UMOJA WA WAZAZI TANZANIA
WARI SECONDARY SCHOOL
Instructions
- This paper consists of Six (6) questions.
- Answer any Five (5) questions.
- Each question carries twenty (20) marks.
- Mathematical tables and non-programmable calculators may be used.
Useful Information
(i) Distinguish between Static pressure and Dynamic pressure. (02 marks)
(ii) A small oil drop falls with terminal velocity of \( 4.0 \times 10^{-4} \, m/s \). What is the new terminal velocity for an oil drop of radius half of the original radius. (04 marks)
(i) Write stoke's equation as applied to motion of a body in viscous medium. Under what conditions is the stoke's equation valid? (03 marks)
(ii) A liquid is kept in a cylindrical vessel which is rotated along its axis. The liquid rises at the sides. If the radius of the vessel is \( 0.05 \, m \) and the speed of rotation is \( 2 \, rev/s \), find the difference in the height of the liquid at the center of the vessel and its sides. (04 marks)
(i) Give two applications of Bernoulli's theorem. (02 marks)
(ii) Liquid mercury flows through a horizontal pipe of internal radius \( 1.88 \, cm \) and length \( 1.26 \, m \). The volume flux is \( 5.35 \times 10^{-2} \, L/min \). Describe how you will show that the flow is laminar and hence calculate the difference in pressure between the two sides of the pipe. (05 marks)
(i) Explain why there is usually a time interval between observing a flash and hearing a thunder? (02 marks)
(ii) How is energy transmitted in wave motion? (02 marks)
The wave function for a standing wave in a string is given by \( y = 0.3 \, \sin(0.25x) \, \cos(120\pi t) \) where \( x \) and \( y \) are in meters and \( t \) in seconds.
(i) Determine the wavelength and frequency of the interfering travelling waves. (03 marks)
(ii) Also, determine the equations of the interfering waves. (03 marks)
(i) Differentiate the two main types of diffraction (02 marks)
(ii) Briefly explain two ways in which young's experiment can be improved. (02 marks)
A light with a wavelength of \( 590 \, nm \) is incident normally on a diffraction grating. The angle of the second order maxima is \( 32^\circ \).
(i) Calculate the spacing of the lines on the grating (03 marks)
(ii) How many lines per centimeter (03 marks)
(i) Differentiate the terms creep and fatigue. (02 marks)
(ii) A stone of \( 0.5kg \) mass is attached to one end of a \( 0.8m \) long aluminum wire of \( 0.7mm \) diameter and supported vertically. The stone is now rotated in a horizontal plane at a rate such that the wire makes an angle of \( 85^\circ \) with the vertical. Find the increase in length of wire. (05 marks)
(i) What is meant by the term most probable velocity? (02 marks)
(ii) At what temperature will root mean square velocity of hydrogen molecules be doubled its value at N.T.P? (05 marks)
(i) Mention three evidences that proves that there is surface tension in water surfaces. (03 marks)
(ii) Estimate the radius of a single droplet when the rain drop of radius 0.5 mm strikes the surface and breaks to 125 droplets of equal size. (03 marks)
(i) State three factors affecting capacitance of a parallel capacitor. (03 marks)
(ii) A battery of 10V is connected to a capacitor of capacitance of 0.1F. The battery is now removed and this capacitor is connected to a second uncharged capacitor. If the charge distribute equally on these two capacitors, find the total energy stored in the two capacitors. Also compare this energy with the initial energy stored in the first capacitor. (04 marks)
Three particles are fixed in place in a horizontal plane, as shown in the figure below.
