- This paper consists of sections A and B with a total number of ten (10) questions.
- Answer ALL questions in section A and two (2) questions from section B.
- Mathematical tables and non-programmable calculators may be used.
- Cellular phones are not allowed in the examination room.
- Write YOUR EXAMINATION NUMBER on every page of your answer booklet(s).
- Acceleration due to gravity \( g = 9.8m/s^2 \)
- Density of water \( (\rho_w) = 1000kg/m^3 \)
- Pie \( \pi = 3.14 \)
- Speed of light \( c = 340m/s \)
- Electron charge \( e = 1.6 \times 10^{-19}C \)
Answer ALL questions from this section
Define the term dimension and give two examples of it.
State the principle of homogeneity of dimensions.
If the escape velocity depends upon acceleration due to gravity of the planet and the radius of the planet, by using dimensional analysis establish the relationship between escape velocity and given quantities. Use proportionality constant value as \(\sqrt{2}\).
The error in measuring mass is 6% and volume is 1%. What will be the error in the measuring of density?
The force acting on objects of mass, \(m\) travelling at velocity \(v\) on a circle of radius \(r\) is given by: \(F = \frac{mv^2}{r}\) the measurements are recorded as
and
find the maximum possible fraction error in measuring force.
Define the term projectile motion, then mention three examples of projectile motion.
Find the range of the ball which when projected with a velocity of 29.4m/s just passes over a pole 4.9m high.
Explain the following phenomena as examples of conservation of linear momentum:
(i) When the bullet is fired, the gun is always held close to the shoulder
(ii) When a man jumps from a boat, the boat slightly moves away from the shore.
Determine the internal energy produced by a bullet of mass 10g travelling horizontally at a speed of \(1.0 \times 10^2ms^{-1}\) which embed itself in a block of wood of mass 9.9 \(\times 10^2g\) suspended freely by two strings.
Define "centripetal force" as applied in circular motion.
Why is the outer rail of a curved railway track generally raised over the inner?
Why does a pilot not fall when his aero plane loops a vertical loop?
The road bridge over a canal is in the form of an arc of a circle of radius 20m, what is the maximum speed with which a car can cross a bridge without leaving contact?
A particle is moving with simple harmonic motion of period 16sec and amplitude 10m.
(i) Find the speed of the particle when it is 6.0m from its equilibrium position.
(ii) How far is the particle from its equilibrium position 1.5sec after passing through it? What is its speed at this time?
State Kepler's Laws of planetary motion.
Find the mass and density of the earth, given that \( g = 9.8m/s^2 \) on the earth's surface, gravitational constant = \( 6.7 \times 10^{-11} \, Nm^2 kg^{-2} \) and the radius of the earth \( r_e = 6400km \).
Distinguish between gravitational field and gravitational field strength.
Why constant volume gas thermometer is considered as a standard gas thermometer?
The resistance \( R_\theta \) of platinum varies with temperature \( \theta^\circ C \) as measured by a constant volume gas thermometer according to the equation \( R_\theta = R(1 + 8000\gamma \theta - \gamma \theta^2) \) where \( \gamma \) is a constant. Calculate the temperature on platinum scale corresponding to \( 45^\circ C \) on the gas scale.
What is meant by the following?
(i) Emissive power
(ii) Solar constant
The sun radiates energy in all directions. The average radiation on the earth surface per second is \( 1400W m^{-2} \). The average earth – sun distance is \( 1.5 \times 10^{11} m \). Calculate the mass lost by the sun per day.
Write down an equation of the first Law of thermodynamics under adiabatic change and define all the symbols used.
If the molar specific heat capacity of oxygen at a constant pressure is \( 29.5Jmol^{-1} K^{-1} \) and \( R = 8.3Jmol^{-1} K^{-1} \), Calculate the amount of heat taken by 5 moles of oxygen when heated at a constant volume from \( 10^\circ C \) to \( 20^\circ C \).
On the same set of axes sketch the curves for adiabatic and isothermal change.
