WAEC SSCE - Physics - 2020

Maelezo ya Majibu

The statement that explains why hot soapy water is more effective in cleaning oil-stained dishes is that "soap and heat decrease the surface tension of oil." When oil gets stuck on dishes, it forms a layer on top of the surface, making it difficult for water to penetrate and remove it. However, soap molecules are designed to break down the surface tension of oil, allowing water to mix with it and wash it away. The addition of heat to the soapy water further helps to reduce the surface tension of the oil, making it easier to clean. Therefore, using hot soapy water is more effective in cleaning oil-stained dishes.

Maelezo ya Majibu

The angle of refraction can be calculated using Snell's Law, which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the refractive indices of the two materials. Mathematically, this can be written as: sin(angle of incidence) / sin(angle of refraction) = refractive index of material 1 / refractive index of material 2 For this problem, we can substitute the given values and solve for the angle of refraction: sin(30\(^o\)) / sin(angle of refraction) = 1.5 / 1.36 Taking the inverse sine of both sides: angle of refraction = sin^-1(sin(30\(^o\)) * 1.36 / 1.5) Using a calculator, the angle of refraction can be calculated to be approximately 33.5\(^o\). So, the angle of refraction is approximately 33.5\(^o\).

Maelezo ya Majibu

The capacitance of a parallel plate capacitor is directly proportional to the surface area of the plates and inversely proportional to the distance between the plates. This relationship can be expressed as C = εA/d, where C is the capacitance, ε is the permittivity of free space, A is the surface area of the plates, and d is the distance between the plates. When the distance of separation between the plates is decreased, the denominator of this equation decreases, causing the overall value of the capacitance to increase. So, the answer is that the capacitance will increase in value.

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

The vacuum between the double walls of a thermos flask helps to reduce heat loss in two ways: conduction and radiation. Conduction is the transfer of heat through a material, such as the metal walls of the thermos flask. When there is a vacuum between the walls, there are no particles to transfer heat through conduction, so heat cannot escape or enter the flask easily. Radiation is the transfer of heat through electromagnetic waves, such as the infrared radiation that is emitted by hot objects. The vacuum between the walls of the thermos flask helps to reduce heat loss through radiation by reflecting the infrared radiation back into the flask, rather than allowing it to escape through the walls. Therefore, the vacuum between the double walls of a thermos flask reduces heat loss through both conduction and radiation, making it an effective insulator for keeping hot liquids hot and cold liquids cold.

Maelezo ya Majibu

The electrical device described is a step-down transformer. A transformer is a device that can change the voltage level of an alternating current (AC) electrical signal. It has two coils, a primary coil and a secondary coil, which are wrapped around a common iron core. In this case, the primary coil has more turns than the secondary coil, which means that the voltage in the secondary coil will be lower than the voltage in the primary coil. This is because transformers work by transferring energy from one coil to the other through the magnetic field created by the alternating current. A step-up transformer would have more turns in the secondary coil than the primary coil, resulting in a higher voltage in the secondary coil. A DC generator produces direct current, which is different from the alternating current used in transformers. An AC generator produces alternating current, but it does not have the two coils of a transformer, so it cannot be a transformer.

Maelezo ya Majibu

When two cells of negligible internal resistances are connected in parallel, their positive terminals are connected together and their negative terminals are connected together. This forms a circuit with two paths for current flow. Since the cells have the same electromotive force (emf) E1 and E2, they are trying to push the same amount of charge through each path. Thus, the total emf of the combination E must be the same as E1 and E2. Therefore, the correct option is E = E1 = E2. is incorrect because emfs in parallel do not add up. is incorrect because it represents the formula for calculating the combined resistance of two resistors in parallel, not emfs. is incorrect because it represents the formula for calculating the ratio of emfs in a series circuit.

Maelezo ya Majibu

A luminous object is one that gives off light of its own. This means it does not need any other source of light to shine. It produces light on its own, making it visible even in the dark. Examples of luminous objects include the sun, stars, fireflies, and light bulbs.

