Instructions
- This paper consists of section A and B with total of ten (10) questions.
- Answer all questions in section A and any two (2) questions from section B.
- Mathematical tables and non-programmable calculator may be used.
- Cellular phones and unauthorized materials are not allowed in the examination room.
- Write your examination number on every page of your answer booklet(s).
The following information may be useful:
- Mass of mars \( (m_m) = 6.4 \times 10^{22} kg \)
- Acceleration due to gravity \( (g) = 9.8 m/s^2 \)
- Universal gravitational constant \( (G) = 6.67 \times 10^{-11} Nm^2 kg^{-2} \)
- Radius of mass \( (R_m) = 3.375 \times 10^6 m \)
- Thermal conductivity of brass \( = 109 W m^{-1} C^{-1} \)
- Heat of vaporization of water \( = 2256 \times 10^3 Jkg^{-1} \)
- Stefan Boltzmann constant \( = 5.67 \times 10^{-8} Wm^{-2} K^{-4} \)
- Average radius of the sun \( = 7.0 \times 10^5 km \)
- Molar gas constant \( (R) = 8.31 Jmol^{-1} K^{-1} \)
- Atomic weight of silver \( = 108 \)
- Density of silver \( = 1.05 \times 10^4 kg/m^3 \)
(a) (i) What type of quantity is Avogadro's number? (01 mark)
(ii) Name two quantities each whose dimensions are \( ML^{-1}T^{-2} \) and \( ML^{2}T^{-1} \). (02 marks)
(b) Consider the following equation of Bernoulli's theorem.
Using the concept of dimension suggest the physical quantity given by \( k / \rho \). (03 marks)
(c) (i) State the causes of gross errors. (02 marks)
(ii) The heat generated in a circuit depend upon the current, resistance and time for which current flows. If the errors in measuring the above are 2%, 1% and 1% respectively, find the maximum error in measuring heat. (02 marks)
(a) (i) How does projectile motion differ from uniform circular motion? (02 marks)
(ii) What is the nature of projectile motion. (01 mark)
(b) From the top of building 45m high, a stone is thrown at an angle of 30° to the horizontal with an initial velocity of 20m/s. Find
(i) The time of flight (1.5 marks)
(ii) The distance from the foot of the building where it strikes the ground. (2 marks)
(c) You may have seen in circus, a motor cyclist driving in vertical loops inside a "death-well" (a hollow spherical chamber with holes so the spectators can watch from outside). Explain clearly why the motor cyclist does not drop down when he is at the upper most point of death well with a no support from below. What is the minimum speed required at the upper most position to perform a vertical loop if the radius of the chamber is 25m. (3.5 marks)
(a) (i) Is there any work done by the car if it moves with a uniform speed on a smooth level road. (01 mark)
(ii) Can linear momentum of a system be changed without changing its kinetic energy? (01 mark)
(b) (i) The momentum of a body is increased by 100%. What is the percentage increase in its kinetic energy? (02 marks)
(ii) A ball of mass "m" moving with a speed of 2m/s collides with a stationary ball of the same mass in a elastic collision. The incident ball is scattered at an angle of 20° from its original direction. Find the final speed of each ball and the direction of the struck ball with the original direction of the striking ball. (03 marks)
(c) (i) Write the three (3) uses of earth's satellite. (1.5 marks)
(ii) Calculate the gravitational intensity on the surface of mass. (1.5 marks)
(a) Give the qualitative meaning of the term moment of inertia. (01 mark)
(b) A fly wheel with axle 1.0cm in diameter is mounted on frictionless bearings and set in motion by applying steady tension of 3.0N to a thin thread wound tightly round the axle.
The moment of inertia of the system about its axis of rotation is \( 5.0 \times 10^{-4} \, \text{kgm}^2 \). Calculate.
(i) The angular acceleration of the flywheel when 1.0m of the thread has been pulled off the axle; and (02 marks)
(ii) The constant retarding couple which must be applied to bring the flywheel to rest in one complete turn when tension in the thread having been removed. (03 marks)
(c) If \( y = 8 \cos \omega t + 7.5 \sin \omega t \), show that it represents S.H.M and obtain its amplitude. (04 marks)
(a) A copper – constant thermocouple with its cold junction at 0°C had an e.m.f of 4.28mV when its other hot junction was at 100°C. The e.m.f became 9.2mV when the temperature of hot junction was 200°C. If the e.m.f is related to the temperature difference \(\theta\) between the hot and cold junction by the equation.
