WAEC SSCE - Physics - 2012

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The device used to determine the relative density of an acid is a hydrometer. A hydrometer is an instrument used to measure the density or specific gravity of a liquid. It works based on Archimedes' principle, which states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. The hydrometer consists of a glass tube with a bulb at the bottom, and a scale to measure the density. When the hydrometer is placed in the acid, it floats and sinks to a certain level depending on the density of the acid. The density is then read off the scale at the level where the surface of the liquid intersects the scale on the hydrometer. Therefore, a hydrometer is used to determine the relative density of an acid.

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As the temperature of the water in the syringe increases, the water molecules transfer some of their thermal energy to the trapped air molecules. This causes the air molecules to move faster, collide more frequently, and exert more pressure on the walls of the syringe. The increase in pressure of the trapped air forces the piston to move upwards, against the external atmospheric pressure. Therefore, the correct statement explaining the observation is "molecules of trapped air expands more".

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The average speed is defined as the total distance traveled divided by the total time taken. Let's first calculate the total distance traveled by the bird. Distance traveled in the first 3 seconds = speed × time = 10 ms\(^{-1}\) × 3 s = 30 m Distance traveled in the next 3 seconds = speed × time = 15 ms\(^{-1}\) × 3 s = 45 m Distance traveled in the next 4 seconds = speed × time = 20 ms\(^{-1}\) × 4 s = 80 m Total distance traveled = 30 m + 45 m + 80 m = 155 m Now let's calculate the total time taken: Total time taken = 3 s + 3 s + 4 s = 10 s Therefore, the average speed of the bird is: Average speed = Total distance traveled / Total time taken = 155 m / 10 s = 15.5 ms\(^{-1}\) So the correct option is: - 15.5 ms\(^{-1}\)

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A thermopile is a device that is used to detect radiant energy. When radiant energy is absorbed by a thermopile, it generates a small voltage, which can be measured by a voltmeter. This is because the thermopile is made up of a series of thermocouples, which are devices that generate a voltage when there is a difference in temperature between their ends. When the thermopile absorbs radiant energy, it causes a temperature difference between the ends of the thermocouples, which in turn generates a voltage. Therefore, the correct option is: radiant energy.

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The magnetic relay is a device used for controlling another circuit carrying a larger current. It consists of an electromagnet, a spring-loaded armature, and a set of contacts. When a current passes through the coil of the electromagnet, it generates a magnetic field which attracts the armature and causes it to move, closing or opening the contacts. This can be used to switch on or off a circuit carrying a larger current, or to switch between different circuits. The magnetic relay is commonly used in applications such as industrial automation, power systems, and telecommunications.

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The SI unit of heat is the joule (J). Heat is a form of energy that is transferred from one body to another as a result of a difference in temperature. The amount of heat transferred is measured in joules, which is the amount of energy required to move an object with a force of one newton for a distance of one meter in the direction of the force. Therefore, the correct option is (a) joule. Kelvins (K) are used to measure temperature, watts (W) measure power and amperes (A) measure electric current.

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The radiation emitted in radioactive decay that has momentum, a fairly high penetrating power, and is deflected by a magnet is the beta particle. Beta particles are fast-moving electrons or positrons emitted from the nucleus of an atom during radioactive decay. They have a negative charge, a mass of about 1/1836th of a proton, and can penetrate several millimeters of matter. Beta particles can be deflected by a magnetic field, which is a characteristic of charged particles in motion. In contrast, alpha particles are heavy particles consisting of two protons and two neutrons, which have a high ionizing power but are not deflected by a magnetic field. Gamma radiation and X-radiation are both electromagnetic radiation and do not have charge or mass, so they are not deflected by a magnetic field.

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In an R-C circuit, a resistor R and a capacitor C are connected in series to an AC voltage source. When an AC voltage is applied to a capacitor in a circuit, it takes some time to fully charge, as the capacitor acts like an open circuit to DC, but passes AC through it. As a result, the current flowing through the circuit will lag the voltage by some angle. This angle is determined by the ratio of the resistance and capacitance in the circuit. Therefore, in a R-C circuit, V^rms leads I^rms by 90^o.