Maelezo ya Majibu

When light travels from a less dense medium (such as air) to a denser medium (such as glass), it slows down. According to Snell's Law of Refraction, the ratio of the speed of light in the two media is proportional to the ratio of their refractive indices. This means that the speed of the light decreases in the denser medium. As the speed of light is proportional to its frequency, a decrease in the speed of light results in a decrease in its frequency. This means that there is a change in the frequency of the light when it travels from the less dense medium to the denser medium. Additionally, when light travels obliquely from a less dense medium to a denser medium, it bends towards the normal (perpendicular) to the surface separating the two media. This is known as refraction and is responsible for the bending of light at the surface of a lens or the surface of a swimming pool. So, the answer is that the light refracts towards the normal and there is a change in the frequency of the light.

Maelezo ya Majibu

The electric force between two charged particles is determined by two factors: the magnitude of the charges and the distance between the particles. Like charges, such as two positively charged particles, repel each other, so an increase in the electric force would mean an increase in the repulsion between the particles. To increase the electric force between two positively charged particles, we need to either increase the magnitude of the charges or decrease the distance between the particles. Decreasing the mass of the particles or increasing the mass of the particles has no effect on the electric force. Similarly, "increasing the assistance" between the particles is not a meaningful term in this context, so it is not a relevant option. Therefore, the correct answer is: decreasing the distance between the particles. This is because the closer the particles are to each other, the stronger the electric force of repulsion between them will be, according to Coulomb's law.

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

The frequency of a note emitted by a vibrating string is determined by the tension in the string, the mass per unit length of the string, and the length of the string. When a string is plucked or struck, it vibrates at a certain frequency, producing a sound wave with a particular pitch. The tension in the string determines how tightly the string is stretched and how fast it vibrates. The mass per unit length of the string affects how easily the string vibrates and the frequency of the sound wave. The longer the string, the lower the frequency of the sound wave, while the shorter the string, the higher the frequency of the sound wave. Therefore, the correct option is: "Mass per unit length of string, tension in string and length of string".

Maelezo ya Majibu

The formula for acceleration is given by: acceleration = change in velocity / change in time. In this case, the change in velocity is 72 km/hr and the change in time is 10s. To use the formula, we first need to convert the velocity from km/hr to m/s. 72 km/hr = 72 x 1000/3600 m/s = 20 m/s So, acceleration = 20 m/s / 10s = 2 m/s\(^2\) Therefore, the acceleration of the motorcycle is 2 m/s\(^2\).

Maelezo ya Majibu

Maelezo ya Majibu

The problem requires us to find the temperature at which the length of the iron bar increases by 0.01%. We can use the formula for linear expansion of a solid: ΔL = α L ΔT where ΔL is the change in length, L is the original length, ΔT is the change in temperature, and α is the coefficient of linear expansion. Rearranging this equation, we get: ΔT = ΔL / (α L) We can substitute the given values into this equation to obtain the temperature change required to produce the given change in length: ΔT = (0.01/100) / (1.2 x 10^(-5) x 100) = 8.33°C Since we are asked for the temperature at which the length of the iron bar will increase by 0.01%, we need to add this temperature change to the initial temperature of 20°C: T_final = T_initial + ΔT = 20 + 8.33 = 28.3°C Therefore, the correct answer is option (C), 28.3°C.

Maelezo ya Majibu

The correct term for the average distance moved by a molecule between collisions is "mean free path". This is the average distance that a molecule travels before it collides with another molecule or object in its surroundings. Think of it like a crowded room where people are moving around. If you're a person in the room, you're going to bump into other people as you move around. The average distance you travel before bumping into someone else is like the mean free path for a molecule in a gas. The mean free path is affected by several factors, including the density of the gas, the temperature, and the size of the molecules. In general, if the gas is denser or the molecules are larger, the mean free path will be shorter because there are more collisions happening. Similarly, if the temperature is higher, the molecules will be moving faster and the mean free path will be longer. Understanding the concept of mean free path is important in fields like physics and chemistry, where scientists study the behavior of gases and other materials at the molecular level.

