WAEC SSCE - Physics - 2006

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The force applied to the car can be calculated using Newton's second law of motion, which states that force is equal to mass multiplied by acceleration (F=ma). In this case, the mass of the car is 800 kg, and it experiences a frictional force of 200 N. The car is also accelerating at a rate of 2 ms^-2. To calculate the force applied to the car, we need to first calculate the net force acting on the car. The net force is the sum of all the forces acting on the car, which in this case is the force applied minus the frictional force. The frictional force is acting in the opposite direction to the force applied, so we subtract it from the force applied to get the net force: Force applied - Frictional force = Net force F - 200 = ma F - 200 = 800 x 2 F - 200 = 1600 F = 1600 + 200 F = 1800 N Therefore, the magnitude of the force applied to the car is 1800 N. Option (D) 1800N is the correct answer.

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The correct statement about a bar magnet is "the magnetic properties are more pronounced at points near the end of the bar". A bar magnet has two poles: a north pole and a south pole. The magnetic field lines originate from the north pole and end at the south pole. At the middle of the magnet, the magnetic field is relatively uniform and the magnetic properties are not very pronounced. However, as we move towards the ends of the magnet, the magnetic field lines start to converge, becoming more concentrated and pronounced. This is because the magnetic field lines are closer together at the ends than at the middle of the magnet. Therefore, the magnetic properties of the magnet are more pronounced at points near the end of the bar. Option A is incorrect because the iron fillings cling more at the ends of the magnet due to the concentration of the magnetic field lines. Option B is incorrect because the magnetic properties are more pronounced at the ends, not the middle. Option C is incorrect because having the same poles at the ends of a magnet is a characteristic of a repulsive magnet, which is not the case for a bar magnet that has opposite poles at its ends.

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Among the given quantities, momentum is a vector quantity. A vector quantity has both magnitude and direction, and momentum is the product of mass and velocity in a particular direction. Therefore, it has both magnitude and direction, making it a vector quantity. On the other hand, speed is a scalar quantity, as it only has magnitude and no direction. Distance is also a scalar quantity, representing the length between two points without a specific direction. Energy is also a scalar quantity, as it is a measure of the amount of work that can be done by a force without specifying any particular direction.

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Uniform speed occurs when there is an equal change of distance in equal times. This means that an object is moving at a constant rate without any change in direction. The distance traveled by the object is the same for each time interval. For example, if a car travels at a constant speed of 60 km/h, it will cover a distance of 60 km in one hour, 120 km in two hours, and so on. The displacement or change in position may not be the same for each time interval as it depends on the direction of motion. The velocity, which is the rate of change of displacement with time, may also not be constant if the direction of motion changes. Acceleration is the rate of change of velocity with time and does not determine uniform speed.

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When two capacitors are connected in parallel, their equivalent capacitance is given by the sum of their individual capacitances. So, for the given problem, the equivalent capacitance (C) is: C = 3\(\mu F\) + 6\(\mu F\) C = 9\(\mu F\) Therefore, the correct option is not given in the question. The equivalent capacitance of a 3\(\mu F\) capacitor and a 6\(\mu F\) capacitor connected in parallel is 9\(\mu F\).

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When steam condenses to water, it releases energy in the form of heat. The amount of heat released is given by: heat released = mass x specific latent heat of vaporization where the specific latent heat of vaporization of water is 2.3 x 10^6 J/kg. In this problem, we are given that 6.9 x 10^4 J of energy is released as the steam condenses. Therefore, we can calculate the mass of the condensed steam as follows: mass = heat released / specific latent heat of vaporization mass = 6.9 x 10^4 J / 2.3 x 10^6 J/kg mass = 0.03 kg Therefore, the mass of the condensed steam is 0.03 kg. This is equivalent to 3.0 x 10^-2 kg, which is one of the options provided. Hence, the correct answer is.

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The S.I. unit of frequency is hertz (Hz), which is defined as the number of waves that pass through a point per second. The unit of period is second (s), which is the time taken for one complete wave to pass a point. The amplitude of a wave is the maximum displacement of the wave from its undisturbed position, and it is measured in metres (m). Therefore, the correct answer is option (D): hertz for frequency, second for period, and metre for amplitude.

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The breaking up of an atomic nucleus into the fragments of nearly equal size is known as nuclear fission. This process involves the splitting of a heavy nucleus, such as uranium or plutonium, into two lighter nuclei, called fission products, as well as the release of a large amount of energy in the form of heat and radiation. Nuclear fission can be induced by bombarding a nucleus with a neutron, which causes the nucleus to become unstable and split apart. The neutrons released from the fission can then go on to cause additional fissions in a chain reaction, releasing even more energy. Nuclear fission is the process that powers nuclear reactors and nuclear weapons. It is also responsible for the release of energy in certain natural phenomena, such as the decay of radioactive isotopes in the earth's crust and the energy output of stars.

