FORM SIX PRE NATIONAL EXAMINATION PHYSICS 1 SERIES 17 (With Comprehensive Solution)

UMOJA WA WAZAZI TANZANIA.

WARI SECONDARY SCHOOL

PRE-NATIONAL EXAMINATION SERIES

PHYSICS 1 – SERIES 17

131/1

TIME: 2:30 HRS

JANUARY- MAY, 2023

Instructions

1. This paper consists of TWO sections A and B
2. Answer SEVEN (7) questions in section A and any two (2) questions from sections B
3. Extra questions attempted will not be marked.
4. Mathematical tables and Calculators may be used
5. Cellular phones are not allowed in the examination room.
6. Write your NAMES on every page of your answer booklets.
7. Unorganized work will not be marked.
8. Failure to follow instructions will lead you to the loss of marks.
9. The following information may be used.

   i. Acceleration due to gravity \( g = 9.8m/s^2 \)	ii. Radius of the earth, \( R = 6370km \)
   iii. Pie, \( \pi = 3.14 \)	iv. Density of water \( \rho = 1000kg/m^3 \)
   v. Stefan-Boltzmann constant, \( \sigma = 5.67 \times 10^{-8}w/m^2k^{-4} \)	vi. Density of sea water \( = 1023kg/m^3 \)
   vii. Mass of the Earth \( = 6.0 \times 10^{24}kg \)	viii. Stefan-Boltzmann constant \( \delta = 5.67 \times 10^{-8}Wm^2K^{-4} \)

SECTION A (70 MARKS)

Answer SEVEN (7) questions from this section

1.

(a) (i) What is dimensional constant? (01 mark)
(ii) In the formula \( X = 3yz^2 \), \( X \) and \( Z \) are the dimension of capacitance and magnetic induction respectively. Find the dimension of y in M KSQ system (where MKSQ stand for metre, kilogram, second and charge.) (04 marks)

(b) (i) What is relative error? (01 mark)
(ii) The specific resistance \(\rho\) of a thin circular wire of radius \( r(cm) \) resistance \( R(Ohm) \) and length \( L(cm) \) is given by \(\rho = \frac{\pi r^2 R}{L} \), if \( r = (0.26 \pm 0.02)cm \), \( R = (32 \pm 1)ohm \) and \( L = (78 \pm 0.01)cm \). Find the percentage error in \(\rho\). (04 marks)

2.

(a) (i) What factors determine the span of jump? (01 mark)
(ii) Show that the path taken by projectile is parabola. (02 marks)

(b) A ball falling freely from a given height H hits an inclined plane in its path at a height h as a result of this impact, the direction of velocity of the ball becomes horizontal, for what value of \( \frac{h}{H} \) the ball will take maximum time to reach the ground? (02½ marks)

(c) (i) Draw the graph to show the variation of gravitational force g with depth/height inside the earth and outside the surface of the earth. (02 marks)
(ii) The Mass of earth is 81 times that of the moon and the distance from the centre of the earth to that of the moon is about \( 4.0 \times 10^5 km \). Calculate the distance from the centre of the earth to where the resultant gravitation force becomes zero when a spacecraft is launched from the earth to the moon. (02½ marks)

3.

(a) (i) If the net force on the body is zero, will the body remain necessarily in rest position? If No, give one reason. (01 mark)
(ii) Why is a cat not hurt while falling on the ground? (01 mark)
(iii) Energy of 484J is spent in increasing the speed of a fly wheel from 60 r.p.m to 360 r.p.m. Find the moment of inertia of the wheel. (03 marks)

(b) (i) What is banking of road? (01 mark)
(ii) A car whose wheels are 1.5m apart laterally and whose centre of gravity is 1.5m above the ground round, a curve of radius 250m. Find the maximum speed at which the car can travel without toppling. (04 marks)

4.

(a) (i) Differentiate between forced oscillation and free oscillation (02 marks)
(ii) Will a simple pendulum vibrate at the centre of the earth? Explain (01 mark)

(b) (i) Show that the total energy of a body executing S.H.M of y = ASinot is independent of time (04 marks)
(ii) A body is moving with SHM with period of 10 seconds. If the distance between extreme positions is 200cm. Find its velocity from the centre of its motion. (03 marks)

5.

