Mock Exam Set 1 - Paper 2

[Maximum mark: 7]

While practising, a table tennis player hits a ball of mass 2.8 $g$ that collides with the floor and then bounces back from the wall. The speed of the ball just before reaching the floor is 8.0 $m \, s^{-1}$. The ball leaves the floor with an angle of 65$\degree$ to the floor as shown.

1. Due to the collision with the floor, the ball losses 30$\%$ of its initial kinetic energy. Show that the leaving speed of the ball from the floor is around 7 $m \, s^{-1}$. [2]
   

2. The ball strikes the wall just as it reaches the highest point of its motion. Determine the horizontal distance between the bounce point and the wall. [3]
   

3. The ball collides elastically with the wall, with the time of contact during the collision is $0.040\, s$. Calculate the average horizontal force exerted by the wall on the ball during the collision. [2]
   

[Maximum mark: 4]

1. An electric heater with a power rating of 2.8 $kW$ heats liquid that flows through it. The temperature of the liquid increases from $18°C$ to $38°C$ while passing through the heater. The flow rate of the liquid $\sigma$ is $0.035\,kg\,s^{-1}$. Calculate the specific heat capacity of the liquid. [2]
   

2. Suggest whether your answer for (a) is above or below the actual value of the specific heat capacity. Justify your answer. [2]
   

[Maximum mark: 11]

1. Outline how a gas exerts pressure on a surface. [3]
   

2. 1. A balloon containing helium gas is released from the ground. The pressure exerted by the helium is approximately equal to the pressure exerted by surrounding air. The balloon expands at constant temperature as it rises.

      $\footnotesize{\textrm{[Source: Created with chemix - https://chemix.org/]}}$

      The balloon contains $16 \,g$ of helium (molar mass $4.0\, g\, mol^{-1}$), has a temperature $T$ of $300\, K$ and a pressure $P_0$ of $100\, kPa$ at ground level.

      Calculate the volume $V_0$ of the balloon at ground level. [2]
      

   2. At heights much less than 10 km, the pressure $P$ inside the balloon is related to its height $h$ (in $m$) by the approximation

      $P\approx P_0\left(1-\dfrac{h}{10\ km}\right)$

      The balloon ascends until its volume is 5% higher than the volume at ground level.

      Estimate the new height of the balloon. [3]
      

   3. The same balloon at $100\,kPa$ is placed inside a sealed, rigid container of volume $3V_0$ and a temperature of $300 K$. A pump is used to remove air from the container. When the air remaining in the container consists of half as many molecules as the number of molecules inside the balloon, the balloon bursts.

      Determine the pressure inside the container at the moment the balloon bursts. [3]
      

[Maximum mark: 9]

A longitudinal wave is travelling in a medium from left to right. The diagram below is a snapshot of plane wave fronts of showing position of particles $X$ (center of compression) and $Y$ (center of rarefaction) in the medium at $t =0 \ s$.

1. Distinguish between a transverse wave and a longitudinal wave. [2]
   

2. In the graph below, the solid line shows the variation of displacement $s$ of particle $X$ with time, and the dotted line shows the variation of displacement of particle $Y$ with time.

   

   1. Show that the angular frequency $\omega$ of the oscillating particles in the medium is 3.9 $rad \ s^{-1}$. [1]
      

   2. Calculate the speed of the wave in the medium. [2]
      

3. 1. State the phase difference between particles $X$ and $Y$. [1]
      

   2. Determine the speed of particle $Y$ when $t = 2.3 \ s$. [3]
      

[Maximum mark: 8]

1. The diagram below shows a small section of wire which is a part of a closed electric circuit.

   

   Due to a potential difference across the circuit, an electric field exists in the wire, as shown. Explain, in terms of the charge carriers, why current exists in the wire. [2]
   

2. The electric field is due to a potential difference of $120 \,V$ across the circuit. If the field performs $600 \,J$ of work moving negative charges over a time period of 2 seconds, show that approximately $3 \times 10^{19}$ electrons move through a cross-section of wire in this time. [2]
   

3. Calculate the current in the wire. [1]
   

4. If the density of mobile electrons in this wire segment is $8 \times 10^{28}$ electrons $m^{-3}$, and the wire has a diameter of $2.6\, mm$, calculate the drift speed of the electrons in the wire. [3]
   

[Maximum mark: 7]

A roller coaster car with a mass of 80 $kg$ moves on the track shown in the diagram starting from point A.

At points B and C, the track approximates a circular path so that the radii at at B and C are 12 $m$ and 18 $m$ respectively.

1. Compare the normal force exerted by the track on the car at points B and C. [2]
   

2. Draw and label the free body diagram of the car at point C. [2]
   

3. Calculate the maximum speed of the car at point C so that it could remain on the track. [3]
   

[Maximum mark: 7]

1. State Newton's law of Gravitation. [2]
   

2. A geostationary orbit is a special orbit in which satellites follow a circular path and remain above a fixed point above the Earth's surface. A satellite orbits in geostationary orbit at a constant speed of $v$ and orbital radius of $R$. The mass of the Earth is $M$.