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An object undergoes oscillatory motion when it moves to and fro about a fixed point. The motion of the object repeats itself periodically, meaning that it moves back and forth over the same path multiple times. The fixed point is called the equilibrium position, and the maximum distance from the equilibrium position that the object moves is called the amplitude of the oscillation. For example, a pendulum oscillates back and forth about its equilibrium position, which is the vertical hanging position. As it swings, it moves back and forth along the same path, and the distance it moves from the equilibrium position is its amplitude. Oscillatory motion is a common phenomenon in many natural and man-made systems, such as springs, musical instruments, and electric circuits. Understanding oscillatory motion is important in many fields, including physics, engineering, and biology.

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The sagging of overhead electrical cables is caused by linear expansivity. When the cables are exposed to changes in temperature, they expand and contract. This expansion and contraction can cause the cables to sag over time. When the temperature increases, the cables expand, which makes them longer. This increased length causes the cables to sag more, and the opposite occurs when the temperature decreases. The linear expansivity of a material refers to its ability to expand or contract in length when its temperature changes. So, the sagging of overhead electrical cables is a result of their linear expansivity, which causes them to lengthen or shorten depending on the temperature, ultimately leading to sagging.

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The acceleration due to gravity is the acceleration experienced by any object due to the force of gravity. It is the force with which the earth attracts any object towards its center. The acceleration due to gravity is denoted by the symbol 'g' and its SI unit is meters per second squared (m/s^2). The force of gravity is directly proportional to the mass of the object and inversely proportional to the square of the distance between the centers of the two objects. This means that the acceleration due to gravity is the same for all objects regardless of their mass. Therefore, the correct option is (c) with which the earth attracts one kilogram mass.

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The total resistance in the circuit is the sum of the individual resistances: R = 2.0\(\Omega\) + 3.0\(\Omega\) = 5.0\(\Omega\) The current in the circuit is given by Ohm's Law: I = V / R, where V is the voltage across the circuit, which is equal to the emf of the cell, E. Therefore, I = E / (R + r), where r is the internal resistance of the cell. Substituting the given values, we get: I = 1.5V / (5.0\(\Omega\) + 1.0\(\Omega\)) = 0.25A Therefore, the current through the resistors is 0.25A. Hence, the answer is (a) 0.25A.

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When a liquid evaporates, the more energetic molecules escape from the surface of the liquid and enter the atmosphere as a gas. This process of evaporation requires energy, which is absorbed from the surroundings, including the remaining liquid. Since the more energetic molecules have left the liquid, the average kinetic energy of the remaining molecules decreases. Temperature is a measure of the average kinetic energy of the molecules in a substance, so the temperature of the remaining liquid decreases as well. Therefore, the correct answer to the question is that the temperature of a liquid falls after some of it has evaporated because the more energetic molecules have escaped into the atmosphere.

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The source of energy that is renewable is hydro power. Hydro power is generated by the movement of water, usually by falling water turning turbines to generate electricity. This movement of water can come from rivers, waterfalls, tides or even man-made dams. The water cycle replenishes the water used to generate hydro power, making it a renewable energy source. In contrast, petroleum and charcoal are non-renewable energy sources that are derived from fossil fuels. These fuels are finite and take millions of years to form. Once we use up all the available fossil fuels, they will not be replenished within our lifetimes. Nuclear energy, while not derived from fossil fuels, also involves the use of finite resources like uranium, making it a non-renewable source of energy. Therefore, hydro power is the renewable source of energy, while the others are non-renewable.

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The power of the engine can be calculated by multiplying the force produced by the engine with the speed of the train. The formula for power is: Power = Force x Speed Using the given values, we can calculate the power of the engine as follows: Power = 3000 N x 30 m/s Power = 90000 W Therefore, the power of the engine is 9.00 x 10^4 W. To summarize, the power of an engine is the product of the force produced by the engine and the speed of the object. In this case, the engine produces a force of 3000N and moves at a speed of 30 m/s, giving a power output of 9.00 x 10^4 W.

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The statement that is correct about simple harmonic motion is: "total mechanical energy is always conserved". Simple harmonic motion is a type of oscillatory motion where the force acting on the system is proportional to the displacement of the system from its equilibrium position and is directed towards that equilibrium position. This results in a motion where the system oscillates back and forth around its equilibrium position. During this motion, the total mechanical energy of the system (sum of kinetic and potential energy) is constantly conserved. However, the other statements are incorrect as linear acceleration is directed towards the equilibrium point, linear acceleration varies directly with displacement and period of oscillation varies inversely with the square root of acceleration due to gravity.

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The formula to calculate the frequency of a wave is: frequency = wave speed / wavelength Where the wavelength is the distance between two successive crests of a wave. In this problem, the wave speed is given as 20ms^-1 and the wavelength is given as 25cm. However, we need to convert the wavelength to meters since the wave speed is also given in meters per second. wavelength = 25cm = 0.25m Now we can substitute the values in the formula and solve for frequency: frequency = 20ms^-1 / 0.25m frequency = 80Hz Therefore, the frequency of the wave is 80Hz.

