a sample of an ideal gas. The samples are in thermal equilibrium with each other.
X has a volume of 0.0260 m³ and contains 0.740 mol of gas at a pressure of 1.20 × 10⁵ Pa. Y has a volume of 0.0430 m³ and contains gas at a pressure of 2.90 × 10⁵ Pa. Data for the two cylinders are shown in Fig. 2.1.
(i) Show that the temperature of the gas in X is 234 °C. [3]
(ii) Determine the number N of molecules of the gas in Y. Explain your reasoning. [3]
(iii) The gas in X consists of molecules that each have a mass that is four times the mass of a molecule of the gas in Y.
Explain how the root-mean-square (r.m.s.) speed of the molecules in X compares with the r.m.s. speed of the molecules in Y. [3]
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(a) State what is meant by the internal energy of a system. [2]
(b) With reference to molecular kinetic and potential energies, describe and explain how the internal energy of the system changes when:
(i) a gas is heated at constant volume so that its temperature increases [3]
(ii) a wire is stretched within its elastic limit at constant temperature. [3]
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A block of mass $m$ oscillates vertically on a spring, as shown in Fig. 4.1.
The acceleration $a$ of the block varies with displacement $x$ from its equilibrium position, as shown in Fig. 4.2.
The amplitude of the oscillations is $3Y$ and the maximum acceleration is $2A$.
(a) Explain how Fig. 4.2 shows that the oscillations of the block are simple harmonic. [2]
(b) Deduce expressions, in terms of some or all of $m$, $A$ and $Y$, for:
(i) the angular frequency $\omega$ of the oscillations [1]
(ii) the maximum speed $v_0$ of the oscillations [2]
(iii) the energy $E$ of the oscillations. [2]
(c) The period of the oscillations is $0.75\, \text{s}$ and the value of $3Y$ is $1.8\, \text{cm}$.
Determine an expression for $x$ in terms of time $t$, where $x$ is in cm and $t$ is in seconds. [2]
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(a) Define electric potential at a point.
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[2]
(b) Two isolated charged metal spheres $X$ and $Y$ are near to each other in a vacuum. The centres of the spheres are 1.2 m apart, as shown in Fig. 5.1.
Point P is on the line joining the centres of spheres $X$ and $Y$ and is at a variable distance $x$ from the centre of $X$.
Fig. 5.2 shows the variation with $x$ of the total electric potential $V$ due to the two spheres.
State extbf{three} conclusions that may be drawn about the spheres from Fig. 5.2. The conclusions may be qualitative or quantitative.
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[3]
(c) A proton is held at rest on the line joining the centres of the spheres in (b) at the position where $x = 0.60 ext{ m}$.
The proton is released.
Describe and explain, without calculation, the subsequent motion of the proton.
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[2]
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(a) Two capacitors X and Y are connected in series to a power supply of voltage V, as shown in Fig. 6.1.
The capacitance of X is $C_X$ and the capacitance of Y is $C_Y$.
Derive an expression, in terms of $C_X$ and $C_Y$, for the combined capacitance $C_T$ of the capacitors in this circuit.
Explain your reasoning.
(b) Two capacitors P and Q are connected in parallel to a power supply of voltage V.
The capacitance of P is 200 $\mu$F. The capacitance $C_Q$ of Q can be varied between 0 and 400 $\mu$F.
When $C_Q = 0$, the total energy stored in the capacitors is 2.5 mJ.
(i) Show that the supply voltage V is 5.0 V. [2]
(ii) Calculate the total energy, in mJ, stored in the capacitors when $C_Q$ has its maximum value.
total energy = .......................................................... mJ [3]
(iii) On Fig. 6.2, sketch the variation of the total energy E stored in the capacitors with $C_Q$, as $C_Q$ varies from 0 to 400 $\mu$F.
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(a) State Faraday’s law of electromagnetic induction.