Show that for an adiabatic change, work done on one mole of the gas is given by \( W = \frac{R}{\gamma - 1}(T_1 - T_2) \). Where \( T_1 \) and \( T_2 \) are temperature in Kelvin.
A gas in an insulated cylinder has a pressure of \( 2.0 \times 10^5 Pa \). If the piston has an area of \( 3.0 \times 10^{-3} m^2 \) and is pulled out suddenly a distance of 10mm. Calculate the change in internal energy of the gas.
Define the term environment as applied in physical system.
Mention three (3) parts of environment necessary for plant growth.
What is mulching?
Give three (3) advantages of mulches.
What are wind belts? Briefly explain how they are formed.
What are the positive effects of wind on plant growth?
Answer any two (2) questions from this section
Define each of the following as used in current electricity:
(i) Current density
(ii) Mobility of the electron
An aluminium wire of diameter 0.24cm is connected in series to a copper wire of diameter 0.16cm. The wires carry an electric current of 10 A. Find the drifting velocity of electrons in the copper wire. Given the number of electrons per cubic metre volume is \(8.4 \times 10^{28}\).
What is meant by "reactive power in a.c circuit"?
An inductor 200mH, capacitor 500\(\mu F\), resistor 10\(\Omega\) are connected in series with a 100V, variable frequency a.c source. Calculate: Frequency at which the power factor of the circuit is unit and current amplitude at this frequency.
Give four any advantage of potentiometer versus that of voltmeter.
In the given circuit find the potential difference across the 8\(\Omega\) resistor.
List three properties of operational amplifier.
Define closed loop gain.
Derive an expression of the closed loop gain for an inverting Op-amp with an input resistor \( R_1 \) and feedback resistor \( R_2 \).
Mention two types of junction transistors.
Which among the transistors mentioned responds quickly to electrical signals. Why?
Calculate the value of the output potential \(V_o\) if the input potential \(V_i\) is \(+2V\).
Draw a NAND gate and its truth table.
Sound signal in the frequency range of 300Hz to 3400Hz are used to amplitude modulate a carrier wave of frequency 200 kHz. Determine the band width of the resultant modulated signal.
How do you distinguish LED from a photodiode?
Give two (2) applications of LED.
Define current gain \(\beta\), then show that \(\beta = \frac{\alpha}{1-\alpha}\).
The circuit below shows a transistor in a common emitter mode. Study the circuit and answer the questions that follow.
• 16V
R = 425KΩ
R_L = 2KΩ
V_o
If the output voltage (\(V_o\)) is at 5V. Calculate:
(i) The collector current
(ii) The base emitter voltage
(iii) The current gain of transistor
What is thermal run away effect and what are its causes?
Explain with the aid of diagram how the thermal run away might be prevented in a circuit.
Instructions
- This paper consists of sections A and B with a total number of ten (10) questions.
- Answer ALL questions in section A and two (2) questions from section B.
- Mathematical tables and non-programmable calculators may be used.
- Cellular phones are not allowed in the examination room.
- Write YOUR EXAMINATION NUMBER on every page of your answer booklet(s).
Useful Information
- Acceleration due to gravity \( g = 9.8m/s^2 \)
- Density of water \( (\rho_w) = 1000kg/m^3 \)
- Pie \( \pi = 3.14 \)
- Speed of light \( c = 340m/s \)
- Electron charge \( e = 1.6 \times 10^{-19}C \)
SECTION A (70 Marks)
Answer ALL questions from this section
Define the term dimension and give two examples of it.
State the principle of homogeneity of dimensions.
If the escape velocity depends upon acceleration due to gravity of the planet and the radius of the planet, by using dimensional analysis establish the relationship between escape velocity and given quantities. Use proportionality constant value as \(\sqrt{2}\).
The error in measuring mass is 6% and volume is 1%. What will be the error in the measuring of density?
The force acting on objects of mass, \(m\) travelling at velocity \(v\) on a circle of radius \(r\) is given by: \(F = \frac{mv^2}{r}\) the measurements are recorded as
and
find the maximum possible fraction error in measuring force.