Diagram Placeholder
\( |Q_1| = 4 \times 10^{-6} C \) 37° 5 m \( Q_3 = -1 \times 10^{-6} C \) 53° \( Q_2 = 1.7 \times 10^{-6} C \)
Particle 3 at the top of the triangle has charge \( Q_3 \) of \(-1 \times 10^{-6} C\), and the electrostatic force F on it due to the charge on two other particles is measured to be entirely in the negative x-direction. The magnitude of the charge \( Q_1 \) on particle 1 is known to be \( 4 \times 10^{-6} C \), and the magnitude of the charge \( Q_2 \) on the particle 2 is known to be \( 1.7 \times 10^{-6} C \), but their signs are not known.
(i) Determine the signs of the charges \( Q_1 \) and \( Q_2 \).
(ii) Draw and label arrows to indicate the direction of the forces \( F_1 \) exerted by particle 1 on particle 3 and the force \( F_2 \) exerted by particle 2 on particle 3.
(iii) Calculate the magnitude of F, the electrostatic force on particle 3.
An electron falls through a distance of 1.5cm in a uniform electric field of magnitude \( 2.0 \times 10^4 N/C \), figure (1). The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance figure (2). Calculate the time of fall in each case.
(i) Wearing a metal bracelet in a region of strong magnetic field could be hazardous. Discuss this statement. (02 marks)
(ii) A piece of aluminum is dropped vertically downward between the poles of an electromagnet. Does the magnetic field affect the velocity of the aluminum? (02 marks)
(iii) Does dropping a magnet down a copper tube produce the current in the tube? Explain. (02 marks)
(i) Briefly explain at which position of the rotating coil in the magnetic field, the induced e.m.f is maximum? (03 marks)
(ii) A coil of a wire of certain radius has 200 turns and inductance of 89mH. What will be the inductance of another similar coil with 120 turns? (03 marks)
(i) Distinguish between diamagnetic, paramagnetic and ferromagnetic materials on the basis of relative permeability. (03 marks)
(ii) An α - particle of mass 6.65 x \(10^{-27}\) kg is travelling at right angles to a magnetic field with a speed of 6 x \(10^5\) m/s. The strength of the magnetic field is 0.2T. Calculate the force on the α-particle and its acceleration. (05 marks)
(i) What are the main two differences between Rutherford's and Bohr's model of an atom. (02 marks)
(ii) By using Bohr's model, find the orbital speed of an electron in the atom of hydrogen when n=1, 2 and 3 levels (03 marks)
(i) Why is a neutron most effective as a bullet in nuclear reaction? (02 marks)
(ii) What is the power output of a \( U^{235} \) reactor if it takes 30days to fuse up 2kg of fuel and if each fission gives 185MeV of usable energy? \( N_A = 6.02 \times 10^{26} \) per kilomole. (06 marks)
The half life of strontium – 90 is 32 years. Find
(i) The initial activity of 5g of strontium (03 marks)
(ii) How much activity will be remaining after 6 half-lives? (04 marks)
COMPREHENSIVE SOLUTIONS
Dynamic Pressure (Pdynamic): Pressure due to fluid motion, Pdynamic = ½ρv²
Original vt = 4.0 × 10⁻⁴ m/s
Since vt ∝ r²:
New vt = (r/2)²/r² × Original vt
= (1/4) × 4.0 × 10⁻⁴
= 1.0 × 10⁻⁴ m/s
Where: η = viscosity, r = radius, v = velocity
- Spherical body
- Laminar flow (Re < 1)
- Infinite fluid medium
- No wall effects
Where: ω = angular velocity, r = radius, g = 10 m/s²
ω = 2πf = 2 × 3.14 × 2 = 12.56 rad/s
Δh = (12.56)² × (0.05)²/(4 × 10)
= (157.75 × 0.0025)/40
= 0.3944/40 = 0.00986 m ≈ 9.86 mm
- Venturi meter for fluid flow measurement
- Aerofoil design for aircraft wings
- Atomizers and sprayers
Volume flux = 5.35 × 10⁻² L/min = 8.92 × 10⁻⁷ m³/s
Step 1: Check for laminar flow
Calculate Reynolds number: Re = ρvd/η
v = Q/A = 8.92×10⁻⁷/(π×(0.0188)²) = 0.000803 m/s
Re = (13600×0.000803×0.0376)/(1.55×10⁻³)
= 0.410/0.00155 ≈ 264.5
Since Re < 2000, flow is laminar.