Calculate
(i) The value of A and B; and (03 marks)
(ii) The temperature for which "E" may be assumed to be proportional to \(\theta\) without incurring an error of more than 1% (02 marks)
(b)(i) Briefly explain how you can determine the thermal conductivity of poor conductors by lee's disc. (02 marks)
(ii) A brass boiler has a base area of 0.15m\(^2\) and thickness of 1cm. It boils water at the rate of 6 kilogram per minute when placed on a gas stove. What is the temperature of the part of the flame in contact with the boiler? (03 marks)
(a) (i) Define the term solar constant. (01 mark)
(ii) The energy arriving per unit area on the Earth's surface per second from the sun is \( 1.34 \times 10^3 \, \text{W/m}^{-2} \). The average distance from the Earth to the sun is 215 times the length of the sun's radius. Given that, both the Earth and the sun are black bodies, estimate the temperature of the sun. (03 marks)
(b) The tungsten filament of an electric lamp has a length of 0.5m and a diameter of \( 6 \times 10^{-2} \, \text{m} \). The power rating of the lamp is 60w. Assuming the radiation from the filament is equivalent to 80% that of a perfect black body radiator at the same temperature, estimate the steady temperature of the filament. (03 marks)
(c) Two moles of an ideal gas are compressed in a cylinder at a constant temperature of 65.0°C until the original pressure is triple.
(i) Sketch a pv diagram for this process. (01 mark)
(ii) Calculate the amount of work done. (02 marks)
(a) (i) A wire is stretched to double its length. Will its resistivity change? (01 mark)
(ii) Assume that there is one free electron per atom. Calculate the number of free electrons in a piece of silver of cross sectional area \( 1.5 \times 10^{-4} \, \text{m}^2 \) and length 2m. (03 marks)
(b) (i) Discuss the successive steps in the appearance of the discharged tube as the gas pressure within the tube is steadily reduced. (02 marks)
(ii) Sketch and discuss the current – voltage curve for electrical conduction in gases. (02 marks)
(c) Give two advantages and disadvantages of a potentiometer (02 marks)
(a) (i) With the help of a circuit diagram describe the working principle of transistor as an amplifier. List any applications of a common base amplifier. (2.5 marks)
(ii) Explain how a transistor in active state exhibits a low resistance at its emitter- base junction and high resistance at its base - collector junction. (2.5 marks)
(b) For a common emitter amplifier, current gain is equal to 50. If the emitter current is \(6.6mA\), calculate the collector and base current. Also calculate the current gain when the emitter is working as a common base amplifier. If the base current is \(100\mu A\) and collector current is \(3mA\) calculate the value of \(I_E\) and \(\beta\) (05 marks)
(c) Simplify the following expressions using law's and rules Boolean algebra- and implement the results a logic gate circuit.
(i) \(Q = x(y + z) + y(y + z) + xy\)
(ii) \(Q = (x + y)(x + y + z)\) (05 marks)
(iii) \(Q = xyz + xy\)
(a) (i) Define the term modulation and discuss different types of modulation. (03 marks)
(ii) Distinguish between amplitude modulation and frequency modulation. (02 marks)
(iii) Give advantages and disadvantages of AM over FM (02 marks)
(b)(i) What is a closed loop voltage gain as it is applied in op-amps? Explain its advantage. (02 marks)
(ii) What is the output wave form of an integrator circuit if the input waveform is a square? (02 marks)
(c) If light falls on LDR in the circuit below:- current flow through \(R_1\) and causes a voltage at the non- inverting input of the operational amplifier. Fig 3.93
What happens when this voltage equals the voltage set by the values of \(R_2\) and \(R_3\) at the inverting input? What happens to the output if \(R_1\) and LDR are interchanged? (04 marks)
(a)(i) What is meant by wind power? (01 mark)
(ii) Briefly explain how wind power is generated (02 marks)
(b)(i) Write down two sources of geothermal energy and tell what determines its conduction to the earth's surface. (04 marks)
(ii) What climatic factors influence the evaporation of water from the soil? (03 marks)
(c) (i) What is the difference between a seismograph and a seismometers. (02 marks)
(ii) At a recording station a difference in time of arrival between P- waves and S- waves was observed to be 1.5s. What is the approximate distance from the station at which the event occurred? Assume P- waves and S- wave's velocity to be 4km/s and 2km/s respectively. (03 marks)
Contact with MITIHANI POPOTE:
WhatsApp: +255 716 655 236
WhatsApp Channel: https://whatsapp.com/channel/0029VbA7XZkBvvsZxay5kl1l
Web: https://mitihanipopote.blogspot.com/
Instagram: https://www.instagram.com/higher_awaits
Facebook: https://www.facebook.com/profile.php?id=100064001654581
Instructions
- This paper consists of section A and B with total of ten (10) questions.