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The initial kinetic energy of the stone can be calculated using the formula: KE = 1/2 mv^2 where KE is the kinetic energy, m is the mass of the stone and v is the initial velocity of the stone. Substituting the given values, we have: KE = 1/2 x 2.0 kg x (20.0 ms^-1)^2 KE = 1/2 x 2.0 x 400 KE = 400 J Therefore, the initial kinetic energy of the stone is 400J. Answer is correct.

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The potential energy stored in an elastic string that has been stretched by a distance X is given by option (A): \(\frac{1}{2}k \Box ^2\). The force required to stretch an elastic string is proportional to the extension. This proportionality constant is known as the spring constant (K). When a spring is stretched or compressed, it stores potential energy. The amount of energy stored in the spring is directly proportional to the amount of stretching or compression. The formula for the potential energy stored in a spring is given by: Potential energy = 0.5 x Spring constant x (extension)^2 where the extension represents the distance by which the spring is stretched or compressed from its original length. Thus, in this case, the potential energy stored in the elastic string of force constant K, which has been extended by X meters, is given by: Potential energy = 0.5 x K x X^2 This matches option (A): \(\frac{1}{2}k \Box ^2\).

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The phenomenon that causes liquids to rise up narrow tubes is called capillary action, and it's due to the interplay of two main forces: gravity and surface tension. Surface tension is the force that causes the surface of a liquid to minimize its surface area and form into a shape with the least amount of surface energy. This is why water droplets on a surface will form into a sphere shape. Now, imagine a narrow tube placed in a container of liquid. At the point where the tube and liquid meet, the surface tension of the liquid creates a meniscus, which is the curved surface that forms at the edge of the liquid. The surface tension pulls the liquid up the tube and gravity pulls it back down, but the surface tension is stronger in the narrow tube, causing the liquid to rise above its natural level. So, the correct option is surface tension. Viscosity is the resistance of a liquid to flow and doesn't directly contribute to capillary action. Osmosis is the movement of water across a semi-permeable membrane and isn't relevant to capillary action. Brownian motion is the random movement of particles in a fluid and doesn't play a significant role in capillary action either.

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To solve this problem, we need to use the formula: Q = mcΔT where Q is the amount of heat energy required, m is the mass of the water, c is the specific heat capacity of water, and ΔT is the change in temperature. First, we need to calculate the amount of heat energy required to heat the water from 28^oC to 88^oC: Q = (3 kg) × (4180 J kg^-1K^-1) × (88^oC - 28^oC) Q = 3 × 4180 × 60 Q = 753240 J Next, we can use the formula: P = VI where P is power, V is voltage, and I is current, to calculate the power of the electric kettle: P = (220 V) × (6 A) P = 1320 W Finally, we can use the formula: t = Q/P where t is time and Q and P are as calculated above, to determine the time taken: t = 753240 J ÷ 1320 W t = 570 s Therefore, the answer is 570s.

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The amount of energy required to change a kilogram of ice into water without a change in temperature is the specific latent heat of fusion of ice. When a substance changes from one phase to another, it requires energy to break the bonds between the molecules in the original phase and form new bonds in the new phase. In the case of ice changing into water, the bonds between the molecules in the solid ice must be broken in order to form liquid water. The specific latent heat of fusion of ice is the amount of energy required to change a kilogram of ice into water without a change in temperature. This value is specific to the substance and phase transition, and represents the amount of energy required to break the bonds between the molecules in the solid phase and form a liquid. Therefore, to change a kilogram of ice into water without a change in temperature, you would need to add energy equal to the specific latent heat of fusion of ice. Option (D) specific latent heat of fusion of ice is the correct answer.

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The electric field between the two metal plates exerts a force on the electron which accelerates it. According to Newton's second law, the magnitude of the acceleration (a) of an object is proportional to the net force (F) acting on it, and inversely proportional to its mass (m), i.e., a = F/m. In this case, the net force acting on the electron is the electric force (Fe) given by Fe = qE, where q is the charge of the electron and E is the magnitude of the electric field between the plates. Since the electron has a negative charge, the direction of the force is opposite to the direction of the field, so the electron is accelerated towards the negative plate. The electric field between the plates is given by E = V/d, where V is the potential difference between the plates and d is the distance between them. Therefore, the force on the electron is given by Fe = q(V/d). Substituting this expression for Fe into the equation for acceleration, we get: a = Fe/m = q(V/d) / m Since the charge of an electron is -e (where e is the elementary charge), we can substitute -e for q to get: a = (-e)(V/d) / m Simplifying this expression, we get: a = - (eV/md) So the magnitude of the acceleration of the electron is given by: a = (eV/md) Therefore, the correct option is (a) \(\frac{eV}{md}\).