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The movement of fluid up or down a narrow tube is capillarity. Capillary action, also known as capillarity, is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity. This occurs because of the combination of two opposing forces: adhesion, the attraction between the liquid molecules and the inner surface of the tube, and cohesion, the attraction between the liquid molecules themselves. When the adhesion force is greater than the cohesion force, the liquid is pulled up the tube, and when the cohesion force is greater, the liquid is pushed down the tube. Capillarity is responsible for many natural phenomena, such as the ability of plants to transport water from their roots to their leaves, as well as some man-made ones, such as the operation of wicks in candles and the ink in fountain pens.

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The penetrating power of radiation is a measure of its ability to penetrate through matter. The more penetrating the radiation is, the farther it can travel through matter. In general, the penetrating power of radiation is inversely proportional to its ionizing power. - Alpha particles (\(\alpha\)): They are positively charged and have a large mass. They have a low penetrating power and are stopped by a sheet of paper or a few centimeters of air. - Beta particles (\(\beta\)): They are negatively charged and have a smaller mass compared to alpha particles. They have a higher penetrating power than alpha particles and can penetrate through a few millimeters of aluminum or several meters of air. - Gamma rays (\(\gamma\)): They are neutral and have no mass. They have the highest penetrating power and can penetrate through several centimeters of lead or many meters of concrete. Therefore, the correct order of decreasing penetrating power is \(\gamma, \beta, \alpha\). lists the radiations in this order.

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The work function of a metal is the minimum amount of energy required to remove an electron from the surface of the metal. When the metal is irradiated with light of energy greater than the work function, electrons are emitted from the metal surface. The excess energy of the light is used to give kinetic energy to the ejected electron. In this case, the metal has a work function of 3.0eV and is irradiated with ultraviolet light of energy 4.0eV. Since the energy of the light is greater than the work function, electrons will be ejected from the metal surface. The kinetic energy of the ejected electron can be found by subtracting the work function from the energy of the incident light: Kinetic energy of ejected electron = Energy of incident light - Work function = 4.0eV - 3.0eV = 1.0eV Therefore, the kinetic energy of the ejected electron is 1.0eV.

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The magnitude of the acceleration is given by the formula: acceleration = resultant force / mass In this question, the resultant force is given as 15N and the mass is 250g which is equivalent to 0.25kg. Therefore, acceleration = 15N / 0.25kg = 60.00ms^-2 Hence, the correct option is 60.00ms^-2.

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An atom in an excited state is one whose electrons have moved to higher energy levels. When an atom absorbs energy, its electrons move to higher energy levels, or orbitals, which are farther from the nucleus. When an electron moves from a lower energy level to a higher energy level, the atom is said to be in an excited state. Eventually, the electrons return to their original energy level by emitting energy in the form of electromagnetic radiation.

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When an object is dropped from rest, it falls under the influence of gravity. The distance it falls in time t is given by the formula: s = (1/2)gt² where g is the acceleration due to gravity. In this question, we are given t = 4s and g = 10 ms⁻². Substituting these values into the formula above, we get: s = (1/2) x 10 ms⁻² x (4 s)² = 80 m So, the height of the tower is 80 meters. Therefore, the correct answer is 80m.

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The tendency for a stationary body to continue to remain at rest when a force is applied to it is known as inertia. Inertia is a fundamental property of matter that resists changes in motion, whether it is a stationary object or an object in motion. It is often described as the resistance of an object to changes in its state of motion, and it is directly related to an object's mass. The greater the mass of an object, the greater its inertia and the more force it will take to move it from a stationary position. This property of inertia is the reason why a heavy object requires more force to move than a light one.

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The time of flight is the total time taken by a projectile to complete its motion before it returns to the same level from which it was projected. The given expression for time of flight involves the initial velocity of the projectile (U), the angle of projection (theta) and the acceleration due to gravity (g). Therefore, the expression for the time of flight of a projectile is: \(\frac{2U sin \theta}{g}\) Option (C) matches the given expression and is the correct answer.

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The core of a transformer is laminated in order to reduce eddy currents. Eddy currents are created by a magnetic field that is changing over time, inducing a current in the core. This current can cause energy loss and heating of the core, leading to inefficiencies in the transformer. Laminating the core creates many thin layers of material, each separated by a thin layer of insulation. This structure reduces the size of the current loops that can form and reduces the resistance of the current flow in the material. This reduction in eddy current also reduces energy loss, leading to greater efficiency of the transformer.