Maelezo ya Majibu

In a collision between two objects, kinetic energy is conserved only if the collision is elastic. An elastic collision is a type of collision in which the total kinetic energy of the two objects before and after the collision remains the same. In other words, no kinetic energy is lost or gained during the collision. In an inelastic collision, on the other hand, some or all of the kinetic energy is converted into other forms of energy, such as heat or sound. This means that kinetic energy is not conserved in an inelastic collision. So, the correct option is "the collision is elastic". If the collision is inelastic, then the kinetic energy is not conserved. The other options are not correct because the conservation of kinetic energy does not depend on whether one of the objects was initially at rest or whether potential energy is converted to work.

Maelezo ya Majibu

Maelezo ya Majibu

The net force acting on the ball is the vector sum of all the forces acting on it along the plank. The forces acting on the ball are the gravitational force and the kinetic frictional force. The gravitational force acting on the ball can be resolved into two components, one perpendicular to the plank and the other parallel to the plank. The component perpendicular to the plank is balanced by the normal force of the plank on the ball. The component parallel to the plank is responsible for the acceleration of the ball down the plank and is given by mg sin θ, where θ is the angle of inclination of the plank. The kinetic frictional force acts in the opposite direction to the motion of the ball and is given by F. Therefore, the net force acting on the ball along the plank is the vector sum of the gravitational force and the kinetic frictional force: Net force = mg sin θ - F Therefore, the correct answer is option (C): mgsin θ - F.

Maelezo ya Majibu

The material used to slow down the neutrons in a nuclear reactor is graphite. In a nuclear reactor, neutrons are released from the fission of uranium atoms, and these neutrons need to be slowed down or moderated so that they can be absorbed by other uranium atoms to sustain the chain reaction. Graphite is an excellent moderator because it contains many carbon atoms that are arranged in a crystalline lattice structure. The neutrons can easily collide with the carbon atoms in the graphite lattice and transfer some of their kinetic energy to the graphite atoms, thereby slowing down the neutrons. Slower neutrons are more likely to be absorbed by uranium atoms and initiate fission reactions. Boron and copper are not typically used as moderators in nuclear reactors. Boron is often used as a neutron absorber to control the rate of the nuclear reaction, while copper is a good conductor of heat and electricity but is not an effective moderator. Uranium is the fuel used in the reactor to sustain the nuclear reaction, but it is not used as a moderator because it is more likely to absorb neutrons than to slow them down.

Maelezo ya Majibu

An AC generator and a DC motor are different in their basic operating principles and construction. To convert an AC generator to a DC motor, certain modifications must be made. The main difference between an AC generator and a DC motor is that an AC generator produces alternating current (AC) while a DC motor runs on direct current (DC). In order to convert an AC generator to a DC motor, the AC output of the generator must be converted to DC. This can be achieved by replacing the slip rings of the AC generator with a split ring, and connecting a DC source (such as a battery) to the split ring. The split ring acts as a rectifier, converting the AC output of the generator to DC. Therefore, to convert an AC generator to a DC motor, you would replace the slip rings with a split ring and connect a DC source.

Maelezo ya Majibu

The reading of the scale in this situation is equal to the weight of the man plus the force due to the acceleration of the elevator. The weight of the man can be calculated as follows: Weight = mass x gravity Weight = 60kg x 10ms\(^{-2}\) Weight = 600N The force due to the acceleration of the elevator can be calculated as follows: Force = mass x acceleration Force = 60kg x 5ms\(^{-2}\) Force = 300N Adding these two forces together, we get the total force acting on the man: Total force = Weight + Force due to acceleration Total force = 600N + 300N Total force = 900N So the reading of the scale would be 900N.

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

The wave that requires a material medium for its propagation is sound waves. Sound waves are a type of mechanical wave, which means they need a material medium, such as air, water, or a solid, to travel through. This is because sound waves involve the compression and rarefaction of the medium, which causes the wave to propagate. Radio waves and light waves are types of electromagnetic waves, which do not require a material medium to travel through. They can travel through a vacuum, such as outer space, and do not involve the compression and rarefaction of a medium. X-rays are also a type of electromagnetic wave, and like other electromagnetic waves, they do not require a material medium to propagate. They can travel through a vacuum and do not involve the compression and rarefaction of a medium.