(a) (i) State Stefan's law (01 mark)
(ii) The sun's rays are focused by a concave mirror of diameter 12cm fixed with its axis towards the sun onto a copper calorimeter, where they are absorbed. If the thermal capacity of the calorimeter and its contents is 247.8 Joule per degree centigrade and the temperature rises by \( 8^\circ C \) in 2 minutes. Calculate the heat received in 1 minute by a square metre of the earth's surface when their rays are incident normally. (04 marks)

(b) (i) State two limitation of Newton's law of Cooling (02 marks)
(ii) A body cools from \( 80^\circ C \) to \( 70^\circ C \) in 10 minutes. What will be the temperature of the body in 15 minutes of surrounding is kept at \( 28^\circ C \) (03 marks)

6.

(a) Define the following process:-
i. Isobaric (01 mark)
ii. Isochoric (01 mark)

(b) What amount of heat is to be transferred to nitrogen in an isobaric heating so that the gas may perform 21 of work? (04 marks)

(c) (i) Absolute zero is not the temperature of zero energy. Explain (02 marks)
(ii) What is the specific heat in an isotherm change? (02 marks)

7.

(a) Explain the following
i. Earthquake focus (1.5 marks)
ii. The epicenter (1.5 marks)

(b) i. List down three sources of earthquake (03 marks)
ii. Define the ionosphere (01 mark)

(c) i. State the ionospheric layer that exists during day time (01 mark)
ii. Give the reason for better wave's reception for light frequencies signals at night than during day time (02 marks)

SECTION B (30 MARKS)

Answer any TWO (2) questions from this section

8.

(a) (i) What is the meaning of super conductor and thermistor? (02 marks)
(ii) Why connecting wires are made of copper? (01 mark)

(b) i) Differentiate between capacitive reactance and inductive reactance. (02 marks)
ii) A 750Hz, 20V source is connected to a resistance of 100Ω and inductance of 0.1803H and a capacitance of 10μF, all are in series. Calculate the time in which the resistance, (thermal capacity = \( 2J/^\circ C \)) will get heated by \( 10^\circ C \). (04 marks)

c) i. Prove that the capacitor acts as conductor for a.c and insulator for d.c. (02 marks)
ii. Study the circuit below and then answer the question that follows

Circuit diagram showing I₁, I₂, R₂, E_T, R₁, E₂

By using Kirchhoff's laws, show that the current passing through \( R_2 \) is given by

\[I_2 = \frac{E_2 (R_1 + R_3) - E_1 R_3}{R_1 R_2 + R_1 R_3 + R_2 R_3} \]

(04 marks)

9.

(a) Carefully answer the following questions:
i. Why emitter terminal is highly doped? (01 mark)
ii. Why base in any transistor is made thin? (01 mark)
iii. Why collector in any transistor is made thick? (01 mark)

(b) Study the diagram below and then answer question that follow

Transistor circuit diagram with IB, RB, VC, 200K, VB, VBB, RE, RC, IC, 3K, 10V, Vcc

Given that current gain is 100.
i) Find \( V_{CE} \) (04 marks)
ii) From above in \( 8b(i) \) comment on which region the transistor works? (01 mark)

c. Calculate the values of \( \beta \) and \( \alpha \) from the transistor shown in the graph below. (03 marks)

Transistor characteristic graph showing I_C vs V_CE with I_B as parameter

(d) (i) Differentiate between amplitude modulation and frequency modulation (02 marks)
(ii) Mention and draw three basic elements of a communication system (02 marks)

10.

(a) What is the meaning of open loop gain and closed loop gain as used in OPAMP? (02 marks)

(b) The figure below shows a Non-Inverting amplifier
i. Find the value of input voltage, if \( R_1 = 9K\Omega \), \( R_2 = 10R_1 \) and \( Vout = 6.0V \) (03 marks)
ii. Why for Non-inverting amplifier the input resistance is not equal to \(\frac{Vin}{I} \) ? (01 mark)

c) (i) What is universal gate? (02 marks)
(ii) Draw the logic gate of OR-gate from NAND gate system (02 marks)

(b) Draw the truth table which is corresponding to the circuit below (02 marks)

Logic circuit with inputs A, B, C, D

(d) Design a warning device that will alert the driver of a car, if any of three doors, boot or bonnet of the four door car is not properly shut. Assume that each of these is fitted with a switch producing logic 1 unless the door/boost/bonnet is used (04 marks)

UMOJA WA WAZAZI TANZANIA.

