WAEC SSCE - Physics - 2020 (Essay)

Answer Details

(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.

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

Answer Details

(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.)

Answer Details

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.

Answer Details

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).

Answer Details

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.

Answer Details

(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 \(763.8\ \text{rad s}^{-1}\)).

Answer Details

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.

Answer Details

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.

Answer Details

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 \(8894.7\text{ N}\).

Answer Details

(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).

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

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

Answer Details

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.