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The given problem is related to optics and involves the use of the mirror formula. According to the mirror formula, 1/f = 1/v + 1/u where, f = focal length of the mirror u = object distance v = image distance The magnification produced by the mirror is given by the ratio of the height of the image to the height of the object. magnification = height of image / height of object From the problem statement, height of object (h) = 2.5 cm distance of object from the mirror (u) = 20 cm focal length of the mirror (f) = 10 cm Using the mirror formula, we can find the image distance (v) as: 1/10 = 1/v + 1/20 => 1/v = 1/10 - 1/20 => v = 20 cm The image is formed at a distance of 20 cm from the mirror. Now, using the magnification formula, we can find the height of the image as: magnification = height of image / height of object => height of image = magnification * height of object The magnification produced by a convex mirror is always less than 1. Therefore, the image formed will be smaller than the object. For a convex mirror, magnification = -v/u Substituting the given values, we get: magnification = -20/20 = -1 Therefore, the height of the image is given by: height of image = magnification * height of object => height of image = -1 * 2.5 cm => height of image = -2.5 cm Since the height of the image cannot be negative, we take its absolute value: height of image = |-2.5 cm| = 2.5 cm Hence, the height of the image is 2.5 cm. Therefore, the correct option is (a) 2.5 cm.

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A telescope is said to be in normal adjustment when the objective focal point coincides with that of the eyepiece. This means that the light rays coming in from the object being observed converge to a point at the same location where the eyepiece is positioned, allowing the viewer to see a magnified image. If the objective focal point is too far away from the eyepiece or if the focal lengths of the objective and eyepiece are not matched, the viewer will not be able to see a clear image. Therefore, adjusting the telescope to achieve normal adjustment is crucial for obtaining a clear, magnified image of the object being observed.

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The formula for the inductive reactance is X_L = 2πfL, where X_L is the inductive reactance in ohms, f is the frequency in Hertz and L is the inductance in Henrys. We are given that the mains supply has a frequency of 50Hz, and that the current through the inductor is 1.5A. We also know that the voltage across the inductor is V = I X_L, where V is the voltage in volts. The voltage across the inductor is given as 24V, so we can use Ohm's law to find the inductive reactance: X_L = V / I = 24V / 1.5A = 16Ω We can now use the formula X_L = 2πfL to find the inductance: L = X_L / 2πf = 16Ω / (2π x 50Hz) = 0.051H ≈ 5.1 x 10^-2H Therefore, the inductance of the inductor is approximately 5.1 x 10^-2H. The correct option is (d).

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The most economical method of transmitting electrical power over long distances is alternating current at high voltage and low current. This is because the power loss in a transmission line is given by the formula P=I²R, where P is the power loss, I is the current and R is the resistance of the transmission line. By transmitting power at high voltage and low current, the current can be reduced, resulting in a lower power loss due to the lower value of I in the formula. This means that more power can be transmitted over longer distances with less loss, resulting in a more efficient and economical method of power transmission.

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The total capacitance of two capacitors connected in parallel is given by the sum of individual capacitances, which is 2μF + 2μF = 4μF. The charge on a capacitor is given by the product of its capacitance and the potential difference across it, which is Q = CV. Using the formula Q = CV and substituting the values, we get: Q = 4μF x 120V = 480μC Since there are two capacitors connected in parallel, the total charge is divided equally between them. Therefore, the charge on each capacitor is: Q/2 = 480μC/2 = 240μC So the answer is 2.4 x 10^-4C.

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When the metal ball falls through the column of glycerine, it experiences both weight and upthrust, which opposes the motion of the ball. Eventually, the forces will be balanced and the velocity of the ball becomes constant, which is called terminal velocity. At this point, the forces on the ball are in equilibrium, which means that the weight of the ball (W) is equal to the upthrust (U) plus the viscous drag force (V). Therefore, the correct answer is: W = U + V Option (B)

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X-rays are produced when a metal target is used to block fast moving electrons. When high-speed electrons are fired at a metal target, they collide with the atoms in the target and produce X-rays. X-rays have short wavelengths and are highly penetrating, which makes them useful in medical imaging and materials analysis. Therefore, the correct statement among the options is "are produced when a metal target is used to block fast moving electrons."