WARI SECONDARY SCHOOL

PRE-NATIONAL EXAMINATION SERIES

PHYSICS 1 – SERIES 17

131/1

TIME: 2:30 HRS

JANUARY- MAY, 2023

Instructions

1. This paper consists of TWO sections A and B
2. Answer SEVEN (7) questions in section A and any two (2) questions from sections B
3. Extra questions attempted will not be marked.
4. Mathematical tables and Calculators may be used
5. Cellular phones are not allowed in the examination room.
6. Write your NAMES on every page of your answer booklets.
7. Unorganized work will not be marked.
8. Failure to follow instructions will lead you to the loss of marks.
9. The following information may be used.

   i. Acceleration due to gravity \( g = 9.8m/s^2 \)	ii. Radius of the earth, \( R = 6370km \)
   iii. Pie, \( \pi = 3.14 \)	iv. Density of water \( \rho = 1000kg/m^3 \)
   v. Stefan-Boltzmann constant, \( \sigma = 5.67 \times 10^{-8}w/m^2k^{-4} \)	vi. Density of sea water \( = 1023kg/m^3 \)
   vii. Mass of the Earth \( = 6.0 \times 10^{24}kg \)	viii. Stefan-Boltzmann constant \( \delta = 5.67 \times 10^{-8}Wm^2K^{-4} \)

SECTION A (70 MARKS)

Answer SEVEN (7) questions from this section

1.

(a) (i) What is dimensional constant? (01 mark)

Solution 1(a)(i):

A dimensional constant is a physical constant that has dimensions (units) and a fixed value. Examples include the gravitational constant (G), Planck's constant (h), and the speed of light (c). These constants have specific units and their values do not change in different systems of measurement.

(ii) In the formula \( X = 3yz^2 \), \( X \) and \( Z \) are the dimension of capacitance and magnetic induction respectively. Find the dimension of y in M KSQ system (where MKSQ stand for metre, kilogram, second and charge.) (04 marks)

Solution 1(a)(ii):

Given: \( X = 3yz^2 \)

Dimensions of capacitance X: [M⁻¹L⁻²T²Q²]

Dimensions of magnetic induction Z: [MT⁻²Q⁻¹]

From the equation: \( [X] = [y][Z]^2 \)

\( [M⁻¹L⁻²T²Q²] = [y][MT⁻²Q⁻¹]^2 \)

\( [M⁻¹L⁻²T²Q²] = [y][M²T⁻⁴Q⁻²] \)

Therefore: \( [y] = [M⁻¹L⁻²T²Q²] / [M²T⁻⁴Q⁻²] = [M⁻³L⁻²T⁶Q⁴] \)

So the dimension of y in MKSQ system is [M⁻³L⁻²T⁶Q⁴]

(b) (i) What is relative error? (01 mark)

Solution 1(b)(i):

Relative error is the ratio of the absolute error to the actual measurement. It gives an indication of how good a measurement is relative to the size of the thing being measured.

Relative Error = Absolute Error / Measured Value

It is often expressed as a percentage (percentage error).

(ii) The specific resistance \(\rho\) of a thin circular wire of radius \( r(cm) \) resistance \( R(Ohm) \) and length \( L(cm) \) is given by \(\rho = \frac{\pi r^2 R}{L} \), if \( r = (0.26 \pm 0.02)cm \), \( R = (32 \pm 1)ohm \) and \( L = (78 \pm 0.01)cm \). Find the percentage error in \(\rho\). (04 marks)

Solution 1(b)(ii):

Given: \(\rho = \frac{\pi r^2 R}{L} \)

The relative error in ρ is given by:

\(\frac{\Delta \rho}{\rho} = 2\frac{\Delta r}{r} + \frac{\Delta R}{R} + \frac{\Delta L}{L}\)

Substituting the values:

\(\frac{\Delta r}{r} = \frac{0.02}{0.26} = 0.0769\)

\(\frac{\Delta R}{R} = \frac{1}{32} = 0.03125\)

\(\frac{\Delta L}{L} = \frac{0.01}{78} = 0.000128\)

\(\frac{\Delta \rho}{\rho} = 2(0.0769) + 0.03125 + 0.000128 = 0.1538 + 0.03125 + 0.000128 = 0.185178\)

Percentage error = \(0.185178 \times 100\% = 18.52\%\)

Therefore, the percentage error in ρ is approximately 18.52%.

2.