- Answer all questions in section A and any two (2) questions from section B.
- Mathematical tables and non-programmable calculator may be used.
- Cellular phones and unauthorized materials are not allowed in the examination room.
- Write your examination number on every page of your answer booklet(s).
The following information may be useful:
- Mass of mars \( (m_m) = 6.4 \times 10^{22} kg \)
- Acceleration due to gravity \( (g) = 9.8 m/s^2 \)
- Universal gravitational constant \( (G) = 6.67 \times 10^{-11} Nm^2 kg^{-2} \)
- Radius of mass \( (R_m) = 3.375 \times 10^6 m \)
- Thermal conductivity of brass \( = 109 W m^{-1} C^{-1} \)
- Heat of vaporization of water \( = 2256 \times 10^3 Jkg^{-1} \)
- Stefan Boltzmann constant \( = 5.67 \times 10^{-8} Wm^{-2} K^{-4} \)
- Average radius of the sun \( = 7.0 \times 10^5 km \)
- Molar gas constant \( (R) = 8.31 Jmol^{-1} K^{-1} \)
- Atomic weight of silver \( = 108 \)
- Density of silver \( = 1.05 \times 10^4 kg/m^3 \)
(a) (i) What type of quantity is Avogadro's number? (01 mark)
Avogadro's number is a dimensionless constant (pure number). It represents the number of atoms or molecules in one mole of a substance.
(ii) Name two quantities each whose dimensions are \( ML^{-1}T^{-2} \) and \( ML^{2}T^{-1} \). (02 marks)
For \( ML^{-1}T^{-2} \):
- Pressure
- Stress
- Young's modulus
For \( ML^{2}T^{-1} \):
- Angular momentum
- Planck's constant
- Action
(b) Consider the following equation of Bernoulli's theorem.
Using the concept of dimension suggest the physical quantity given by \( k / \rho \). (03 marks)
Dimensions of ρ = [ML⁻³]
Dimensions of ½ρv² = [ML⁻³][LT⁻¹]² = [ML⁻¹T⁻²]
Dimensions of ρgh = [ML⁻³][LT⁻²][L] = [ML⁻¹T⁻²]
Therefore, dimensions of K = [ML⁻¹T⁻²]
Dimensions of K/ρ = [ML⁻¹T⁻²]/[ML⁻³] = [L²T⁻²]
This corresponds to the dimensions of velocity squared (v²).
Therefore, K/ρ represents a quantity with dimensions of velocity squared.
(c) (i) State the causes of gross errors. (02 marks)
Causes of gross errors include:
- Improper use of measuring instruments
- Recording wrong observations
- Calculation mistakes
- Using incorrect formulas or theories
- Personal bias or carelessness
(ii) The heat generated in a circuit depend upon the current, resistance and time for which current flows. If the errors in measuring the above are 2%, 1% and 1% respectively, find the maximum error in measuring heat. (02 marks)
Heat generated H = I²Rt
Relative error: ΔH/H = 2(ΔI/I) + ΔR/R + Δt/t
ΔH/H = 2(2%) + 1% + 1% = 4% + 1% + 1% = 6%
Maximum error in measuring heat = 6%
(a) (i) How does projectile motion differ from uniform circular motion? (02 marks)
Projectile Motion:
- Path is parabolic
- Acceleration is constant (g, downward)
- Speed changes continuously
- Velocity direction changes continuously
Uniform Circular Motion:
- Path is circular
- Acceleration is constant in magnitude but changes direction (centripetal)
- Speed is constant
- Velocity magnitude is constant but direction changes
(ii) What is the nature of projectile motion. (01 mark)
Projectile motion is a two-dimensional motion under constant acceleration due to gravity. It follows a parabolic path.