Step 2: Pressure difference (Poiseuille's Law)
ΔP = 8ηLQ/(πr⁴)
= (8 × 1.55×10⁻³ × 1.26 × 8.92×10⁻⁷)/(π×(0.0188)⁴)
= (1.392×10⁻⁸)/(π×1.247×10⁻⁸)
= 1.392/(π×1.247) ≈ 0.355 Pa
Pressure difference = 0.355 Pa
λ = 2π/k = 2π/0.25 = 8π ≈ 25.12 m
From cos(120πt): ω = 120π rad/s
f = ω/(2π) = 120π/(2π) = 60 Hz
y₁ = 0.15 sin(0.25x - 120πt)
y₂ = 0.15 sin(0.25x + 120πt)
- Fresnel Diffraction: Source and screen at finite distances
- Fraunhofer Diffraction: Source and screen at infinite distances (parallel rays)
- Use monochromatic light source to reduce chromatic effects
- Increase slit-to-screen distance for clearer pattern
d = nλ/sinθ = (2 × 5.9×10⁻⁷)/sin32°
= (1.18×10⁻⁶)/0.5299
= 2.227×10⁻⁶ m
Lines/cm = 449,000/100 = 4490 lines/cm
Fatigue: Progressive structural damage under cyclic loading
θ = 85°, Young's modulus for Al = 7×10¹⁰ Pa
Step 1: Find tension T
T sinθ = mv²/r (centripetal force)
T cosθ = mg (vertical equilibrium)
From vertical: T = mg/cosθ = 0.5×10/cos85°
= 5/0.08716 = 57.37 N
Step 2: Calculate extension
Area A = π(d/2)² = π(3.5×10⁻⁴)² = 3.848×10⁻⁷ m²
ΔL = (T × L)/(A × Y) = (57.37 × 0.8)/(3.848×10⁻⁷ × 7×10¹⁰)
= 45.896/(2.6936×10⁴) = 1.704×10⁻³ m
v2 = 2v1
√T2 = 2√T1
T2 = 4T1
At NTP: T1 = 273 K
T2 = 4 × 273 = 1092 K
- Water droplets form spherical shapes
- Insects can walk on water surface
- Capillary rise in thin tubes
(4/3)πR³ = 125 × (4/3)πr³
R³ = 125r³
r = R/∛125 = R/5
R = 0.5 mm, so r = 0.5/5 = 0.1 mm
- Area of plates (C ∝ A)
- Distance between plates (C ∝ 1/d)
- Dielectric constant (C ∝ εr)
C₁ = 0.1 F, V = 10 V
Q = C₁V = 0.1 × 10 = 1 C
E₁ = ½C₁V² = ½ × 0.1 × 100 = 5 J
Step 2: After connection
Equal charge distribution: Q/2 on each capacitor
For each: V' = (Q/2)/C₁ = 0.5/0.1 = 5 V
E₂ (each) = ½C₁(5)² = ½ × 0.1 × 25 = 1.25 J
Total energy = 2 × 1.25 = 2.5 J
Step 3: Comparison
Energy ratio = 2.5/5 = 0.5
Energy is halved (50% remains)
1. Q₁ must be positive (attracts Q₃ with force having +x and +y components)
2. Q₂ must be negative (repels Q₃ with force having -x and +y components)
F₁ (from Q₁ on Q₃): directed toward Q₁ (attraction)