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(b) Fig. 7.1 shows a coil at rest in a uniform magnetic field that is parallel to the axis of the coil.
The coil is connected to a centre-zero voltmeter.
The flux density $B$ of the uniform magnetic field varies with time $t$ as shown in Fig. 7.2.
The coil consists of 340 turns, each of cross-sectional area $3.2 \times 10^{-4}\,\text{m}^2$.
(i) Calculate the maximum magnetic flux through one turn of the coil.
maximum magnetic flux = ................................................ Wb [2]
(ii) Determine the maximum rate of change of magnetic flux linkage in the coil.
maximum rate of change of flux linkage = ........................................ Wb $s^{-1}$ [3]
(iii) State the maximum electromotive force (e.m.f.) $V_0$ induced across the coil.
$V_0$ = ............................................................ V [1]
(iv) On Fig. 7.3, sketch the variation of the e.m.f. $V$ induced across the coil with $t$ from $t = 0$ to $t = 6.0\,\text{ms}$.
[3]
(v) The variation of $V$ with $t$ can be described by $V = A \sin Bt$ where $A$ and $B$ are constants. Determine the values of $A$ and $B$. Give units with your answers.
$A = ..................................................\text{unit} ..................
B = ..................................................\text{unit} .................. [3]
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Fig. 8.1 shows part of the emission spectrum of visible radiation emitted by hydrogen gas in a star in a distant galaxy.
The galaxy is moving away from the Earth at a speed of $6.2 \times 10^6 \text{ m s}^{-1}$.
(a) (i) Explain how the positions of the lines in the emission spectrum seen by an observer on the Earth differ from the positions shown in Fig. 8.1.
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[2]
(ii) On Fig. 8.1, draw the three lines in possible positions in the spectrum seen by the observer.
[2]
(b) The lines in Fig. 8.1 correspond to electron transitions down to the energy level $-3.40 \text{ eV}$. One of the lines represents emitted radiation of wavelength $488 \text{ nm}$.
(i) Calculate the energy of a photon of this radiation.
photon energy = ........................................................ J [2]
(ii) Determine the energy, in eV, of the energy level from which the electron transition originates to cause the emission of this radiation.
energy level = ........................................................... eV [2]
(iii) Determine the wavelength, in nm, of this radiation as detected by the observer on the Earth.
wavelength = ........................................................ nm [2]
(c) A value for the Hubble constant is $2.3 \times 10^{-18} \text{ s}^{-1}$.
Determine the distance of the galaxy from the Earth.
distance = ........................................................... m [2]
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(a) State what is meant by the binding energy of a nucleus.
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(b) Table 9.1 shows the masses of two sub-atomic particles and a polonium-212 ($^{212}_{84}\text{Po}$) nucleus.
[Table_1]
For the polonium-212 nucleus, determine:
(i) the mass defect $\Delta m$, in kg
$$\Delta m = \text{....................................................... kg [3]}$$
(ii) the binding energy
$$\text{binding energy} = \text{........................................................ J [2]}$$
(iii) the binding energy per nucleon.
$$\text{binding energy per nucleon} = \text{........................................ J [1]}$$
(c) (i) On Fig. 9.1, sketch the variation with nucleon number $A$ of binding energy per nucleon for values of $A$ from 1 to 250.
[2]
(ii) On your line in Fig. 9.1, draw an X to show the approximate position of polonium-212. [1]
(iii) Polonium-212 is radioactive and undergoes alpha-decay. Suggest and explain, with reference to Fig. 9.1, why the alpha-decay of polonium-212 results in a release of energy.
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(a) Describe how reflected ultrasound pulses may be used to obtain diagnostic information about internal structures.
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(b) (i) Define specific acoustic impedance of a medium.
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(ii) Table 10.1 shows some data for water and for glass.
[Table_1]
Determine the intensity reflection coefficient for ultrasound that is incident on a water–glass boundary.
intensity reflection coefficient = .............................................................. [3]
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