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

To convert a moving-coil galvanometer into a voltmeter, an external resistance is added in series with the galvanometer. The value of the external resistance is chosen so that the full-scale deflection of the galvanometer corresponds to the maximum voltage that the voltmeter is intended to measure. In this case, the moving-coil galvanometer gives a full-scale deflection with a current of 0.005A. To convert it into a voltmeter that reads up to 5V, an external resistance of 975\(\Omega\) is used. Using Ohm's Law, we can calculate the resistance of the galvanometer (meter resistance) as follows: Meter resistance = (maximum voltage / full-scale current) - external resistance Meter resistance = (5 V / 0.005 A) - 975\(\Omega\) Meter resistance = 1000\(\Omega\) - 975\(\Omega\) Meter resistance = 25\(\Omega\) Therefore, the resistance of the meter is 25.00\(\Omega\). Option C is the correct answer.

Maelezo ya Majibu

The wavelength of a moving particle, such as an electron, is given by the de Broglie equation: λ = h/mv where λ is the wavelength, h is Planck's constant, m is the mass of the particle, and v is its velocity. Substituting the given values, we get: λ = (6.6 x 10^-34 J s) / (9.1 x 10^-31 kg) x (2.0 x 10^6 m/s) Simplifying this expression, we get: λ = 3.63 x 10^-10 m Therefore, the associated wavelength of the electron is 3.63 x 10^-10 m Explanation: This question requires the application of the de Broglie equation, which relates the wavelength of a moving particle to its momentum. In this case, we are given the mass and velocity of an electron and asked to calculate its associated wavelength. We can do this by simply plugging in the given values into the de Broglie equation and solving for λ. The answer we obtain is the wavelength of the electron.

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

When the diver steps off the diving platform, they start to fall towards the water due to the force of gravity. As the diver falls, their height above the water decreases, and therefore their gravitational potential energy decreases. At the same time, the diver's speed increases as they fall, which means their kinetic energy increases. Since there is no air resistance during the fall, the total mechanical energy of the diver (which is the sum of their kinetic energy and gravitational potential energy) remains constant. This is known as the conservation of energy. However, momentum is not directly affected by the decrease in height or the increase in speed. Momentum is defined as the product of an object's mass and velocity, and it only changes if there is a net force acting on the object. In the absence of external forces, the momentum of the diver remains constant. Therefore, the correct answer is: the diver's gravitational potential energy decreases, while their kinetic energy increases, and their momentum remains constant.

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

Maelezo ya Majibu

The earpiece of a telephone handset converts energy from one form to another. In this case, the earpiece converts electrical energy into sound energy that can be heard by the person using the telephone. When a person speaks into the telephone, their voice is converted into an electrical signal which is sent along the telephone wire to the other person's telephone. This electrical signal is received by the earpiece, which then converts it back into sound energy that can be heard by the person holding the telephone. Therefore, the correct answer is option (A), "electrical to sound".

Maelezo ya Majibu

The depolarizing agent in a Leclanche cell is manganese (IV) oxide. A Leclanche cell is a type of battery that uses a chemical reaction to generate electricity. Inside the cell, there are two electrodes, a zinc plate and a carbon rod, immersed in an electrolyte solution of ammonium chloride. When the cell is in use, the zinc plate undergoes a chemical reaction with the ammonium chloride, releasing electrons and forming zinc chloride and hydrogen gas. This process creates a buildup of positive charge on the zinc plate, which can eventually inhibit the flow of electrons and reduce the cell's efficiency. To counteract this buildup of positive charge, the manganese (IV) oxide acts as a depolarizing agent. It reacts with the hydrogen gas produced by the reaction on the zinc plate, converting it to water and preventing the buildup of positive charge. This allows the cell to continue generating electricity for a longer period of time.

Maelezo ya Majibu

The impedance of the circuit can be calculated using the formula: Z = √(R^2 + X^2) Where R is the resistance of the inductor and X is the reactance of the inductor, which is given by: X = 2πfL Where f is the frequency of the AC source and L is the inductance of the inductor. Substituting the given values, we get: X = 2π(60)(0.5) = 188.5 Ω R = 50 Ω Z = √(50^2 + 188.5^2) = 195.0 Ω Therefore, the impedance of the circuit is 195.0 Ω. To explain it in simpler terms, the impedance of the circuit is like the total resistance that the AC source "sees" when it is connected to the circuit. It takes into account both the resistance and the reactance (which is like a resistance but specific to AC circuits) of the circuit. In this case, the inductor has a reactance of 188.5 Ω which, when combined with the resistance of 50 Ω, gives an impedance of 195.0 Ω.

