- This paper consists of two sections A and B.
- Answer nine (9) questions, choosing ALL questions from sections A and only two (2) questions from section B.
- Mathematical tables and non-programmable calculators may be used.
- Any unauthorized materials are not allowed in the examination room.
- Write your examination number on your answer booklet(s) on every page.
- Thermal conductivity of plaster = \(0.60 \text{W} m^{-1} k^{-1}\)
- Thermal conductivity of fiberglass = \(0.040 \text{W} m^{-1} k^{-1}\)
- Acceleration due to gravity, \(g = 9.8 \text{m/s}^2\)
- The radius of the earth = \(6.38 \times 10^6 \text{m}\)
- Mass of the earth = \(5.98 \times 10^{24} kg\)
- Universal gravitational constant = \(6.67 \times 10^{-11} \text{N} m^2 kg^{-2}\)
- Heat of vaporization of water = \(2256 \times 10^3 \text{J/kg}\)
- 1 atm = \(1.01 \times 10^5 Pa\)
- Pie, \(\pi = 3.14\)
- Density of air = \(1.3 \, \text{kg/m}^3\)
- Electronic charge, e = \(1.6 \times 10^{-19} \, \text{C}\)
- Thermal conductivity of ice = \(2.3 \, \text{Wm}^{-1} \text{K}^{-1}\)
- Molar gas constant is \(8.31 \, \text{Jmol}^{-1} \text{K}^{-1}\)
Answer ALL questions from this section
Define the term dimension of a physical quantity.
Show that the expression \( V = V_0 + at \) is dimensionally correct, where \( V \) and \( V_0 \) represent velocities, 'a' is acceleration and t is a time interval.
What are the basic rules of dimensional analysis?
A gas bubble from an explosion under water oscillates with period T proportional to \( P^a d^b E^c \) where 'P' is the static pressure, 'd' is the density of water and 'E' is the total energy of the explosion. Find the values of a, b and c.
What is meant by Absolute Error?
The period of oscillation of a simple pendulum in an experiment is recorded as 2.63s, 2.56s, 2.42s, 2.71s and 2.80s respectively. Find the mean time period, absolute error in each observation and percentage error.
Explain why a horse cannot pull a cart and run in empty space?
A man of 90kg jumps into a swimming pool from a height of 5m. If it takes 0.4 seconds for the water in a pool to reduce its velocity to zero, what average force did the water exert on the man?
An object at rest explodes into three pieces of equal masses. One moves east at 20m/s, a second moves south-east at 30m/s. What is the velocity of the third piece?
A block is released from rest from a height of 1m on a 45° rough inclined surface. It comes to rest after covering 1.8m on the horizontal part of the surface. Find the coefficient of friction between the block and the surface.
What do you understand by the following terms:
i. Oscillation
ii. Simple harmonic motion
If the displacement of N is given by \( x = r \cos \theta \) and \( \theta = \omega t \), Sketch the graph of the following components:
(i) Showing the variation of the displacement of N with time
(ii) Showing the variation of the velocity of N with time
(iii) Showing the variation of the acceleration of N with time
(iv) From the graphs, obtain the phase difference present between the velocity and the acceleration
A simple pendulum has a period of 2.0s and an amplitude of swing 5.0cm. Calculate the maximum magnitudes of:
(i) The velocity of the bob
(ii) The acceleration of the bob
Define the term angular momentum.
A shaft rotating at \(3.0 \times 10^3\) revolutions per minute is transmitting a power of 10 Kilowatts. Find the magnitude of driving couple.
State the Newton Law of universal gravitation.
A satellite of mass M moves in circular orbit about the Earth with a constant speed of V and a height of \(h = 1000km\) above the Earth's surface. Find the orbital speed of satellite.
A rocket is launched vertically upward from the surface of the earth with an initial velocity of \(V_0\). Show that its velocity V at a height h is given by:
where R is the radius of the earth and g is the acceleration due to gravity at the earth's surface.
Briefly explain why there is no unique value of moment of inertia of a given body.
If polar ice caps melts and spread uniformly, how will the length of the day be affected?
A mass of 0.3kg hangs from the rim of a wheel of radius 0.1cm by a light string which is wound round the wheel. When released from rest, the mass falls through a distance of 5m in 10 seconds rotating the wheel. Find the moment of inertia of the wheel.
With what velocity must a body be thrown upwards from the surface of the earth so that it reaches a height of 10R (R is the radius of the earth)?