(a) (i) What factors determine the span of jump? (01 mark)

Solution 2(a)(i):

The span of a jump (range) is determined by:

1. Initial velocity of projection
2. Angle of projection
3. Acceleration due to gravity
4. Height from which the jump is initiated (if different from landing height)

(ii) Show that the path taken by projectile is parabola. (02 marks)

Solution 2(a)(ii):

For a projectile launched with initial velocity u at an angle θ to the horizontal:

Horizontal motion: \( x = u\cos\theta \cdot t \) ⇒ \( t = \frac{x}{u\cos\theta} \)

Vertical motion: \( y = u\sin\theta \cdot t - \frac{1}{2}gt^2 \)

Substituting t from horizontal motion:

\( y = u\sin\theta \cdot \frac{x}{u\cos\theta} - \frac{1}{2}g\left(\frac{x}{u\cos\theta}\right)^2 \)

\( y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta} \)

This is of the form \( y = ax - bx^2 \), which is the equation of a parabola.

Therefore, the path of a projectile is parabolic.

(b) A ball falling freely from a given height H hits an inclined plane in its path at a height h as a result of this impact, the direction of velocity of the ball becomes horizontal, for what value of \( \frac{h}{H} \) the ball will take maximum time to reach the ground? (02½ marks)

Solution 2(b):

Let's analyze the motion:

Before hitting the plane, the ball falls through height (H-h) with initial velocity 0.

Velocity just before hitting: \( v = \sqrt{2g(H-h)} \)

After hitting, the velocity becomes horizontal with magnitude v.

Time to fall from height h with horizontal velocity: \( t = \sqrt{\frac{2h}{g}} \)

Total time T = time to fall (H-h) + time to fall h after impact

\( T = \sqrt{\frac{2(H-h)}{g}} + \sqrt{\frac{2h}{g}} \)

To maximize T, we differentiate with respect to h and set dT/dh = 0:

\( \frac{dT}{dh} = -\frac{1}{\sqrt{2g(H-h)}} + \frac{1}{\sqrt{2gh}} = 0 \)

\( \frac{1}{\sqrt{2gh}} = \frac{1}{\sqrt{2g(H-h)}} \)

\( \sqrt{h} = \sqrt{H-h} \) ⇒ \( h = H-h \) ⇒ \( 2h = H \) ⇒ \( \frac{h}{H} = \frac{1}{2} \)

Therefore, the ball takes maximum time to reach the ground when \( \frac{h}{H} = \frac{1}{2} \).

(c) (i) Draw the graph to show the variation of gravitational force g with depth/height inside the earth and outside the surface of the earth. (02 marks)

Solution 2(c)(i):

The variation of g with depth and height:

[Graph showing g vs distance from Earth's center]
Y-axis: g (acceleration due to gravity)
X-axis: Distance from Earth's center

The graph would show:
- Inside Earth (0 to R): g decreases linearly from g₀ to 0 at the center
- At surface (R): g = g₀ (maximum value)
- Outside Earth (R to ∞): g decreases following inverse square law (g ∝ 1/r²)

Inside the Earth: \( g = g_0 \left(1 - \frac{d}{R}\right) \) where d is depth, R is Earth's radius

Outside the Earth: \( g = g_0 \left(\frac{R}{R+h}\right)^2 \) where h is height above surface

(ii) The Mass of earth is 81 times that of the moon and the distance from the centre of the earth to that of the moon is about \( 4.0 \times 10^5 km \). Calculate the distance from the centre of the earth to where the resultant gravitation force becomes zero when a spacecraft is launched from the earth to the moon. (02½ marks)

Solution 2(c)(ii):

Let M be mass of Moon, then mass of Earth = 81M

Let the distance from Earth where gravitational force is zero be x from Earth

Then distance from Moon = (4.0×10⁵ - x) km

At this point, gravitational force due to Earth = gravitational force due to Moon

\( \frac{G(81M)}{x^2} = \frac{GM}{(4.0\times10^5 - x)^2} \)

\( \frac{81}{x^2} = \frac{1}{(4.0\times10^5 - x)^2} \)

Taking square root: \( \frac{9}{x} = \frac{1}{4.0\times10^5 - x} \)

\( 9(4.0\times10^5 - x) = x \)

\( 36\times10^5 - 9x = x \)

\( 36\times10^5 = 10x \)

\( x = 3.6\times10^5 \) km

Therefore, the distance from Earth's center where gravitational force is zero is 3.6×10⁵ km.

3.