Maelezo ya Majibu

Maelezo ya Majibu

The correct observation would be "Deflected, not deflected, not deflected." \(\alpha\)-particles are positively charged particles and are therefore deflected when they pass through a magnetic field. The direction of deflection is determined by the right-hand rule and depends on the direction of the magnetic field and the direction of the velocity of the particle. \(\beta\)-particles are also charged particles, but they are negatively charged, so they are not deflected by a magnetic field. \(\gamma\)-rays are not charged particles and therefore do not experience a force when they pass through a magnetic field. Thus, they are not deflected. In summary, the deflection of charged particles in a magnetic field depends on the charge and velocity of the particle, while uncharged particles are not deflected.

Maelezo ya Majibu

Maelezo ya Majibu

The statement "The volume of a fixed mass of a gas varies inversely as the pressure on it provided the temperature is constant" is describing Boyle's law. Boyle's law states that the volume of a fixed amount of gas at a constant temperature is inversely proportional to the pressure on it. In other words, if the pressure on a gas increases, its volume will decrease, and vice versa. So, the statement is Boyle's law.

Maelezo ya Majibu

Maelezo ya Majibu

The quantity of motion of a body is its momentum. Momentum is a measure of how much "oomph" an object has when it is moving. It is calculated by multiplying the object's mass by its velocity, or speed in a particular direction. Think of a heavy truck moving slowly and a small car moving quickly. Even though the car is lighter than the truck, it can still have a greater momentum because it is moving faster. Momentum is important in physics because it helps us understand how objects interact with each other during collisions and other events. Acceleration refers to how quickly an object's velocity changes over time, while displacement is a measure of how far an object has moved from its original position. Velocity, on the other hand, is a measure of how fast an object is moving in a particular direction.

Maelezo ya Majibu

The work done on the box is equal to the force exerted on the box multiplied by the distance it moves in the direction of the force. In this case, the force exerted on the box is the force required to overcome friction, which is equal to the normal force multiplied by the coefficient of kinetic friction. First, we need to calculate the normal force, which is the force exerted on the box by the floor in a direction perpendicular to the surface. Since the box is at rest, the normal force is equal to the weight of the box, which is 500N. Next, we can calculate the force required to overcome friction, which is equal to the normal force multiplied by the coefficient of kinetic friction. force of friction = 500N x 0.22 = 110N Finally, we can calculate the work done on the box by multiplying the force required to overcome friction by the distance moved. work done = force of friction x distance moved = 110N x 8m = 880J Therefore, the work done on the box is 880J, and the correct option is (a) 880J.

Maelezo ya Majibu

Properties of lasers that make them preferable to an ordinary light beam:

• Monochromatic: laser light is of a single (or very narrow band of) wavelength/colour, unlike ordinary light which is a mixture of many wavelengths.
• Coherent: the waves are all in phase with one another, both in space and time.
• Highly directional and narrow (collimated): the beam is very parallel and spreads very little over long distances, giving high intensity concentrated in a small area (it is also very intense/bright).

Maelezo ya Majibu

Functions of the parts of a solar (flat-plate) heat collector:

• (a) Metal flat plate (absorber): it is painted dull black to absorb the incident solar radiation and convert it into heat, then conducts this heat to the water flowing in the tubes attached to it.
• (b) Thermal insulator: it is placed behind and around the plate to reduce heat loss (by conduction and convection) to the surroundings, so that the collected heat is retained and transferred to the water.
• (c) Tubes: they carry the water (or fluid) in good thermal contact with the hot absorber plate, so that the water is heated and then conveyed away to the storage tank for use.

Maelezo ya Majibu

(a)(i) Completed table

The mass evaporated is the decrease in balance reading:

\[m=203.22-\text{balance reading}\]

Time / s	0	100	200	300	400
Balance reading / g	203.22	201.62	199.79	198.26	196.50
Mass of water evaporated / g	0.00	1.60	3.43	4.96	6.72
Energy supplied / J	0	3800	7600	11400	15200

(ii) The energy supplied is \(E=Pt\), where \(P=38\text{ J s}^{-1}\). Thus, for example, at \(100\text{ s}\), \(E=38\times100=3800\text{ J}\).