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Polarization is the property of waves that is exclusive to transverse waves. Polarization refers to the orientation of the oscillations of the wave, which can be either perpendicular or parallel to the direction of wave propagation. In a transverse wave, the oscillations are always perpendicular to the direction of propagation. This means that transverse waves can be polarized, while longitudinal waves cannot. Reflection, interference, and diffraction can occur in both transverse and longitudinal waves.

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For a diverging lens, the image formed is always virtual, upright and smaller than the object. When the object is placed between the focal point and the optical center of the lens, the image formed will always be virtual, upright and will appear on the same side of the object. Also, the image will be formed by the diverging lens itself, so it will be between the lens and the focal point F on the same side as the object. Therefore, the correct option is (B) X and C.

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The sound heard by a person after the reflection of the sound generated by him is called an "echo." An echo occurs when sound waves bounce off a hard surface and return to the listener's ear after a noticeable delay. The delay between the original sound and the reflected sound is called the "echo time." The quality of the echo is determined by the distance between the sound source and the reflecting surface, as well as the shape and size of the reflecting surface.

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The secondary colors of light are the colors that result from mixing two primary colors of light. The three primary colors of light are red, green, and blue. Option i consists of two primary colors of light (red and blue) and a secondary color of light (green). Therefore, it is not a set of secondary colors of light. Option ii consists of one primary color of light (blue) and two secondary colors of light (cyan and magenta). Therefore, it is a set of secondary colors of light. Option iii consists of two secondary colors of light (magenta and yellow) and a primary color of light (green). Therefore, it is not a set of secondary colors of light. Option iv consists of three secondary colors of light (cyan, magenta, and yellow). Therefore, it is a set of secondary colors of light. Thus, the correct answer is option ii and iv only.

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The period of a simple pendulum is the time it takes for one complete oscillation. In this case, we are given that the pendulum makes 50 oscillations in one minute. To find the period, we can use the formula: period = time / number of oscillations We know that the time for one minute is 60 seconds. So, we have: period = 60 seconds / 50 oscillations period = 1.2 seconds Therefore, the period of oscillation is 1.20 seconds, which corresponds to.

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Pressure is defined as force per unit area. Mathematically, pressure (P) can be expressed as P = F/A, where F is force and A is the area. The unit of force is newton (N) and the unit of area is square meter (m²). Therefore, the unit of pressure is N/m². This can be further simplified as kg m/s² / m² which gives kg m^-1s^-2. Therefore, option (B) is the correct answer.

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(a) Surface tension

Surface tension is the force per unit length acting along (perpendicular to) a line drawn on the surface of a liquid, tending to make the surface behave like a stretched elastic skin and to contract to the smallest possible area. Its unit is the newton per metre (\( \text{N m}^{-1} \)).

(b) Two methods of reducing surface tension

• Raising the temperature of the liquid (surface tension falls as temperature rises).
• Adding a surface-active substance such as soap, detergent or oil (impurities) to the liquid.

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(a) Critical angle. The critical angle is the angle of incidence in the optically denser medium for which the corresponding angle of refraction in the less dense medium is exactly \(90^\circ\). For an angle of incidence greater than this value, total internal reflection occurs.

(b) Two factors that determine the deviation by a triangular prism.

• The angle of incidence of the ray on the first face.
• The refracting (apex) angle of the prism, and the refractive index of the glass (which itself depends on the colour/wavelength of the light).

(c) Least deviation. The ray is least deviated when it passes symmetrically through the prism, so the two internal angles are equal to the given angle of refraction: \(r_1=r_2=r=29.0^\circ\). The refracting angle is then

\[A=r_1+r_2=2(29.0^\circ)=58.0^\circ.\]

At the first face, using \(n=\dfrac{\sin i}{\sin r}\) with \(n=1.5\):

\[\sin i = n\sin r = 1.5\times\sin 29.0^\circ = 1.5\times0.4848 = 0.7272,\]

\[i=\sin^{-1}(0.7272)=46.7^\circ.\]

The angle of minimum (least) deviation is

\[D_{min}=2i-A = 2(46.7^\circ)-58.0^\circ = 93.4^\circ-58.0^\circ \approx 35.4^\circ.\]

(d) From the ray diagram the apex angle \(A=60^\circ\), the angle of incidence on the left face is \(45^\circ\), and the ray is refracted through \(30^\circ\) inside the glass.