(a) (i) If the net force on the body is zero, will the body remain necessarily in rest position? If No, give one reason. (01 mark)

Solution 3(a)(i):

No, the body will not necessarily remain at rest if the net force is zero.

According to Newton's First Law of Motion, if the net force on a body is zero, the body will either remain at rest or continue moving with constant velocity (uniform motion).

Reason: A body in motion will stay in motion with the same speed and direction unless acted upon by an unbalanced force.

(ii) Why is a cat not hurt while falling on the ground? (01 mark)

Solution 3(a)(ii):

A cat is not hurt while falling due to several factors:

1. Cats have a righting reflex that allows them to orient themselves during a fall
2. They spread their body to increase air resistance, reducing terminal velocity
3. Their flexible bodies and limbs act as shock absorbers upon impact
4. They have a low body mass relative to their surface area

(iii) Energy of 484J is spent in increasing the speed of a fly wheel from 60 r.p.m to 360 r.p.m. Find the moment of inertia of the wheel. (03 marks)

Solution 3(a)(iii):

Given: Energy spent = 484J

Initial speed = 60 rpm = 60/60 = 1 revolution per second = 2π rad/s

Final speed = 360 rpm = 360/60 = 6 revolutions per second = 12π rad/s

Energy spent = Change in rotational kinetic energy

\( \Delta KE = \frac{1}{2}I(\omega_f^2 - \omega_i^2) \)

\( 484 = \frac{1}{2}I((12\pi)^2 - (2\pi)^2) \)

\( 484 = \frac{1}{2}I(144\pi^2 - 4\pi^2) \)

\( 484 = \frac{1}{2}I(140\pi^2) \)

\( 484 = 70\pi^2 I \)

Using π² ≈ 9.8696

\( 484 = 70 \times 9.8696 \times I \)

\( 484 = 690.872 \times I \)

\( I = \frac{484}{690.872} = 0.7 kg\cdot m^2 \)

Therefore, the moment of inertia of the wheel is 0.7 kg·m².

(b) (i) What is banking of road? (01 mark)

Solution 3(b)(i):

Banking of road is the process of raising the outer edge of a curved road above the inner edge. This is done to provide the necessary centripetal force for vehicles to safely navigate the curve.

The banking angle is calculated such that at a certain speed, no friction is required for the centripetal force, which is provided by the horizontal component of the normal reaction.

(ii) A car whose wheels are 1.5m apart laterally and whose centre of gravity is 1.5m above the ground round, a curve of radius 250m. Find the maximum speed at which the car can travel without toppling. (04 marks)

Solution 3(b)(ii):

Given: Distance between wheels (track width) = 1.5m

Height of center of gravity = 1.5m

Radius of curve = 250m

For no toppling condition: \( \tan\theta = \frac{v^2}{rg} \)

Also, \( \tan\theta = \frac{\text{track width}}{2 \times \text{height of CG}} = \frac{1.5}{2 \times 1.5} = \frac{1.5}{3} = 0.5 \)

Therefore: \( 0.5 = \frac{v^2}{250 \times 9.8} \)

\( v^2 = 0.5 \times 250 \times 9.8 = 1225 \)

\( v = \sqrt{1225} = 35 m/s \)

Converting to km/h: \( 35 \times \frac{3600}{1000} = 126 km/h \)

Therefore, the maximum speed without toppling is 35 m/s or 126 km/h.

SECTION B (30 MARKS)

Answer any TWO (2) questions from this section

8.

(a) (i) What is the meaning of super conductor and thermistor? (02 marks)

Solution 8(a)(i):

Superconductor: A material that exhibits zero electrical resistance and perfect diamagnetism when cooled below a certain critical temperature. This means electric current can flow through it indefinitely without any energy loss.

Thermistor: A type of resistor whose resistance varies significantly with temperature. There are two main types: NTC (Negative Temperature Coefficient) thermistors whose resistance decreases with increasing temperature, and PTC (Positive Temperature Coefficient) thermistors whose resistance increases with increasing temperature.

(ii) Why connecting wires are made of copper? (01 mark)

Solution 8(a)(ii):

Connecting wires are made of copper because:

1. Copper has very low electrical resistivity (1.68×10⁻⁸ Ω·m), allowing efficient conduction of electricity with minimal energy loss
2. It is ductile and can be easily drawn into wires
3. It has good mechanical strength
4. It is relatively abundant and cost-effective compared to other good conductors like silver
5. It has good thermal conductivity which helps dissipate heat

END OF EXAMINATION

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