(iii) Graph of energy supplied against mass evaporated

(iv) Slope of the graph

Using two well-separated points on the line of best fit, \((0.8\text{ g},2000\text{ J})\) and \((6.1\text{ g},14000\text{ J})\):

\[\text{Slope}=\frac{\Delta E}{\Delta m}=\frac{(14000-2000)\text{ J}}{(6.1-0.8)\text{ g}}=\frac{12000}{5.3}=2.26\times10^3\text{ J g}^{-1}\]

\[\therefore\ \text{Slope}=2.26\times10^6\text{ J kg}^{-1}.\]

(v) The slope represents the specific latent heat of vapourisation of water. It is the heat energy required to change unit mass of water at its boiling point into vapour at the same temperature.

Thus, approximately \(2.26\times10^3\text{ J}\) is required to evaporate \(1\text{ g}\) of water, or \(2.26\times10^6\text{ J}\) is required for \(1\text{ kg}\) of water.

Maelezo ya Majibu

(a)(i) The torque (moment) of a force about a point is the turning effect of the force about that point; it is the product of the force and the perpendicular distance of its line of action from the point (the axis of rotation): \(\tau=F\times d\).

(a)(ii) Factors that determine a torque:

• The magnitude of the applied force.
• The perpendicular distance between the line of action of the force and the pivot (axis).
• The angle at which the force is applied relative to the arm (line of action direction).

(b)(i) Free fall is the motion of a body falling only under the action of gravity, with no other force (such as air resistance) acting on it; its acceleration is g.

(b)(ii) Take upward positive, u = 10.0 m/s, tower height 40.0 m so displacement to the ground = -40.0 m, g = 10.0 m/s\(^2\).

\[s=ut-\tfrac{1}{2}gt^{2}\Rightarrow-40=10t-5t^{2}.\]

\[5t^{2}-10t-40=0\Rightarrow t^{2}-2t-8=0\Rightarrow(t-4)(t+2)=0.\]

Taking the positive root, \(t=4\,\text{s}\).

(c) Cube side 8.0 cm, so face area \(A=8\times8=64\,\text{cm}^2\). It floats with 2.0 cm below the oil-water interface (in water) and \(8-2=6\,\text{cm}\) in the oil. By the principle of flotation, the weight of the wood equals the total upthrust from oil and water.

Mass \(=\rho_{water}V_{water}+\rho_{oil}V_{oil}\) (using densities in g/cm\(^3\): water 1.00, oil 0.72).

\(V_{water}=64\times2=128\,\text{cm}^3\); \(V_{oil}=64\times6=384\,\text{cm}^3\).

\[m=(1.00\times128)+(0.72\times384)=128+276.48=404.48\,\text{g}\approx404.5\,\text{g}.\]

Maelezo ya Majibu

Projectile fired at an angle. For a projectile launched with speed u at angle \(\theta\) to the horizontal, the time of flight is

\[T=\dfrac{2u\sin\theta}{g}.\]

Given \(\theta=30^{\circ}\), \(T=25\,\text{s}\) and \(g=10\,\text{m s}^{-2}\):

\[25=\dfrac{2u\sin30^{\circ}}{10}=\dfrac{2u(0.5)}{10}=\dfrac{u}{10}.\]

Therefore \(u=25\times10=250\,\text{m s}^{-1}\).

The velocity of projection is \(250\,\text{m s}^{-1}\) at \(30^{\circ}\) to the horizontal.

Maelezo ya Majibu

(a)(i) The (critical) optical angle is the angle of incidence in the denser medium for which the angle of refraction in the less dense medium is exactly 90 degrees; at this angle the refracted ray grazes the surface, and beyond it total internal reflection occurs.

(a)(ii) Conditions for total internal reflection:

• Light must travel from an optically denser medium to an optically less dense medium (from high to low refractive index).
• The angle of incidence in the denser medium must be greater than the critical angle for the two media.

(a)(iii) Applications: optical fibres (communication/endoscopes); totally reflecting prisms in periscopes and binoculars; the shining/sparkle of diamonds; the formation of mirages.

