EP015 Note #KMKK

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FLUID
MECHANICS

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CHAPTER 12
FLUID MECHANICS
LEARNING OUTCOMES
12.1 Hydrostatic Presure
(a) Relate and use atmospheric pressure, gauge pressure and absolute pressure.
12.2 Buoyancy
(a) Use diagram and hydrostatic pressure to explain buoyancy.
(b) State and apply the Archimedes’ principle.
12.3 Fluid Dynamics
(a) Illustrate fluid flow.
(b) Use continuity and Bernoulli’s equations
12.4 Viscosity
(a) Define viscosity
(b) State and use Stokes’ law
(c) Sketch the graph of velocity-time, v-t to explain terminal velocity in liquid

(use Speed of sound = 343 m s1)

159

12.1 HYDROSTATIC PRESSURE

1. The pressure inside a television tube is 50 Pa and the atmospheric pressure is 1.0x105 Pa.
What is the force on the screen of the television tube which has an area of 0.3 m2.
[3 x 104 N]

2. Assume the density of seawater is 1024 kgm-3 and the air exerts a pressure of 101.3 kPa.
(a) Calculate the absolute pressure at an ocean depth of 1 km.
[1.015 x 107 Pa]
(b) Calculate the gauge pressure at the ocean of 500 m depth.
[5.02 x 106 Pa]

3.

oil P1 1.4 m

water P2 2m

FIGURE 12.1

Figure 12.1 shows a tank containing two liquids of different densities. Calculate the

pressure at

(a) Point P1

[1.13 x 107 Pa]

(b) Point P2

[1.32 x 105 Pa]

Given , ρwater = 1 × 103 kgm−3and ρoil = 0.82 × 103 kgm−3

12.2 BUOYANCY

1. A rock which weighs 1400 N in air, has an apparent weight of 900 N when submerged in

water. Calculate the volume of the rock.
[5.10x10-2 m3]

2. A sea lion of mass 200 kg stands on an ice floe of 100 cm thick. What is the minimum

area must the floe have for the sea lion to be able to stand on it without getting it feet wet
in the salt water of density 1030 kg m–3? Given the density of ice is 980 kg m–3.

[4 m2]

160

3. A block of aluminum with mass 1.0 kg and density 2700 kg m-3 is suspended from a
string and then completely immersed in a container of water as in FIGURE 12.2
Calculate the tension in the string
a) before
[9.81 N]
b) after the aluminium is immersed
[6.17 N]

FIGURE 12.2
4. A solid cube of edge 0.75 cm floats in oil of density 800 kg m-3 has one third of the cube

in air. Calculate the
a) buoyant force on the cube.
[2.21 10-3 N]
b) density of the cube.
[533 kg m-3]

5. Figure 12.3 shows a cylinder of diameter d = 1.2 cm, height hc = 30 cm and weight W =
0.245 N is floating vertically, partially submerged in water (ρw = 1000 kg m–3 ).
a) What is the depth of the cylinder that is beneath the water.
[22 cm]

FIGURE 12.3

161

b) Figure 12.4 shows a layer of paraffin (ρp = 800 kg m–3) of height hp=10 cm is poured
into the container described above. What is the depth of the cylinder that is beneath
the level of the water/paraffin?
[14 cm]

FIGURE 12.4
6. A huge rising balloon has a volume of 2300 m3 and is filled with hot air with a density of

0.92 kg m–3. The cold air surrounding the balloon has a density of 1.29 kg m–3. How
much load can the balloon carry?

[851 kg]

12.3 FLUID DYNAMICS
1. A Venturi tube is used as a fluid flow meter (see FIGURE 12.5). If the difference in

pressure is 21.0 kPa, calculate the fluid flow rate (in cubic meters per second ). The
radius of the outlet tube is 1.00 cm, the radius of the inlet tube is 2.00 cm and the fluid is
gasoline whose density is 700 kg m-3.

[2.5110-3 m3 s-1]
Inlet Outlet

FIGURE 12.5

162

2. Water flows from the nozzle attached to a hose at rate of 3.6 x 10-3 m3 s-1. The diameter
of the hose is 3.20 cm and that of the nozzle is 1.20 cm. what is the speed of the water
a) in the hose
[4.48 ms-1]
b) emerging from the nozzle
[319 ms-1]

3. Water flows through a horizontal pipe at 3.5 ms-1 under a pressure of 280 kPa. If the pipe
narrows to half of its original diameter, calculate for this narrow section
a) The speed of water.
[ − ]
b) The water pressure

[188 kPa]

4. A pipe with radius 0.02 m contained water flowing with a speed of 0.2 ms-1. A smaller

tube with diameter 0.01 m is then joined to the pipe.

a) Write the continuity equation for the fluid flowing in the pipe.

b) Calculate the speed of the water flowing in the smaller tube.