(i) Angles P, Q and R.

\(P\) is the angle of refraction at the first face, read directly from the diagram: \(P=r_1=30^\circ\).

Since \(r_1+r_2=A\), the angle of incidence at the second face is \(R=r_2=A-r_1=60^\circ-30^\circ=30^\circ\).

\(Q\) is the angle between the two normals (drawn at P and R) where they meet inside the prism. In the triangle formed by the internal ray PR and the two normals, \[P+R+Q=180^\circ,\] \[Q=180^\circ-30^\circ-30^\circ=120^\circ.\]

So \(P=30^\circ,\; Q=120^\circ,\; R=30^\circ.\)

(ii) Refractive index n. At the first face,

\[n=\frac{\sin 45^\circ}{\sin 30^\circ}=\frac{0.7071}{0.5000}=1.41.\]

(iii) Angle of emergence e. At the second face the angle of incidence inside is \(r_2=30^\circ\), so

\[\sin e = n\sin r_2 = 1.41\times\sin 30^\circ = 1.41\times0.5 = 0.707,\]

\[e=\sin^{-1}(0.707)=45^\circ.\]

(iv) Total deviation D.

\[D=(i_1+e)-A=(45^\circ+45^\circ)-60^\circ=30^\circ.\]

Because \(i_1=e=45^\circ\) and \(r_1=r_2=30^\circ\), the ray passes symmetrically, so this \(30^\circ\) is in fact the minimum deviation.

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Relative density from spring extensions

By Hooke's law the extension of the spring is proportional to the force (weight) it supports.

In air the spring supports the full weight of the metal, so its extension is proportional to the weight:

\[ e_1 = 10.5\,\text{cm} \propto W \]

In water the spring supports the apparent weight (weight minus upthrust), so:

\[ e_2 = 6.8\,\text{cm} \propto (W - \text{upthrust}) \]

Therefore the extension due to the upthrust is proportional to

\[ e_1 - e_2 = 10.5 - 6.8 = 3.7\,\text{cm} \]

The upthrust equals the weight of water displaced, so

\[ \text{relative density} = \frac{\text{weight of metal in air}}{\text{weight of equal volume of water}} = \frac{e_1}{e_1 - e_2} \]

\[ = \frac{10.5}{3.7} = 2.84 \]

The relative density of the metal is about 2.84.

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(a) Wave-particle duality of light

Wave-particle duality means that light has a dual nature: it can behave both as a wave and as a stream of particles (photons). In some experiments light shows wave properties, while in others it shows particle properties. No single model fully describes all the behaviour of light; the two descriptions complement each other.

(b) Observable phenomena illustrating this

Wave nature is shown by:

• Interference (for example Young's double-slit fringes).
• Diffraction of light around edges and through narrow slits.
• Polarization of light.

Particle nature is shown by:

• The photoelectric effect, where light ejects electrons only if its frequency is high enough (energy comes in photon packets \( E = hf \)).
• The Compton effect, where an X-ray photon collides with an electron like a particle.

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Why an air conditioner is installed near the ceiling

An air conditioner cools the air passing through it. Cold air is denser (heavier) than warm air, so it tends to fall, while warm air is less dense and rises.

When the air conditioner is placed near the ceiling, the cold air it produces sinks downwards through the room, pushing the warmer air upwards towards the unit, where it is cooled in turn. This sets up a continuous convection current that circulates the air and cools the whole room evenly and quickly.

If it were fixed near the floor, the cold air produced would simply remain at floor level (having nowhere to fall to), the warm air would stay trapped near the ceiling, and there would be no effective circulation, so the room would be cooled poorly.

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(a) Principle of a fluorescent tube

A fluorescent tube works on the principle of fluorescence. An electric discharge is passed through mercury vapour in the tube, which emits ultraviolet (UV) radiation. This UV radiation strikes the fluorescent (phosphor) powder coating the inside wall of the tube, causing it to fluoresce and give out visible light.

(b) Two factors that determine the colour of the light produced

• The nature (chemical composition) of the fluorescent powder (phosphor) coating the inside of the tube.
• The type of gas or vapour used in the tube.