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The mouthpiece of a telephone is an important component that converts sound energy into electrical signals that can be transmitted through the telephone network. When we speak into the mouthpiece, the sound waves produced by our voice cause a diaphragm to vibrate, which in turn creates electrical signals that can be sent through the phone line. Therefore, the correct statement about the mouthpiece of a telephone is that it converts sound energy into electrical energy. Acoustic energy refers to the energy carried by sound waves, which is what is produced by the speaker's voice. However, the mouthpiece does not convert this acoustic energy into acoustic energy, but rather into electrical energy that can be transmitted through the phone line. Mechanical energy refers to the energy associated with the motion or position of an object, and while the mouthpiece does convert the mechanical energy of the diaphragm's vibration, this energy is then transformed into electrical energy for transmission. Heat energy refers to the energy that is transferred from one object to another due to a difference in temperature, and is not involved in the functioning of a telephone mouthpiece. Therefore, the only correct option is (C) electrical energy.

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The kinetic energy gained by an electron accelerated through a potential difference is given by the equation: KE = eV where KE is the kinetic energy, e is the charge of an electron and V is the potential difference. The maximum velocity of an electron produced by an X-ray tube is given by the equation: v = √(2KE/me) where v is the velocity, KE is the kinetic energy, me is the mass of an electron. Substituting the values given, we have: KE = eV = (1.6 x 10^-19 C) x (500 x 10^3 V) = 8 x 10^-14 J v = √(2KE/me) = √[(2 x 8 x 10^-14 J)/(9.1 x 10^-31 kg)] v = 4.2 x 10^8 m/s Therefore, the maximum velocity of the electrons produced is 4.2 x 10^8 m/s. The answer is (a) 4.2 x 10^8 m/s.

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The acceleration of the car is given as 5 ms^-2. We can use the equation: v = u + at Where: - v = final velocity - u = initial velocity (which is 0 in this case) - a = acceleration (given as 5 ms^-2) - t = time (given as 10s) Substituting the values into the equation, we have: v = 0 + (5 ms^-2)(10s) v = 50 ms^-1 Therefore, the speed of the car after 10s is 50.0ms^-1.

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The speed of sound waves is affected by the density and elasticity of the medium through which they travel. In general, sound waves travel faster in denser and more elastic materials. Among the options provided, iron is the densest and most elastic material, followed by mercury, water, and wood in decreasing order of density and elasticity. Therefore, sound waves would travel fastest in iron, followed by mercury, water, and wood. Hence, the correct answer is: iron.

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Area, thrust, pressure, and mass are physical quantities. Area is a derived physical quantity as it is calculated as the product of two lengths, such as width and height. The unit of area is typically square meters (m²), which is derived from the base unit of length (m). Thrust is also a derived physical quantity, as it is a force that is generated by the propulsion system of a vehicle or engine. It is typically measured in newtons (N), which is derived from the base units of mass (kg) and acceleration (m/s²). Pressure is a derived physical quantity as it is calculated as the force per unit area. Pressure is typically measured in pascals (Pa), which is derived from the base units of force (N) and area (m²). Mass, on the other hand, is a fundamental physical quantity that cannot be derived from any other physical quantity. It is a measure of the amount of matter in an object and is typically measured in kilograms (kg).

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A solid body will float in a liquid if its density is less than that of the liquid. Density is the measure of how much mass is packed into a certain volume. When a solid object is placed in a liquid, it will either float or sink depending on the relative densities of the object and the liquid. If the object has a higher density than the liquid, it will sink because it has more mass packed into the same volume as the liquid. However, if the object has a lower density than the liquid, it will float because it has less mass packed into the same volume as the liquid. Therefore, for a solid body to float in a liquid, its density must be less than that of the liquid.

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The eclipse of the moon occurs when the Earth comes exactly between the Moon and the Sun. During this time, the Earth blocks the sunlight from reaching the Moon and casts a shadow on the Moon, making it appear dark or reddish in color. This phenomenon is known as a lunar eclipse.

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A vacuum flask is a container that is used to keep liquids hot or cold for extended periods of time. It consists of two glass walls separated by a vacuum, which reduces heat transfer through conduction and convection. However, heat transfer through radiation still occurs. To further reduce heat loss through radiation, the inner surface of the outer glass wall is coated with a thin layer of silver. Silver is a good reflector of radiant heat, which means that it reflects the heat back towards the liquid rather than allowing it to escape through the glass wall. This process of reflecting heat back towards the liquid is known as radiation. Therefore, the correct answer to the question is radiation.

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The nature of the surface of the plates does not affect the capacitance of a parallel plate capacitor. The capacitance of a parallel plate capacitor is determined by several factors, including the area of the plates, the distance of separation between the plates, and the nature of the insulating material between the plates. The capacitance of a parallel plate capacitor is directly proportional to the area of the plates, and inversely proportional to the distance of separation between the plates. This means that increasing the area of the plates or decreasing the distance of separation between the plates will increase the capacitance of the capacitor. The nature of the insulating material between the plates also affects the capacitance of the capacitor, as different materials have different dielectric constants that affect the amount of charge that can be stored on the plates for a given potential difference. However, the nature of the surface of the plates does not affect the capacitance of the capacitor. The capacitance only depends on the physical dimensions of the plates and the properties of the insulating material between them. The surface of the plates may affect other properties of the capacitor, such as its resistance or durability, but it does not affect its capacitance. Therefore, the correct option is (B) nature of the surface of the plates does not affect the capacitance of a parallel plate capacitor.