(b) Effects of refraction: a stick partly immersed in water appears bent; a pool of water or swimming pool appears shallower than it really is (real depth greater than apparent depth). (Also the twinkling of stars and apparent raising of the sun at sunset.)

(c)(i) A progressive (travelling) wave is one that moves outward from a source, transferring energy from one point to another through a medium, with the disturbance advancing continuously.

(c)(ii) Comparing with \(y=A\sin(\omega t-kx)\), the angular frequency is \(\omega=1000\pi\,\text{rad s}^{-1}\) and the wave number is \(k=\dfrac{100\pi}{17}\,\text{m}^{-1}\) (from the given equation).

(\(\alpha\)) Frequency: \(f=\dfrac{\omega}{2\pi}=\dfrac{1000\pi}{2\pi}=500\,\text{Hz}\).

(\(\beta\)) Wavelength: \(\lambda=\dfrac{2\pi}{k}=\dfrac{2\pi\times17}{100\pi}=0.34\,\text{m}\).

(\(\gamma\)) Speed: \(v=f\lambda=500\times0.34=170\,\text{m s}^{-1}\).

Maelezo ya Majibu

(a)(i) Resonance in a series RLC circuit

Resonance frequency, \(f_0\), is the frequency at which the inductive reactance equals the capacitive reactance:

\[X_L=X_C\]

At this frequency the resultant reactance is zero, so the impedance is minimum and equal to the resistance, \(Z=R\). The current is therefore maximum.

(a)(ii) Variation of \(R\), \(X_C\) and \(X_L\) with frequency

At the resonant frequency \(f_0\), the \(X_L\) and \(X_C\) curves intersect. The resistance \(R\) remains constant as frequency changes.

(a)(iii) Phase relationship between current and supply voltage

1. When \(f=f_0\), \(X_L=X_C\); the current is in phase with the supply voltage.
2. When \(f<f_0\), \(X_C>X_L\); the circuit is capacitive and the current leads the supply voltage.
3. When \(f>f_0\), \(X_L>X_C\); the circuit is inductive and the current lags the supply voltage.

(b)(i) Mutual inductance

Mutual inductance is the induced e.m.f. in one coil per unit rate of change of current in a nearby magnetically linked coil:

\[M=\frac{E}{\mathrm{d}I/\mathrm{d}t}\]

(b)(ii) Angular speed of the generator coil

For a rotating coil generator,

\[E_0=NBA\omega\]

Radius of coil:

\[r=\frac{8.0\times10^{-2}}{2}=4.0\times10^{-2}\ \text{m}\]

Area of coil:

\[A=\pi r^2=3.142(4.0\times10^{-2})^2=5.0272\times10^{-3}\ \text{m}^2\]

Hence,

\[\omega=\frac{E_0}{NBA}=\frac{480}{500\times0.25\times5.0272\times10^{-3}}\]

\[\omega=7.64\times10^2\ \text{rad s}^{-1}\]

Angular speed = \(7.64\times10^2\ \text{rad s}^{-1}\) (approximately \(763.8\ \text{rad s}^{-1}\)).

Maelezo ya Majibu

Labels: isotopes, radioactive tracers, electromagnetic waves, nuclear reactor.

(a) Isotopes are atoms of the same element with the same number of protons but a different number of neutrons.

(b)

(i) Radioactive tracers in medicine are used for diagnosis and treatment of diseases. Two uses are:

1. PET scans: Positron Emission Tomography (PET) scans use radioactive tracers to produce three-dimensional images of the body. This helps in the detection and diagnosis of diseases such as cancer, heart diseases, and brain disorders.
2. Radiation therapy: Radioactive tracers can be used to destroy cancer cells. This treatment is called radiation therapy.

(ii) Radioactive tracers in industry are used for various purposes. Two uses are:

1. Oil and gas exploration: Radioactive tracers are used to locate oil and gas deposits underground. This helps in identifying the most profitable locations for drilling.
2. Quality control: Radioactive tracers are used to detect flaws in materials and products during the manufacturing process. This helps to ensure that the final product meets the required quality standards.