A satellite in stable orbit contains two closed vessels one of these is filled with water while the other is filled with hot steam. Explain why the water exerts very little pressure on its container but steam exerts almost the same pressure as it would on earth when at the same temperature.
What is meant by adiabatic process?
Are melting and boiling isothermal changes?
Let 1.0kg of liquid water of \(100^\circ C\) be converted to steam at \(100^\circ C\) by boiling at standard atmospheric pressure. The volume of that water changes from an initial value of \(1.0 \times 10^{-3} m^3\) as a liquid to \(1.671 m^3\) as steam.
(i) How much work is done by the system during this process?
(ii) How much energy is transferred as heat during the process?
Explain why a gas has two specific heats where as a liquid and solid have only one.
Use the first law of thermodynamics, explain why the specific latent heat of vaporisation is greater than the specific latent heat of fusion for a particular substance.
Why do two layers of cloth of equal thickness provide warmer covering than a single layer of cloth of double the thickness?
Explain why, the cooling unit of a refrigerator is fitted near the top.
One face of a sheet of cork 3mm thick is placed in contact with a glass sheet 5mm thick, both sheets being 20cm square. The outer face of this square composite sheet is maintained at \(100^\circ C\) and \(20^\circ C\), the cork being at higher temperature. Find the temperature of Glass-cork interface. Given, the thermal conductivity of cork = \(6.3 \times 10^{-2} Wm^{-1} K^{-1}\) and thermal conductivity of glass = \(7.2 \times 10^{-1} Wm^{-1} K^{-1}\).
What is meant by aerial environment? Give two examples.
Describe three ways at which the aerial environment is threatened.
Describe the importance of shading, mulching and sheltering in agriculture.
Describe the four major components of soil.
Explain the formation of an earth quake using the elastic rebound theory.
Answer any two (02) questions from this section
Explain why, when a current is taken from a battery, the reading given by a high resistance voltmeter connected across the battery terminals decreases as current taken increases.
Under what condition is the heat produced in an electric circuit inversely proportional to resistance?
A storage battery of electromotive force of 8.0V and internal resistance 0.5Ω is being charged by a 120V d.c supply using a series resistor of 15.5Ω.
What is the terminal voltage of the battery during charging?
What is the purpose of having a series resistor in the charging circuit?
Which is more dangerous between an electric shock suffered from an a.c source or a d.c source rated at the same voltage? Give two reasons to support your answer.
Define the term OPAMP stating its three important characteristics.
Give three advantages of negative feedback to an OPAMP.
Show that, the closed loop gain of the non-inverting amplifier with negative feedback is given by
where each letter carries its usual meaning.
Calculate output voltage in the following OPAMP circuit.
12V
10
3
10kΩ
Vout
Why are integrated circuits more reliable?
Explain how OR gate is realised using NAND gate.
The Attorney general is to elected by the majority voting system for three judges A, B and C. Each judge presses a switch for a Yes vote causing the respective input to rise to logic state 1. However, the seniority judge is to be considered, whereby the most senior judge C has weighted vote equivalent to four votes of the junior judge A. The senior judge B's vote is taken to be three votes of the junior judge A. If the majority YES vote will be passed only when the vote, in favour, exceeds half of the total points, (assume the junior judge casts a vote of 1 point equivalent) then;
(i) Determine the truth table defining the variables,
(ii) Deduce the simplified Boolean expression for the problem.
(iii) Design the circuit including logic gates.
Distinguish between "Frequency Modulated" and "Amplitude Modulated" radio waves.
Explain why FM radio broadcasting is currently more preferred than AM radio broadcasting.
Instructions
- This paper consists of two sections A and B.
- Answer nine (9) questions, choosing ALL questions from sections A and only two (2) questions from section B.
- Mathematical tables and non-programmable calculators may be used.
- Any unauthorized materials are not allowed in the examination room.
- Write your examination number on your answer booklet(s) on every page.