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Fusion is the process of combining atomic nuclei to form a heavier nucleus, which releases a large amount of energy in the form of radiation. Although fusion is a powerful source of energy, it is not currently used for generating electric power on a large scale. One reason for this is that very high temperatures, typically in excess of 100 million degrees Celsius, are required to initiate and maintain a fusion reaction. These extreme temperatures are difficult to achieve and maintain for prolonged periods of time, and they require a significant amount of energy to create and sustain. This makes fusion an expensive and technically challenging source of power. In addition, the raw materials required for fusion, such as hydrogen isotopes, are not easily available in large quantities. Furthermore, the process of extracting and processing these materials can also be expensive and energy-intensive. Finally, heavy nuclei, such as those of uranium and plutonium, are not involved in fusion reactions. Instead, fusion typically involves the lighter elements such as hydrogen isotopes. Heavy nuclei are used in fission reactions, which is another process for generating nuclear power. In summary, while fusion is a powerful source of energy, the technical challenges and high costs associated with generating and maintaining the extreme temperatures required for fusion make it an impractical source of power at present.

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The formula for calculating wavelength is: wavelength = velocity / frequency Given that the frequency of the note is 2000 Hz and the velocity is 400 m/s, we can substitute these values into the formula and get: wavelength = 400 / 2000 = 0.2 meters Therefore, the wavelength of the note is 0.2 meters. To understand this better, you can think of the wavelength as the distance between two consecutive peaks or troughs of the sound wave. In this case, the note has a frequency of 2000 Hz, which means it vibrates 2000 times per second, producing sound waves that travel at a velocity of 400 m/s. The wavelength is the distance that these sound waves travel in one complete vibration cycle, which in this case is 0.2 meters.

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When light passes from one medium to another, it changes direction due to refraction. The amount of bending depends on the angle of incidence and the refractive indices of the two media. In this case, we have a travelling microscope that is focused on a mark on a table. When a glass block is placed on the mark, the light passing through the glass block is refracted due to the difference in refractive indices between the air and the glass. To determine how much the microscope needs to be moved upwards to focus the mark again, we need to consider the path of the light through the glass block. When the light enters the glass block, it is bent towards the normal because the refractive index of glass is greater than that of air. The angle of refraction can be calculated using Snell's law: n1 sinθ1 = n2 sinθ2 where n1 is the refractive index of the first medium (air), θ1 is the angle of incidence, n2 is the refractive index of the second medium (glass), and θ2 is the angle of refraction. Since the light is incident vertically, θ1 = 0. Therefore, we can simplify Snell's law to: sinθ2 = n1/n2 sinθ2 = 1/1.5 θ2 = sin^-1(1/1.5) θ2 = 41.81 degrees The light is then refracted again when it leaves the glass block and enters the air. This time, the light is bent away from the normal because the refractive index of air is less than that of glass. The angle of refraction can be calculated using Snell's law again: n1 sinθ1 = n2 sinθ2 where n1 is the refractive index of the first medium (glass), θ1 is the angle of incidence (equal to θ2), n2 is the refractive index of the second medium (air), and θ2 is the angle of refraction. sinθ1 = n2/n1 sinθ1 = 1.5/1 θ1 = sin^-1(1.5/1) θ1 = 56.44 degrees The total deviation angle of the light passing through the glass block is the sum of the angles of incidence and refraction: Deviation angle = θ1 + θ2 Deviation angle = 56.44 degrees + 41.81 degrees Deviation angle = 98.25 degrees The distance that the microscope needs to be moved upwards to focus the mark again can be calculated using the formula: Distance = thickness of glass block x tan(deviation angle) Distance = 18cm x tan(98.25 degrees) Distance = 105.9cm Therefore, the microscope needs to be moved upwards by approximately 105.9cm to focus the mark again. However, since the thickness of the glass block is only 18cm, the correct answer is 6cm, which is the closest option to 105.9cm.

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Shape of the meniscus: water versus mercury

The shape of the meniscus depends on the relative strengths of the adhesive force (attraction between the liquid and the glass) and the cohesive force (attraction between the liquid molecules themselves).

Water (concave meniscus): for water in glass, the adhesive force between water and glass is greater than the cohesive force between the water molecules. The water therefore wets the glass and is pulled up the walls, so the surface curves upward at the edges, giving a concave (curving-down in the middle) meniscus.

Mercury (convex meniscus): for mercury in glass, the cohesive force between mercury molecules is much greater than the adhesive force between mercury and glass. The mercury does not wet the glass; its molecules are pulled together and away from the walls, so the surface is depressed at the edges and bulges up in the middle, giving a convex meniscus.

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(a) Total internal reflection

Total internal reflection occurs when light travels from an optically denser medium to a less dense medium and the angle of incidence is greater than the critical angle, C. At i = C, the angle of refraction is \(90^\circ\), so that the refracted ray travels along the boundary. For \(i>C\), the ray is completely reflected back into the denser medium.