(iii) Radioactive tracers in agriculture are used to improve crop production. Two uses are:

1. Soil analysis: Radioactive tracers can be used to determine the nutrients present in the soil. This helps in determining the type and amount of fertilizer needed to improve crop yield.
2. Pest control: Radioactive tracers can be used to study the behavior of pests such as insects and rodents. This helps in developing effective pest control strategies.

(c) Three features of electromagnetic waves are:

1. They are transverse waves.
2. They can travel through a vacuum.
3. They all travel at the same speed in a vacuum.

(d) Four components of the nuclear reactor are:

1. Fuel rods: These contain the fuel (e.g., uranium) that undergoes fission to release energy.
2. Control rods: These are made of materials such as boron or cadmium that can absorb neutrons and control the rate of the nuclear reaction.
3. Moderator: This is a material (e.g., water, graphite) that slows down the neutrons released during fission, making them more likely to cause further fission.
4. Coolant: This is a fluid (e.g., water, liquid sodium) that circulates through the reactor, removing heat and preventing overheating.

(i) The functions of the components are:

1. Fuel rods: To provide the fuel for nuclear fission reactions.
2. Control rods: To regulate the rate of nuclear fission reactions.
3. Moderator: To slow down the neutrons released during fission, increasing the likelihood of further fission.
4. Coolant: To remove heat and prevent overheating of the reactor.

Maelezo ya Majibu

Escape velocity. A body just escapes the earth gravitational field when its total mechanical energy is zero, i.e. its initial kinetic energy equals the work needed to move it from the earth surface to infinity (the gravitational potential energy magnitude).

\[\tfrac{1}{2}mV_E^{2}=\dfrac{GMm}{R}.\]

At the earth surface the acceleration due to gravity is \(g=\dfrac{GM}{R^{2}}\), so \(GM=gR^{2}\). Substituting:

\[\tfrac{1}{2}mV_E^{2}=\dfrac{gR^{2}m}{R}=mgR\Rightarrow V_E^{2}=2gR.\]

Therefore \(V_E=\sqrt{2gR}\). With \(g=10\,\text{m s}^{-2}\):

\[V_E=\sqrt{2\times10\times R}=\sqrt{20R}.\]

Hence \(V_E=\sqrt{20R}\) as required.

Maelezo ya Majibu

Dimensional derivation of the speed of a wave on a stretched string.

Assume the speed v depends on the tension T and the mass per unit length \(\mu\):

\[v=k\,T^{a}\mu^{b}\]

where k is a dimensionless constant. Writing the dimensions:

• \([v]=LT^{-1}\)
• \([T]=MLT^{-2}\) (force)
• \([\mu]=ML^{-1}\) (mass per unit length)

So \(LT^{-1}=(MLT^{-2})^{a}(ML^{-1})^{b}=M^{a+b}L^{a-b}T^{-2a}\).

Equating powers:

• M: \(a+b=0\)
• L: \(a-b=1\)
• T: \(-2a=-1\Rightarrow a=\tfrac{1}{2}\)

From these, \(b=-\tfrac{1}{2}\). Therefore

\[v=k\,T^{1/2}\mu^{-1/2}=k\sqrt{\dfrac{T}{\mu}}.\]

Thus the speed of the wave is proportional to the square root of the tension divided by the mass per unit length.

Maelezo ya Majibu

Principle (Hooke's law). Within the elastic limit the applied force is directly proportional to the extension produced:

where \(K\) is the force constant of the elastic material and \(e\) is the extension.

Given data

• Original length, \(L = 3\text{ m}\)
• Force constant, \(K = 988.3\text{ N/m}\)
• Extension is three times the original length: \(e = 3 \times L = 3 \times 3 = 9\text{ m}\)

Calculation

The force required to produce the extension is \(8894.7\text{ N}\).

Maelezo ya Majibu

(a) The core of an optical fibre is best made of a transparent material of high refractive index, such as very pure (high-quality) glass or silica. It must be optically denser than the surrounding cladding so that light can be totally internally reflected within it.

(b) For signals to be confined to the core, the light must strike the core-cladding boundary at an angle of incidence greater than the critical angle, so that total internal reflection occurs at every point along the boundary. (This requires the core to be optically denser than the cladding.)