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None

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(a)(i) What is a machine? A machine is a device in which an effort applied at one point is used to overcome a load at another point, thereby making work easier to do.

(a)(ii) Two uses of gears:

• To change the speed of rotation (and to change the direction of motion).
• To magnify force, i.e. to increase the turning effect (torque) transmitted from one shaft to another.

(a)(iii) Velocity ratio of a pair of gear wheels. It is the ratio of the number of teeth on the driven wheel to the number of teeth on the driving wheel:

(a)(iv) How the mechanical advantage is increased. Make the driving gear smaller, with fewer teeth than the driven gear; this raises the velocity ratio and hence the mechanical advantage.

(b)(i) Velocity ratio of the bicycle gear system. The driven gear has 12 teeth and the driving gear has 18 teeth, so:

(b)(ii) Effort to overcome a load of 70 N at 90% efficiency.

Since \(M.A. = \dfrac{\text{Load}}{\text{Effort}}\):

(b)(iii) Why the calculated effort is less than the actual effort required. The calculation ignores the extra force lost to friction between the chain and the gears and between the tyres and the load; in practice this friction means a larger effort is actually needed.

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(a) Magnetic line of force

A magnetic line of force is a line (curve) drawn in a magnetic field such that the tangent to it at any point gives the direction of the magnetic field at that point; equivalently, it is the path along which a free (isolated) north pole would move if placed in the field.

(b) Force on the current-carrying wire

Data: \( L = 10\,\text{cm} = 0.10\,\text{m} \), \( I = 4.0\,\text{A} \), \( B = 2.0\,\text{T} \). The force is \( F = BIL\sin\theta \).

(i) Wire parallel to the field (\( \theta = 0^\circ \))

\[ F = BIL\sin 0^\circ = 0\,\text{N} \]

(ii) Maximum force (\( \theta = 90^\circ \))

\[ F = BIL = 2.0 \times 4.0 \times 0.10 = 0.80\,\text{N} \]

(iii) Wire at \( 60^\circ \) to the field

\[ F = BIL\sin 60^\circ = 0.80 \times 0.866 = 0.69\,\text{N} \]

(c) Use of keepers

Bar magnets are stored in pairs, laid side by side with the north pole of one next to the south pole of the other, and short bars of soft iron (keepers) are placed across the two ends. The keepers become magnetized by induction, and together with the magnets they form a closed loop of magnetic flux. This keeps the molecular magnets (domains) aligned and prevents self-demagnetization, so the magnets retain their strength.

(d) True bearing

The compass reads N 10° W relative to magnetic north, and magnetic north itself lies N 15° W of true north (angle of declination). Since both deviations are to the west, they add:

\[ 10^\circ + 15^\circ = 25^\circ \text{ west of true north} \]

The true bearing of the place is N 25° W.

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(a) Principle of conservation of linear momentum

In a closed (isolated) system on which no net external force acts, the total linear momentum of the bodies before an interaction (such as a collision) is equal to the total linear momentum after the interaction.

(b) Mode of action of a propelled rocket

A rocket burns fuel and expels a large mass of hot exhaust gases backwards at very high speed. By the conservation of momentum, the backward momentum given to the gases is balanced by an equal and opposite forward momentum gained by the rocket, so the rocket is driven forward. (This is also Newton's third law: action and reaction.)

(c) Two reasons the to-and-fro motion of the ball is not simple harmonic

• There is no restoring force that is proportional to, and directed opposite to, the displacement from a fixed equilibrium position.
• The motion is not periodic about a fixed point with a fixed period; the ball travels between the players at roughly constant speed rather than with sinusoidally varying speed.

(d)(i) Yes, both balls hit the ground at the same time.

(d)(ii) The vertical motion is independent of the horizontal motion. Both balls start with the same vertical velocity (zero) and fall under the same acceleration due to gravity g through the same height, so \( h = \tfrac{1}{2}g t^2 \) gives the same time of fall for each, regardless of the horizontal projection.

(e) Take the initial direction of motion as positive; the ball rebounds in the opposite direction.

(i) Change in momentum

\[ \Delta p = m(v - u) = 0.10 \times (-13 - 17) = 0.10 \times (-30) = -3.0\,\text{kg m s}^{-1} \]

The change in momentum is 3.0 kg m s\(^{-1}\) (directed away from the wall).