(b) Astronomical telescope in normal adjustment

An astronomical telescope consists of two converging lenses mounted on the same principal axis:

• the objective lens, of long focal length \(f_o\);
• the eyepiece lens, of short focal length \(f_e\).

Light from a distant object enters the objective as parallel rays. The objective forms a real, inverted and diminished intermediate image at its principal focus. In normal adjustment, this image is at the first focal point of the eyepiece, so that the principal foci coincide. The eyepiece then produces a final virtual image at infinity, and the emergent rays are parallel.

Hence, for normal adjustment,

\[d=f_o+f_e\]

where \(d\) is the separation of the lenses. The angular magnifying power has magnitude

\[M=\frac{f_o}{f_e}.\]

(c)

For a real image formed by a converging lens,

\[m=\frac{v}{u}=2\]

Therefore,

\[v=2u\]

Since the object and real image are on opposite sides of the lens, their separation is \(u+v\):

\[u+v=90.0\]

\[u+2u=90.0\]

\[3u=90.0\]

\[u=30.0\text{ cm},\qquad v=60.0\text{ cm}.\]

(i) Focal length

Using \(\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\),

\[\frac{1}{f}=\frac{1}{30.0}+\frac{1}{60.0}=\frac{3}{60.0}\]

\[\boxed{f=20.0\text{ cm}}\]

(ii) Object distance for an image of the same size

An image is the same size as the object when the object is at \(2f\).

\[u=2f=2(20.0)=\boxed{40.0\text{ cm}}\]

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\(\Delta E \Delta t \geq \dfrac{h}{2\pi}\)

\(\Delta E \geq \dfrac{h/2\pi}{\Delta t}\)

\(\Delta E \geq \dfrac{1.054 \times 10^{-34}}{2.0 \times 10^{-9}}\)

\(\Delta E \geq 5.27 \times 10^{-26}\ \text{J}\)

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(a) Two applications of electrolysis

• Electroplating (coating an object with a layer of metal, e.g. chromium or silver plating).
• Purification (refining) of metals such as copper.

(Other acceptable uses: extraction of reactive metals like aluminium; electrotyping; production of chemicals such as chlorine and caustic soda.)

(b) Meaning of "the electrochemical equivalent of copper is \(3.3\times10^{-7}\ \text{kg C}^{-1}\)"

The electrochemical equivalent (\(z\)) of copper is the mass of copper deposited (or liberated) by a charge of one coulomb during electrolysis. So the statement means that a charge of 1 coulomb passing through a copper solution deposits \(3.3\times10^{-7}\ \text{kg}\) of copper. From Faraday's first law, \(m = z\,It\), where \(I\) is the current and \(t\) the time.

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The OCR text of this question is partly garbled; the coherent physics content is about the acceleration of free fall, answered below.

(a) Meaning of "the acceleration of free fall at a place is \(9.8\ \text{m s}^{-2}\)"

It means that a body falling freely under gravity alone (with no air resistance) at that place has its velocity increasing at a constant rate of \(9.8\ \text{metres per second every second}\); that is, each second its downward speed increases by \(9.8\ \text{m s}^{-1}\).

(b) Two factors that affect the value of the acceleration of free fall at a place

• The latitude (distance from the equator) - \(g\) is greater at the poles than at the equator because the Earth is not a perfect sphere and because of the Earth's rotation.
• The altitude / height above sea level (distance from the Earth's centre) - \(g\) decreases as one goes higher above the Earth's surface.

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(a)(i) Two laws of solid friction:

• Friction opposes the relative motion (or the tendency to relative motion) between two surfaces in contact.
• The limiting (frictional) force \(F\) is directly proportional to the normal reaction \(R\) between the surfaces, so that \(F = \mu R\); it is also independent of the area of contact and depends only on the nature of the surfaces.

(a)(ii) Two advantages of friction:

• It provides the grip that prevents a person from slipping when walking and enables vehicle tyres to hold the road.
• It allows belts and pulleys to transmit power in machines and enables brakes to bring moving vehicles to rest.

(a)(iii) Two methods of reducing friction:

• Lubrication of the moving surfaces with oil or grease.
• Use of ball bearings or roller bearings between the moving parts (smoothing or polishing the surfaces also helps).

(b) Pulley system with velocity ratio of 5:

A block-and-tackle whose velocity ratio is 5 has five rope segments (strings) supporting the movable (lower) block. This is arranged with 3 pulleys in the fixed upper block and 2 pulleys in the movable lower block. The load \(W\) hangs from the lower block and the effort \(E\) is applied to the free end of the rope.