(b) From the top of building 45m high, a stone is thrown at an angle of 30° to the horizontal with an initial velocity of 20m/s. Find
(i) The time of flight (1.5 marks)
Vertical component of velocity: u_y = 20 sin30° = 10 m/s
Using equation: y = u_yt - ½gt²
-45 = 10t - ½(9.8)t²
4.9t² - 10t - 45 = 0
Solving quadratic equation: t = [10 ± √(100 + 882)]/(2×4.9)
t = [10 ± √982]/9.8 = [10 ± 31.34]/9.8
Taking positive root: t = (10 + 31.34)/9.8 = 41.34/9.8 = 4.22 s
Time of flight = 4.22 seconds
(ii) The distance from the foot of the building where it strikes the ground. (2 marks)
Horizontal component of velocity: u_x = 20 cos30° = 17.32 m/s
Range = u_x × time of flight = 17.32 × 4.22 = 73.1 m
Distance from foot of building = 73.1 meters
(c) You may have seen in circus, a motor cyclist driving in vertical loops inside a "death-well" (a hollow spherical chamber with holes so the spectators can watch from outside). Explain clearly why the motor cyclist does not drop down when he is at the upper most point of death well with a no support from below. What is the minimum speed required at the upper most position to perform a vertical loop if the radius of the chamber is 25m. (3.5 marks)
The motorcyclist doesn't fall at the uppermost point because:
- The weight (mg) provides the necessary centripetal force
- Normal reaction from the track is zero at minimum speed
- Centripetal force = mg = mv²/r
Minimum speed calculation:
mg = mv²/r ⇒ v² = gr
v = √(gr) = √(9.8 × 25) = √245 = 15.65 m/s
Minimum required speed = 15.65 m/s
(a) (i) Is there any work done by the car if it moves with a uniform speed on a smooth level road. (01 mark)
No, if the car moves with uniform speed on a smooth level road, the net work done is zero. The engine does work against friction and air resistance, but no net work is done to change the kinetic energy.
(ii) Can linear momentum of a system be changed without changing its kinetic energy? (01 mark)
Yes, linear momentum can be changed without changing kinetic energy. This happens in elastic collisions where objects change direction but maintain the same speed, or when a force acts perpendicular to velocity (as in uniform circular motion).
(b) (i) The momentum of a body is increased by 100%. What is the percentage increase in its kinetic energy? (02 marks)
Kinetic energy K = p²/(2m)
If momentum increases by 100%, new momentum p' = 2p
New kinetic energy K' = (2p)²/(2m) = 4p²/(2m) = 4K
Percentage increase = [(K' - K)/K] × 100 = [(4K - K)/K] × 100 = 300%
Percentage increase in kinetic energy = 300%
(ii) A ball of mass "m" moving with a speed of 2m/s collides with a stationary ball of the same mass in a elastic collision. The incident ball is scattered at an angle of 20° from its original direction. Find the final speed of each ball and the direction of the struck ball with the original direction of the striking ball. (03 marks)
For elastic collision of equal masses:
- The balls move at right angles after collision
- Angle between final velocities = 90°
If one ball is deflected by 20°, the other is deflected by 90° - 20° = 70°
Using conservation of momentum and energy:
v₁ = u cosθ = 2 cos20° = 2 × 0.9397 = 1.879 m/s
v₂ = u sinθ = 2 sin20° = 2 × 0.3420 = 0.684 m/s
Final speeds: First ball = 1.88 m/s, Second ball = 0.684 m/s
Direction of struck ball = 70° from original direction
(c) (i) Write the three (3) uses of earth's satellite. (1.5 marks)
- Communication (TV, radio, telephone)
- Weather forecasting and monitoring
- Remote sensing and earth observation
- Navigation (GPS)
- Scientific research
(ii) Calculate the gravitational intensity on the surface of mass. (1.5 marks)
Gravitational intensity g = GM/R²
g = (6.67×10⁻¹¹ × 6.4×10²²)/(3.375×10⁶)²
g = (4.2688×10¹²)/(1.139×10¹³) = 0.375 m/s²
Gravitational intensity on Mars = 0.375 m/s²
(a) Give the qualitative meaning of the term moment of inertia. (01 mark)
Moment of inertia is the rotational analogue of mass in linear motion. It measures an object's resistance to changes in its rotational motion. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation.
(b) A fly wheel with axle 1.0cm in diameter is mounted on frictionless bearings and set in motion by applying steady tension of 3.0N to a thin thread wound tightly round the axle.
The moment of inertia of the system about its axis of rotation is \( 5.0 \times 10^{-4} \, \text{kgm}^2 \). Calculate.