Useful Information
- Thermal conductivity of plaster = \(0.60 \text{W} m^{-1} k^{-1}\)
- Thermal conductivity of fiberglass = \(0.040 \text{W} m^{-1} k^{-1}\)
- Acceleration due to gravity, \(g = 9.8 \text{m/s}^2\)
- The radius of the earth = \(6.38 \times 10^6 \text{m}\)
- Mass of the earth = \(5.98 \times 10^{24} kg\)
- Universal gravitational constant = \(6.67 \times 10^{-11} \text{N} m^2 kg^{-2}\)
- Heat of vaporization of water = \(2256 \times 10^3 \text{J/kg}\)
- 1 atm = \(1.01 \times 10^5 Pa\)
- Pie, \(\pi = 3.14\)
- Density of air = \(1.3 \, \text{kg/m}^3\)
- Electronic charge, e = \(1.6 \times 10^{-19} \, \text{C}\)
- Thermal conductivity of ice = \(2.3 \, \text{Wm}^{-1} \text{K}^{-1}\)
- Molar gas constant is \(8.31 \, \text{Jmol}^{-1} \text{K}^{-1}\)
SECTION A (70 Marks)
Answer ALL questions from this section
Define the term dimension of a physical quantity.
Show that the expression \( V = V_0 + at \) is dimensionally correct, where \( V \) and \( V_0 \) represent velocities, 'a' is acceleration and t is a time interval.
What are the basic rules of dimensional analysis?
A gas bubble from an explosion under water oscillates with period T proportional to \( P^a d^b E^c \) where 'P' is the static pressure, 'd' is the density of water and 'E' is the total energy of the explosion. Find the values of a, b and c.
What is meant by Absolute Error?
The period of oscillation of a simple pendulum in an experiment is recorded as 2.63s, 2.56s, 2.42s, 2.71s and 2.80s respectively. Find the mean time period, absolute error in each observation and percentage error.
Explain why a horse cannot pull a cart and run in empty space?
A man of 90kg jumps into a swimming pool from a height of 5m. If it takes 0.4 seconds for the water in a pool to reduce its velocity to zero, what average force did the water exert on the man?
An object at rest explodes into three pieces of equal masses. One moves east at 20m/s, a second moves south-east at 30m/s. What is the velocity of the third piece?
A block is released from rest from a height of 1m on a 45° rough inclined surface. It comes to rest after covering 1.8m on the horizontal part of the surface. Find the coefficient of friction between the block and the surface.
Comprehensive Answers
1. (a) (i) Definition of dimension of a physical quantity:
The dimension of a physical quantity is the expression that shows how it is related to the fundamental quantities (mass, length, time, etc.). Dimensions represent the nature of a physical quantity and are independent of its magnitude. For example, the dimension of velocity is [LT⁻¹], which means it is derived from length divided by time.
1. (a) (ii) Dimensional correctness of \( V = V_0 + at \):
To check dimensional correctness, we analyze the dimensions of each term:
Left-hand side: V (velocity) has dimensions [LT⁻¹]
Right-hand side:
- V₀ (velocity) has dimensions [LT⁻¹]
- a (acceleration) has dimensions [LT⁻²]
- t (time) has dimensions [T]
- Therefore, at has dimensions [LT⁻²] × [T] = [LT⁻¹]
Since all terms have the same dimensions [LT⁻¹], the equation is dimensionally correct.
1. (b) (i) Basic rules of dimensional analysis:
- Principle of homogeneity: All terms in a physical equation must have the same dimensions.
- Dimensionless quantities: Quantities like angles, trigonometric functions, and exponential functions are dimensionless.
- Addition and subtraction: Only quantities with the same dimensions can be added or subtracted.
- Multiplication and division: Dimensions are multiplied or divided when quantities are multiplied or divided.
1. (b) (ii) Finding a, b, and c for T ∝ PᵃdᵇEᶜ:
Given: T ∝ PᵃdᵇEᶜ
Dimensions:
- T (time) = [T]
- P (pressure) = [ML⁻¹T⁻²]
- d (density) = [ML⁻³]
- E (energy) = [ML²T⁻²]
Equating dimensions:
Equating powers:
- For M: a + b + c = 0
- For L: -a - 3b + 2c = 0
- For T: -2a - 2c = 1
Solving these equations:
From T: -2a - 2c = 1 ⇒ a + c = -½
From M: b = -a - c = ½
From L: -a - 3(½) + 2c = 0 ⇒ -a + 2c = 1.5
Solving: a = -5/6, b = ½, c = 1/3
Therefore: a = -5/6, b = ½, c = 1/3
1. (c) (i) Absolute Error:
Absolute error is the magnitude of the difference between the measured value and the true value of a quantity. It represents the actual error in a measurement without considering the size of the quantity being measured. For example, if the true length is 10.0 cm and you measure it as 10.2 cm, the absolute error is 0.2 cm.