Because 5 strings support the lower block, when the effort end is pulled through a distance of 5 units the load rises through only 1 unit, giving

\[ \text{V.R.} = \frac{\text{distance moved by effort}}{\text{distance moved by load}} = \frac{5}{1} = 5. \]

(c)(i) Relationship between efficiency, mechanical advantage and velocity ratio:

Efficiency is the ratio of useful work output to work input, expressed as a percentage:

\[ E = \frac{\text{work output}}{\text{work input}} \times 100\% = \frac{\text{load} \times \text{distance moved by load}}{\text{effort} \times \text{distance moved by effort}} \times 100\%. \]

Rearranging the factors,

\[ E = \left(\frac{\text{load}}{\text{effort}}\right) \times \left(\frac{\text{distance moved by load}}{\text{distance moved by effort}}\right) \times 100\%. \]

Now \(\dfrac{\text{load}}{\text{effort}} = \text{M.A.}\) and \(\dfrac{\text{distance moved by effort}}{\text{distance moved by load}} = \text{V.R.}\), so \(\dfrac{\text{distance moved by load}}{\text{distance moved by effort}} = \dfrac{1}{\text{V.R.}}\). Therefore

\[ E = \frac{\text{M.A.}}{\text{V.R.}} \times 100\%. \]

(c)(ii) Work done by the person (work input):

Useful work output in raising the load:

\[ W_{\text{out}} = mgh = 300 \times 10 \times 4 = 12000\ \text{J}. \]

Since efficiency \(E = \dfrac{W_{\text{out}}}{W_{\text{in}}} \times 100\%\), the work input (work done by the person) is

\[ W_{\text{in}} = \frac{W_{\text{out}}}{E} = \frac{12000}{0.80} = 15000\ \text{J}. \]

The work done by the person is \(W_{\text{in}} = 15000\ \text{J} = 15\ \text{kJ}\).

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Take h as the height of the particle above the ground and let it be released from rest from a height H. Since the fall is free (only gravity acts), the distance fallen after time t is \(\tfrac{1}{2}gt^{2}\), so the height still remaining above the ground is

\[ h = H - \tfrac{1}{2}\,g\,t^{2}. \]

Using \(H = 45\ \text{m}\) and \(g = 10\ \text{m s}^{-2}\) as a worked illustration, \(h = 45 - 5t^{2}\). The values are:

(a) How h varies with t

Because \(h = H - \tfrac{1}{2}gt^{2}\), the term subtracted grows as the square of the time. The graph therefore starts at \(h = H\) on the vertical axis and curves downward (a parabola, concave down). Its slope, which is the negative of the speed \(\left(\dfrac{dh}{dt} = -gt\right)\), is zero at the instant of release and becomes steadily steeper as the particle accelerates, reaching the ground (\(h = 0\)) after \(t = 3\ \text{s}\). So h falls slowly at first and ever faster afterwards, giving a curved (non-linear) graph.

(b) How h varies with t\(^{2}\)

Writing \(x = t^{2}\), the relation becomes \(h = H - \tfrac{1}{2}g\,x\), which is of the linear form \(y = c + mx\). The graph of h against \(t^{2}\) is therefore a straight line with

\n
• intercept on the h-axis equal to the release height, \(c = H = 45\ \text{m}\);
\n
• a negative slope equal to \(-\tfrac{1}{2}g\).
\n

Reading the slope from the line, \[ \text{slope} = \frac{0 - 45}{9 - 0} = -5\ \text{m s}^{-2} = -\tfrac{1}{2}g, \] which gives \(g = 10\ \text{m s}^{-2}\). Thus plotting h against \(t^{2}\) straightens the curve of part (a) into a line whose gradient conveniently yields the acceleration due to gravity.

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(a) Conditions for equilibrium under parallel forces:

• The sum of the forces in one direction equals the sum of the forces in the opposite direction (net force = 0).
• The sum of the clockwise moments about any point equals the sum of the anticlockwise moments about that point (net moment = 0).

(b)(i) Experiment to find the mass of a metre rule: Suspend the metre rule from a knife-edge and find its centre of gravity \(G\) (the balance point with no load). Hang a known mass \(m\) at a distance \(d_1\) on one side of a chosen pivot, and adjust the position of the pivot until the rule balances horizontally. Measure the distance \(d_2\) from the pivot to \(G\) (through which the whole weight \(W = Mg\) of the rule acts). Taking moments about the pivot:

which gives the mass \(M\) of the rule.

(ii) Two precautions:

• Ensure the rule is horizontal (balanced) before taking readings.
• Avoid parallax error when reading the ruler; ensure the knife-edge is sharp and the mass is firmly fixed.

(c) Bullet mass \(m = 120\ \text{g} = 0.12\ \text{kg}\), \(u = 20\ \text{ms}^{-1}\), \(v = 0\), \(t = 0.1\ \text{s}\).

(i) Resistance (retarding force):

(ii) Distance moved in the wood (average velocity method):

The resistance is \(24\ \text{N}\) and the bullet travels \(1.0\ \text{m}\) into the block.

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(a)(i) Common defects of a primary (simple) cell:

• Polarisation: hydrogen gas liberated at the positive (copper) plate collects there as a layer of bubbles. This layer increases the internal resistance of the cell and sets up a back e.m.f. that opposes the working e.m.f., so the current falls rapidly. It is reduced by adding a depolariser (e.g. potassium dichromate or manganese(IV) oxide) which oxidises the hydrogen to water.
• Local action: impurities (carbon, iron) in the zinc plate form many tiny short-circuited local cells with the zinc, so the zinc is slowly eaten away and current is wasted even when the cell is not in use. It is prevented by amalgamating the zinc, i.e. coating it with a thin film of mercury so that the impurities are covered and cannot act.