(i) The angular acceleration of the flywheel when 1.0m of the thread has been pulled off the axle; and (02 marks)
Radius of axle r = 0.5 cm = 0.005 m
Torque τ = F × r = 3.0 × 0.005 = 0.015 Nm
Moment of inertia I = 5.0 × 10⁻⁴ kgm²
Angular acceleration α = τ/I = 0.015/(5.0×10⁻⁴) = 30 rad/s²
Angular acceleration = 30 rad/s²
(ii) The constant retarding couple which must be applied to bring the flywheel to rest in one complete turn when tension in the thread having been removed. (03 marks)
Initial angular velocity ω₀ = ? (not given, assume from previous motion)
From part (i): After pulling 1m of thread, angle turned θ = length/r = 1/0.005 = 200 rad
Using ω² = ω₀² + 2αθ
ω² = 0 + 2×30×200 = 12000
ω = √12000 = 109.5 rad/s
To stop in one complete turn (2π radians):
Using ω² = ω₀² + 2αθ ⇒ 0 = (109.5)² + 2α(2π)
α = -(109.5)²/(4π) = -12000/12.57 = -954.7 rad/s²
Retarding torque τ = Iα = 5.0×10⁻⁴ × (-954.7) = -0.477 Nm
Retarding couple = 0.477 Nm (magnitude)
(c) If \( y = 8 \cos \omega t + 7.5 \sin \omega t \), show that it represents S.H.M and obtain its amplitude. (04 marks)
y = 8 cosωt + 7.5 sinωt
This can be written as: y = R sin(ωt + φ)
Where R = √(8² + 7.5²) = √(64 + 56.25) = √120.25 = 10.97
And tanφ = 8/7.5 = 1.0667 ⇒ φ = 46.8°
So y = 10.97 sin(ωt + 46.8°)
This is the equation of SHM with:
Amplitude = 10.97 units
Angular frequency = ω
Initial phase = 46.8°
(a) A copper – constant thermocouple with its cold junction at 0°C had an e.m.f of 4.28mV when its other hot junction was at 100°C. The e.m.f became 9.2mV when the temperature of hot junction was 200°C. If the e.m.f is related to the temperature difference \(\theta\) between the hot and cold junction by the equation.
Calculate
(i) The value of A and B; and (03 marks)
For θ = 100°C, E = 4.28 mV: 4.28 = A×100 + B×10000 ...(1)
For θ = 200°C, E = 9.2 mV: 9.2 = A×200 + B×40000 ...(2)
Multiply equation (1) by 2: 8.56 = 200A + 20000B ...(3)
Subtract equation (3) from equation (2):
9.2 - 8.56 = (200A - 200A) + (40000B - 20000B)
0.64 = 20000B ⇒ B = 0.64/20000 = 3.2×10⁻⁵
From equation (1): 4.28 = 100A + 10000×(3.2×10⁻⁵)
4.28 = 100A + 0.32 ⇒ 100A = 3.96 ⇒ A = 0.0396
A = 0.0396 mV/°C, B = 3.2×10⁻⁵ mV/°C²
(ii) The temperature for which "E" may be assumed to be proportional to \(\theta\) without incurring an error of more than 1% (02 marks)
Error = (Bθ²)/(Aθ + Bθ²) × 100% ≤ 1%
Bθ/(A + Bθ) ≤ 0.01
Bθ ≤ 0.01A + 0.01Bθ
0.99Bθ ≤ 0.01A
θ ≤ 0.01A/(0.99B) = 0.01×0.0396/(0.99×3.2×10⁻⁵)
θ ≤ 0.000396/(3.168×10⁻⁵) = 12.5°C
Maximum temperature for 1% error = 12.5°C
(b)(i) Briefly explain how you can determine the thermal conductivity of poor conductors by lee's disc. (02 marks)
Lee's disc method:
- A disc of the poor conductor is sandwiched between a steam chamber and a brass disc
- Steam is passed to maintain constant temperature at one end
- The system reaches steady state where heat flow is constant
- Temperature difference across the sample is measured
- Rate of heat flow is calculated from the temperature rise of the brass disc
- Thermal conductivity k = (Q×d)/(A×ΔT×t) where Q is heat, d is thickness, A is area, ΔT is temperature difference, t is time
(ii) A brass boiler has a base area of 0.15m\(^2\) and thickness of 1cm. It boils water at the rate of 6 kilogram per minute when placed on a gas stove. What is the temperature of the part of the flame in contact with the boiler? (03 marks)
Rate of boiling water = 6 kg/min = 0.1 kg/s
Heat required to vaporize water Q = mL = 0.1 × 2256×10³ = 225600 J/s
Heat conduction equation: Q = kA(ΔT)/d
225600 = 109 × 0.15 × ΔT / 0.01
225600 = 109 × 15 × ΔT
225600 = 1635 × ΔT
ΔT = 225600/1635 = 138°C
Temperature of flame = boiling point + ΔT = 100 + 138 = 238°C
Temperature of flame = 238°C
Note: Answers for Section B questions would follow the same format as Section A.
Contact with MITIHANI POPOTE:
WhatsApp: +255 716 655 236
WhatsApp Channel: https://whatsapp.com/channel/0029VbA7XZkBvvsZxay5kl1l
Web: https://mitihanipopote.blogspot.com/
Instagram: https://www.instagram.com/higher_awaits
Facebook: https://www.facebook.com/profile.php?id=100064001654581
No comments
Post a Comment