   
   

   1. Show that $v = \sqrt{\frac{GM}{R}}$ [2]
      

   2. The radius of the satellite's orbit is 4.23 $\times 10^7\, m$. Use this information to estimate mass of the Earth. [3]
      

[Maximum mark: 7]

1. A kaon that has a quark structure of $u \bar{s}$ decays into an antimuon and muon neutrino.

   $K^{+} \rightarrow μ^+ + \nu_\mu$

   1. Explain how it can be deduced that this decay happens through weak interaction. [2]
      

   2. Using the quark structure for the kaon, draw a Feynman diagram for this decay. [2]
      

   

2. Explain, by reference to properties of nuclear and electrostatic forces, why small stable nuclei have equal numbers of protons and neutrons but in large nuclei there is an imbalance in the number of protons and neutrons. [3]
   

[Maximum mark: 7]

The government of a medium sized city plans to replace coal power plants with photovoltaic cells to power the city.

The following data are available:

• $\bullet \hspace{1em}$ Total area available for photovoltaic cells = 62.6 $km^2$
• $\bullet \hspace{1em}$ Maximum intensity of light incident on photovoltaic cells = $1.36 \times 10^3\ W \, m^{-2}$
• $\bullet \hspace{1em}$ Average efficiency of energy conversion in photovoltaic cells = 20 $\%$
• $\bullet \hspace{1em}$ Hours of sunshine in one month = 150 hours

1. Explain why the government's choice to reduce its reliance on coal would be beneficial to the Earth's climate. [2]
   

2. Outline, in terms of energy changes, how electrical energy is generated from coal. [3]
   

3. Using the data given, calculate in $GJ$ the maximum expected energy generation for one month. [2]
   

[Maximum mark: 10]

A satellite of mass $m$ is launched from the surface and placed in an orbit around a planet of mass $M$ and radius $R$. The height of its orbit from the surface of the planet is $h$.

The initial speed of the satellite is equal to $\dfrac{4}{5}v_{esc}$.

The planet has no atmosphere.

1. Define gravitational potential energy [2]
   

2. 1. Show that the initial total energy of the satellite is given by $-\dfrac{9GMm}{25R}$. [2]
      

   2. The satellite is placed in an orbit with orbital radius $r$.

      Determine the expression for the total energy of the satellite in its orbit in terms of $G, M, m$ and $r$. [2]
      

   3. Use your answers to parts b(i) and b(ii) to determine the height of orbit above the surface of the planet in terms of $R$. [2]
      

3. The satellite ejects excess fuel opposite to its direction of motion. Suggest the effect of this change on the average orbital radius of the satellite. [2]
   

[Maximum mark: 9]

An uncharged capacitor $C$ is connected to a circuit in combination with a cell of $24\,V$ with negligible internal resistance. The permittivity of the dielectric material between plates of the capacitor is twice that of a vacuum. A resistor is also connected to the circuit and the circuit is controlled by switches $S_1$, $S_2$ and $S_3$.

Initially switches $S_1$ and $S_3$ are closed and switch $S_2$ is open. The graph below shows the variation of current passing through the resistor with time.

1. Draw a graph below to show the variation of voltage across the resistor with time. [2]
   

   

2. 1. Show that the time constant of the circuit when switches $S_1$ and $S_3$ are closed and switch $S_2$ is open is approximately $10 \ ms$ [2]
      

   2. Calculate the capacitance of the capacitor. [2]
      

   3. Calculate the charge on one of the plates of the capacitor when it is fully charged. [1]
      

3. When the capacitor fully charged, all three switches are opened and the dielectric material is removed. Explain the effect on energy stored in the capacitor. [2]
   

[Maximum mark: 8]

1. Define decay constant. [1]
   

2. Mushrooms contain a radioactive element called caesium-137, $\ce{^{137}_{55}Cs}$ . The element has a half-life of $30\, yr$ and a sample of living mushrooms has a constant decay rate of $2000\, Bq\,$ for every $kg$ of mass. A sample of $50\, g$ is extracted from a dried mushroom and the activity is measured as $0.80\, Bq$.

   1. Caesium-137 decays to barium-137, $\ce{^{137}_{56}Ba}$, by beta emission. Complete the nuclear reaction equation of this decay, showing both particles emitted in the decay.

      $\ce{^{137}_{55}Cs}$ → ______ + $\ce{^{137}_{56}Ba}$ + ______ [2]
      

   2. Show that the decay constant of caesium-137 is about $0.023\, {yr}^{-1}$ [2]
      

   3. Given that the amount of caesium in the mushroom stops being replenished once the mushroom is no longer living, determine the age of the dried mushroom. [3]