1. (c) (ii) Calculations for pendulum experiment:
Observations: 2.63s, 2.56s, 2.42s, 2.71s, 2.80s
Mean time period:
Absolute errors:
- ΔT₁ = |2.63 - 2.624| = 0.006 s
- ΔT₂ = |2.56 - 2.624| = 0.064 s
- ΔT₃ = |2.42 - 2.624| = 0.204 s
- ΔT₄ = |2.71 - 2.624| = 0.086 s
- ΔT₅ = |2.80 - 2.624| = 0.176 s
Mean absolute error:
Percentage error:
2. (a) (i) Why a horse cannot pull a cart in empty space:
A horse cannot pull a cart and run in empty space due to the absence of friction. According to Newton's third law of motion, when the horse pulls the cart, it exerts a force on the ground backward. The ground, in turn, exerts an equal and opposite forward force (friction) on the horse's hooves, which enables the horse to move forward and pull the cart.
In empty space, there is no medium (like ground) to provide this reaction force. Without this external force to push against, the horse and cart system would not be able to accelerate or move in any particular direction, regardless of how hard the horse pulls.
2. (a) (ii) Average force exerted by water on the man:
Given:
- Mass of man, m = 90 kg
- Height, h = 5 m
- Time to stop, t = 0.4 s
Velocity just before hitting water:
Deceleration in water:
Average force exerted by water:
The average force exerted by water on the man is 2227.5 N.
2. (b) (i) Velocity of the third piece after explosion:
Let the mass of each piece be m.
Initial momentum = 0 (object at rest)
Let's define directions:
- East: +x direction
- North: +y direction
First piece: v₁ = 20 m/s east ⇒ p₁ = (20m, 0)
Second piece: v₂ = 30 m/s southeast (45° south of east)
Components: v₂â‚“ = 30 cos 45° = 21.21 m/s
v₂ᵧ = -30 sin 45° = -21.21 m/s (negative for south)
p₂ = (21.21m, -21.21m)
By conservation of momentum:
p₁ + p₂ + p₃ = 0
p₃ = -p₁ - p₂ = (-20m - 21.21m, 0 + 21.21m) = (-41.21m, 21.21m)
Velocity of third piece:
v₃â‚“ = -41.21 m/s, v₃ᵧ = 21.21 m/s
Magnitude: v₃ = √(41.21² + 21.21²) = √(1698.3 + 449.9) = √2148.2 = 46.35 m/s
Direction: θ = tan⁻¹(21.21 / 41.21) = tan⁻¹(0.5146) = 27.2° north of west
The third piece moves at 46.35 m/s at 27.2° north of west.
2. (b) (ii) Coefficient of friction:
Given:
- Height on incline, h = 1 m
- Angle of incline, θ = 45°
- Distance on horizontal, s = 1.8 m
Velocity at bottom of incline:
Using conservation of energy:
On horizontal surface:
Initial velocity, u = 4.43 m/s
Final velocity, v = 0
Distance, s = 1.8 m
Using v² = u² + 2as:
Friction force:
Also, F = μN = μmg
The coefficient of friction is 0.556.
3. (a) Definitions:
i. Oscillation: Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. It refers to any motion that repeats itself in equal intervals of time.
ii. Simple Harmonic Motion (SHM): Simple Harmonic Motion is a special type of oscillatory motion in which the restoring force is directly proportional to the displacement from the mean position and acts in the direction opposite to that of displacement. The motion is sinusoidal in time and demonstrates a single resonant frequency.
3. (b) Graphs for SHM:
Given: x = r cos θ, θ = ωt ⇒ x = r cos(ωt)
(i) Displacement vs time: A cosine curve starting at maximum amplitude r at t=0, oscillating between +r and -r.
(ii) Velocity vs time: v = dx/dt = -rω sin(ωt). A sine curve phase-shifted by 90° from displacement.
(iii) Acceleration vs time: a = dv/dt = -rω² cos(ωt). A cosine curve phase-shifted by 180° from displacement.
(iv) Phase difference: Velocity leads displacement by 90° (Ï€/2 radians). Acceleration leads velocity by 90° (Ï€/2 radians). Therefore, acceleration and velocity have a phase difference of 90°.
3. (c) Simple pendulum calculations:
Given: T = 2.0 s, A = 5.0 cm = 0.05 m
Angular frequency:
(i) Maximum velocity:
(ii) Maximum acceleration:
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