F₂ (from Q₂ on Q₃): directed away from Q₂ (repulsion)
F₁ = k|Q₁Q₃|/r₁₃² = (9×10⁹×4×10⁻⁶×1×10⁻⁶)/(5²)
= 36×10⁻³/25 = 1.44×10⁻³ N
F₂ = k|Q₂Q₃|/r₂₃² = (9×10⁹×1.7×10⁻⁶×1×10⁻⁶)/(5²)
= 15.3×10⁻³/25 = 0.612×10⁻³ N
Net force components:
Fnet,x = -F₁cos53° - F₂cos37°
= -(1.44×0.6 + 0.612×0.8)×10⁻³
= -(0.864 + 0.490)×10⁻³ = -1.354×10⁻³ N
Fnet = 1.354 × 10⁻³ N in -x direction
ae = eE/me = (1.6×10⁻¹⁹×2×10⁴)/(9.1×10⁻³¹)
= 3.2×10⁻¹⁵/9.1×10⁻³¹ = 3.516×10¹⁵ m/s²
te = √(2s/a) = √(2×0.015/3.516×10¹⁵)
= √(3×10⁻²/3.516×10¹⁵) = √(8.53×10⁻¹⁸)
= 2.92×10⁻⁹ s
For proton:
ap = eE/mp = (1.6×10⁻¹⁹×2×10⁴)/(1.67×10⁻²⁷)
= 3.2×10⁻¹⁵/1.67×10⁻²⁷ = 1.916×10¹² m/s²
tp = √(2×0.015/1.916×10¹²)
= √(3×10⁻²/1.916×10¹²) = √(1.566×10⁻¹⁴)
= 1.25×10⁻⁷ s
89×10⁻³/L₂ = (200/120)² = (5/3)² = 25/9
L₂ = (89×10⁻³)×(9/25) = 0.089×0.36 = 0.03204 H = 32.04 mH
- Diamagnetic: μr < 1 (slightly repelled)
- Paramagnetic: μr > 1 (slightly attracted)
- Ferromagnetic: μr ≫ 1 (strongly attracted)
F = (3.2×10⁻¹⁹)×(6×10⁵)×0.2 = 3.84×10⁻¹⁴ N
m = 6.65×10⁻²⁷ kg
a = F/m = 3.84×10⁻¹⁴/6.65×10⁻²⁷ = 5.774×10¹² m/s²
Acceleration = 5.77 × 10¹² m/s²
- Bohr introduced quantized orbits; Rutherford had continuous orbits
- Bohr explained atomic spectra; Rutherford couldn't
For n=2: v₂ = v₁/2 = 1.094×10⁶ m/s
For n=3: v₃ = v₁/3 = 7.293×10⁵ m/s
Molar mass = 235 g/mol = 0.235 kg/kmol
Moles in 2 kg = 2/0.235 = 8.51 kmol
Atoms = 8.51 × 6.02×10²⁶ = 5.124×10²⁷ atoms
Step 2: Total energy released
Energy per fission = 185 MeV = 185×10⁶ eV
= 185×10⁶ × 1.6×10⁻¹⁹ = 2.96×10⁻¹¹ J
Total energy = 5.124×10²⁷ × 2.96×10⁻¹¹
= 1.517×10¹⁷ J
Step 3: Power output
Time = 30 days = 30×24×3600 = 2.592×10⁶ s
Power = Energy/Time = 1.517×10¹⁷/2.592×10⁶
= 5.85×10¹⁰ W = 58.5 GW
T½ = 32 years = 1.009×10⁹ s
λ = ln2/1.009×10⁹ = 6.87×10⁻¹⁰ s⁻¹
Molar mass Sr-90 = 90 g/mol
Moles in 5 g = 5/90 = 0.0556 mol
Atoms N = 0.0556 × 6.02×10²³ = 3.346×10²²
A₀ = λN = 6.87×10⁻¹⁰ × 3.346×10²²
= 2.299×10¹³ Bq
(ii) After 6 half-lives:
A = A₀(½)⁶ = 2.299×10¹³ × (1/64)
= 3.592×10¹¹ Bq
Activity after 6 half-lives = 3.59×10¹¹ Bq
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