(ii) Average force

\[ F = \frac{\Delta p}{t} = \frac{3.0}{0.15} = 20\,\text{N} \]

The average force exerted on the ball is 20 N (directed away from the wall).

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None

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(a) Showing that \(\mu = \tan\theta\)

The diagram shows a body of mass m on a plane inclined at angle \(\theta\). The forces on it are the weight \(Mg\) acting vertically downward, the normal reaction \(R\) perpendicular to the plane, and the frictional force \(F_p\) acting up the plane (opposing the tendency to slide down).

Resolve the weight into components parallel and perpendicular to the plane:

• Component along the plane (down the slope): \(Mg\sin\theta\)
• Component perpendicular to the plane: \(Mg\cos\theta\)

Perpendicular to the plane there is no motion, so

When the body is just on the point of sliding (or slides down at constant velocity), the frictional force is limiting and balances the component of weight down the plane:

But the limiting friction is \(F_p = \mu R\). Therefore

Substituting \(R = Mg\cos\theta\):

Hence the coefficient of friction equals \(\tan\theta\), where \(\theta\) is the angle of repose.

Relative density of the metal

Because the spring obeys Hooke's law, the extension is proportional to the load (force) on it.

Weight of metal in air \(\propto\) extension in air \(= 10.5\,\text{cm}\).

Apparent weight in water \(\propto\) extension in water \(= 6.8\,\text{cm}\).

Upthrust = loss in weight \(\propto (10.5 - 6.8) = 3.7\,\text{cm}\).

The upthrust equals the weight of water displaced, so

Maelezo ya Majibu

(a) Electrodes in electric cells

Electrodes are the two conducting plates or rods through which electric current enters and leaves the electrolyte in a cell. The electrode by which conventional current enters the electrolyte is the anode (positive), and the one by which it leaves is the cathode (negative).

(b) Calculating the current

By Faraday's first law of electrolysis, the mass deposited is \( m = Z I t \), where Z is the electrochemical equivalent, I the current and t the time.

Given: \( m = 200\,\text{g} \), \( Z = 8.33 \times 10^{-4}\,\text{g C}^{-1} \), \( t = 2\,\text{min} = 120\,\text{s} \).

\[ I = \frac{m}{Z t} = \frac{200}{(8.33 \times 10^{-4}) \times 120} \]

\[ I = \frac{200}{0.09996} \approx 2.0 \times 10^{3}\,\text{A} \]

The current passed is about 2000 A.

Maelezo ya Majibu

Finding the angle of incidence \( \theta \)

The reflected ray makes an angle \( \theta \) with the normal (equal to the angle of incidence), and the refracted ray makes an angle \( r \) with the normal on the other side. If the angle between the reflected ray and the refracted ray is \( 90^\circ \), then, measuring from the normal on each side,

\[ \theta + 90^\circ + r = 180^\circ \quad\Rightarrow\quad \theta + r = 90^\circ \quad\Rightarrow\quad r = 90^\circ - \theta \]

Applying Snell's law at the air-glass boundary (\( n = 1.50 \)):

\[ \sin\theta = n\sin r = n\sin(90^\circ - \theta) = n\cos\theta \]

\[ \frac{\sin\theta}{\cos\theta} = n \quad\Rightarrow\quad \tan\theta = 1.50 \]

\[ \theta = \tan^{-1}(1.50) = 56.3^\circ \]

The angle of incidence is \( \theta \approx 56.3^\circ \). (This is the polarizing, or Brewster, angle, for which \( \tan\theta = n \).)

Maelezo ya Majibu

Why sound waves cannot be plane polarized

Sound waves are longitudinal waves: the particles of the medium vibrate along the same direction in which the wave travels (backwards and forwards, forming compressions and rarefactions).

Polarization is the restriction of the vibrations of a wave to a single plane. This is only possible when the vibrations are perpendicular (transverse) to the direction of travel, because only then are there sideways vibrations that can be confined to one plane.

Since the vibrations of a sound wave are parallel to the direction of propagation and are the same in every plane containing that direction, there is no sideways component that can be selected out. Therefore a sound wave cannot be plane polarized. Only transverse waves (such as light) can be polarized.