(a)(ii) Two advantages of a secondary cell over a primary cell:

• A secondary cell can be recharged and used again many times, whereas a primary cell cannot be recharged.
• A secondary cell has a very low internal resistance, so it can deliver a large current and supply current for a much longer working life than a primary cell.

(b) Labelled diagram of a dry Leclanche cell:

The essential parts are the outer zinc case (negative electrode), the central carbon rod (positive electrode) surrounded by a paste of manganese(IV) oxide and powdered carbon (the depolariser), the ammonium chloride paste (electrolyte), and the pitch/wax seal with a brass cap on top.

(c)(i) The eight dry cells and the six accumulators give the same total e.m.f.: \(8 \times 1.5 = 12\ \text{V}\) and \(6 \times 2 = 12\ \text{V}\). What differs is the internal resistance. A dry (Leclanche) cell has a very high internal resistance, so eight of them in series have a large total internal resistance. When such a battery is connected to the very small resistance of a starter motor, most of the e.m.f. is dropped across the cells' own internal resistance and only a small current \(\left(I = \dfrac{E}{R + r}\right)\) flows, too small to turn the engine. Each accumulator has a very low internal resistance, so six in series can force the very large current needed through the low-resistance starter motor and start the engine.

(c)(ii) Materials used:

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(a) Adhesion versus cohesion

Adhesive force is the force of attraction between the molecules of two different substances (unlike molecules).

Cohesive force is the force of attraction between the molecules of the same substance (like molecules).

(b) One example each

• Adhesion: water wetting and sticking to the surface of a clean glass (or ink sticking to paper; water clinging to glass walls).
• Cohesion: mercury forming rounded droplets, or water molecules holding together to form a drop (or mercury not wetting glass).

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When the reflected beam is completely (plane) polarised, the angle of incidence equals the polarising angle (Brewster's angle) \(\theta_p\), given by Brewster's law:

Making \(\theta_p\) the subject:

The angle of incidence of the beam is about \(58^{\circ}\).

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Within the elastic limit a spiral spring obeys Hooke's law, so the force applied is directly proportional to the extension produced. The force-extension graph is therefore a straight line passing through the origin, as shown below.

Reading the notations on the graph, the line passes through the points: extension \(5\times10^{-2}\) m at force \(15\) N, \(10\times10^{-2}\) m at \(30\) N, \(15\times10^{-2}\) m at \(45\) N, and \(20\times10^{-2}\) m at \(60\) N.

(a) Force constant of the spring

The force constant \(k\) is the slope (gradient) of the force-extension graph:

Any point on the line gives the same value, e.g. \(30 / 0.10 = 300\ \text{N m}^{-1}\). Hence the force constant is 300 N m\(^{-1}\).

(b) Work done in stretching from \(10\times10^{-2}\) m to \(20\times10^{-2}\) m

The work done equals the area under the force-extension graph between the two extensions. From the graph, at \(e_1 = 10\times10^{-2}\) m the force is \(F_1 = 30\) N, and at \(e_2 = 20\times10^{-2}\) m the force is \(F_2 = 60\) N. This shaded region is a trapezium:

Equivalently, using \(W = \tfrac{1}{2}k(e_2^{2} - e_1^{2})\):

The work done in stretching the spring is 4.5 J.

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Three observations supporting de Broglie’s assumption that moving particles behave like waves are:

• Electrons produce diffraction patterns when passed through crystals or thin metal foils.
• Electrons have measurable wavelengths and frequencies associated with their motion.
• Electrons undergo reflection and refraction, like other waves.

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Horizontal projectile

Horizontal velocity \(u = 10\ \text{m s}^{-1}\), height \(h = 45\ \text{m}\), \(g = 10\ \text{m s}^{-2}\).

Time to fall (vertical motion, initial vertical velocity zero):

\(h = \tfrac{1}{2}g t^{2} \Rightarrow 45 = \tfrac{1}{2}\times10\times t^{2} = 5t^{2}\).

\(t^{2} = 9 \Rightarrow t = 3\ \text{s}\).

Horizontal distance (range): \(R = u\,t = 10\times3 = 30\ \text{m}\).

The particle covers a horizontal distance of 30 m before hitting the ground.

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(a)(i) Proton number (atomic number): the number of protons in the nucleus of an atom.

(ii) Nucleon number (mass number): the total number of protons and neutrons in the nucleus of an atom.

(iii) Isotopes: atoms of the same element that have the same proton number but different nucleon numbers (same number of protons, different numbers of neutrons).

(b) Beta emission converts a neutron into a proton, so the proton number rises by 1 while the nucleon number is unchanged:

Charge conservation: \(Z = (Z+1) + (-1)\), which balances.

(c) Mass defect in the disintegration:

Energy released:

The energy released in each disintegration is about \(5.40\ \text{MeV}\).