Define the term projectile motion, then mention three examples of projectile motion.
Find the range of the ball which when projected with a velocity of 29.4m/s just passes over a pole 4.9m high.
Explain the following phenomena as examples of conservation of linear momentum:
(i) When the bullet is fired, the gun is always held close to the shoulder
(ii) When a man jumps from a boat, the boat slightly moves away from the shore.
Determine the internal energy produced by a bullet of mass 10g travelling horizontally at a speed of \(1.0 \times 10^2ms^{-1}\) which embed itself in a block of wood of mass 9.9 \(\times 10^2g\) suspended freely by two strings.
Comprehensive Answers
1. (a) (i) Definition of dimension and examples:
Dimension refers to the physical nature of a quantity and how it is expressed in terms of fundamental quantities (mass, length, time, etc.). Dimensions represent the qualitative aspect of physical quantities.
Examples:
- Velocity has dimensions of [LT⁻¹] (length/time)
- Force has dimensions of [MLT⁻²] (mass × length/time²)
- Energy has dimensions of [ML²T⁻²] (mass × length²/time²)
1. (a) (ii) Principle of homogeneity of dimensions:
The principle of homogeneity of dimensions states that for an equation to be physically correct, all terms in the equation must have the same dimensions. This means we can only add, subtract, or compare quantities that have identical dimensional formulas.
For example, in the equation s = ut + ½at²:
- s (displacement) has dimensions [L]
- ut (velocity × time) has dimensions [LT⁻¹][T] = [L]
- ½at² (½ × acceleration × time²) has dimensions [LT⁻²][T²] = [L]
All terms have the same dimensions [L], so the equation is dimensionally correct.
1. (b) Escape velocity relationship:
Let escape velocity v depend on:
- Acceleration due to gravity (g)
- Radius of the planet (R)
We can write: v ∝ gᵃ Rᵇ
Or: v = k gᵃ Rᵇ, where k is a constant (given as √2)
Dimensions:
- v (velocity) = [LT⁻¹]
- g (acceleration) = [LT⁻²]
- R (radius) = [L]
Equating dimensions:
Equating powers:
- For L: a + b = 1
- For T: -2a = -1 ⇒ a = ½
Substituting a = ½ into a + b = 1:
Therefore: v = k g¹ᐟ² R¹ᐟ² = k √(gR)
With k = √2: v = √(2gR)
The relationship is: v = √(2gR)
1. (c) (i) Error in density measurement:
Density ρ = mass/volume = m/V
The fractional error in density:
Given: Δm/m = 6% = 0.06, ΔV/V = 1% = 0.01
The error in measuring density is 7%.
1. (c) (ii) Maximum possible fraction error in force:
Given: F = mv²/r
The fractional error in F:
Given values:
- m = 3.5 kg, Δm = 0.1 kg ⇒ Δm/m = 0.1/3.5 = 0.0286
- v = 20 m/s, Δv = 1 m/s ⇒ Δv/v = 1/20 = 0.05
- r = 12.5 m, Δr = 0.5 m ⇒ Δr/r = 0.5/12.5 = 0.04
Maximum possible fractional error:
The maximum possible fraction error in measuring force is 0.1686 or 16.86%.
2. (a) (i) Projectile motion definition and examples:
Projectile motion is the motion of an object thrown or projected into the air, subject only to acceleration due to gravity. The path followed by a projectile is called a trajectory, which is typically parabolic.
Examples of projectile motion:
- A ball thrown in any direction (except vertically)
- A bullet fired from a gun
- A javelin thrown by an athlete
- A stone projected from a catapult
- A football kicked during a game
2. (a) (ii) Range of ball passing over pole:
Given:
- Initial velocity u = 29.4 m/s
- Height of pole h = 4.9 m
The ball just passes over the pole, so at the pole:
Vertical displacement y = 4.9 m
Using the equation of trajectory: y = x tanθ - (gx²)/(2u²cos²θ)
Where x is the horizontal distance to the pole.
For maximum range, θ = 45°
At θ = 45°, tanθ = 1, cosθ = 1/√2
Solving this quadratic equation:
Taking the positive root that gives maximum range:
Or the other root:
The range is the larger value: 82.98 m
2. (b) Conservation of linear momentum examples:
(i) When a bullet is fired, the gun is held close to the shoulder:
According to the law of conservation of linear momentum, the total momentum before firing is zero (both gun and bullet are at rest). After firing, the bullet moves forward with high velocity, so to conserve momentum, the gun must move backward with a certain velocity (recoil).
If the gun is held loosely, the recoil velocity can cause injury. When held close to the shoulder, the shooter's body absorbs the recoil, reducing the effective recoil velocity and preventing injury.
(ii) When a man jumps from a boat, the boat moves away from the shore:
Initially, both the man and boat are at rest, so total momentum is zero. When the man jumps toward the shore, he gains momentum in that direction. To conserve momentum, the boat must gain equal momentum in the opposite direction, causing it to move away from the shore.
2. (c) Internal energy produced by bullet impact:
Given:
- Mass of bullet m₁ = 10 g = 0.01 kg
- Velocity of bullet u₁ = 100 m/s
- Mass of wood block m₂ = 9.9 × 10² g = 990 g = 0.99 kg
- Initial velocity of block u₂ = 0 m/s (at rest)
Using conservation of momentum:
Initial kinetic energy:
Final kinetic energy:
Internal energy produced = Loss in kinetic energy:
The internal energy produced is 49.5 J.
3. (a) (i) Centripetal force definition:
Centripetal force is the force that acts on a body moving in a circular path, directed toward the center around which the body is moving. It is responsible for keeping the body in circular motion.
Mathematically: F = mv²/r, where m is mass, v is velocity, and r is radius.
3. (a) (ii) Outer rail raised on curved railway track:
The outer rail of a curved railway track is raised over the inner rail (a practice called banking) to provide the necessary centripetal force for trains to safely navigate the curve.
When a train moves on a curved track, it requires a centripetal force to keep it moving in a circular path. Banking the track causes the normal reaction force to have a horizontal component that provides this centripetal force, reducing the reliance on friction between the wheels and rails.
The angle of banking is calculated such that: tanθ = v²/(rg), where θ is the banking angle, v is the speed of the train, r is the radius of curvature, and g is acceleration due to gravity.
3. (b) (i) Pilot not falling in vertical loop:
A pilot does not fall when the airplane loops a vertical loop because of centripetal force. At the top of the loop:
- The pilot experiences a downward gravitational force (mg)
- The seat exerts a downward normal reaction force on the pilot
- These two forces together provide the necessary centripetal force to keep the pilot moving in a circular path
If the airplane is moving fast enough, the normal reaction force is sufficient to prevent the pilot from falling, even when upside down at the top of the loop.
The minimum speed at the top of the loop is given by: v = √(rg), where r is the radius of the loop and g is acceleration due to gravity.
3. (b) (ii) Maximum speed on curved bridge:
Given: Radius r = 20 m
For a car to not leave contact with the bridge at the highest point, the centripetal force must be provided entirely by gravity:
The maximum speed with which a car can cross the bridge without leaving contact is 14 m/s.
3. (c) Simple harmonic motion calculations:
Given: Period T = 16 s, Amplitude A = 10 m
Angular frequency ω = 2π/T = 2π/16 = π/8 rad/s
(i) Speed when 6.0 m from equilibrium:
Speed = π m/s ≈ 3.14 m/s
(ii) Position and speed 1.5 s after passing through equilibrium:
Using x = A sin(ωt) for motion starting from equilibrium:
Position = 5.556 m from equilibrium
Speed v = ω√(A² - x²) = (π/8)√(100 - 30.86) = (π/8)√69.14 = (π/8)×8.315 = 3.265 m/s
Speed = 3.265 m/s
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