[ . − ]

c) Calculate the water flow rate.

[ . × − − ]

5. A horizontal pipe 10 cm in diameter has a smooth reduction to a pipe 5 cm in diameter. If
the pressure of the water in the larger pipe is 8 × 104 Pa and the pressure in the smaller
pipe is 6 × 104 Pa, calculate the rate water flows through the pipe.
[0.0128 m3 s-1]

6. An ideal fluid is moving at 3.0 m s–1 in a section of a pipe of radius 0.20 m. If the radius
in another section is 0.35 m, what is the flow speed there?

[0.98 ms-1]

12.4 VISCOSITY

1. A fish blows a small air bubble when it swims below the surface of a lake. As the
bubbles rises to the surface of the lake, the diameter of the bubble is
A Increase
B Decrease
C Unchanged
D None of the answer above

163

2. Water flows through a pipe. The diameter of the pipe at point B is larger than at point A.
The speed of water at point B
A is greater at point A
B is greater at point B
C is the same at both points
D none of the answer above

3. a) What are the factors that influence the viscous force when a solid sphere is
released from rest into a fluid?
b) What causes the existence of viscous force?

4. An air bubble of radius 2.0 mm rises from the bottom of the tank of water. Assuming that
the volume of the bubble remains constant, what is the terminal velocity of the bubble?
(ρ = 8.0 x 10-4)
[10.9 ms-1]

5. A sphere of radius r and density  is released from rest into a liquid with density  and
coefficient of viscosity .
a) Describe the motion of the sphere by referring to the forces that act on it.
b) Sketch a graph to show the changes in the speed of the sphere with time.
c) Write an equation for terminal velocity.

6. A steel ball of diameter 5.00 mm and density 8.0 × 103 kg m-3 falls vertically with its
terminal velocity in oil of density 9.0 × 102 kg m-3. The time taken for the ball to fall a
distance of 20.0 cm is 0.52 s. Calculate
a) the terminal velocity of the ball
[0.385 ms-1]
b) the drag force on the ball
[4.56 x 10-3 N]
c) the viscosity of the oil
[0.251 Pa s]

7. a) State Stokes’ law.
b) A 4 g sphere of diameter 1 cm falls through a liquid at terminal velocity. If the
viscosity of the liquid is 0.83 Ns m2, calculate the terminal velocity of the sphere.
Neglect buoyancy effect.
[ . − ]

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13.1 Heat Conduction and Thermal Expansion
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13.2 Thermal Expansion
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CHAPTER 13
HEAT CONDUCTION AND THERMAL EXPANSION

LEARNING OUTCOMES

13.1 Thermal Conduction

a) Define heat conduction

b) Solve problems related to rate of heat transfer = – through a cross-sectional



area (maximum two objects in series)

c) Discuss graphs of temperature-distance, T-x for heat conduction through insulated and
non-insulated rods (maximum two rods in series)

13.2 Thermal Expansion
a) Define coefficient of linear, area and volume thermal expansion.

b) Solve problems related to thermal expansion of linear, area and volume (include
expansion of liquid in a container): l  l0T , = 2 , = 3

170

13.1 THERMAL CONDUCTION

1. Which of the following statements best represents the characteristic of heat as a form of
energy?
A. Heat needs a medium
B. The magnitude of heat depends on its density
C. Heat is transferred from a point or region to another
D. Heat is transferred from a high pressure region to low pressure region

2. Thermal conductivity, k depends on
A. the triple point of the material
B. the type of material
C. the boiling point of the material
D. the shape of the material

3. a) Define heat and state its SI unit.

b) Two metallic rods X and Y with similar length and cross-sectional area are joined
together and insulated. At a steady state, the temperatures of rod X and rod Y are
100 ºC and 40 ºC respectively at each end as shown in FIGURE 13.1. The thermal
conductivity of X is twice the value of Y. Calculate the temperature at the junction
between X and Y.
(80 ºC)

100 oC XY 40 oC
FIGURE 13.1

c) An aluminum rod has a diameter of 3 cm and thickness of 0.6 m. One end of the
rod is placed in boiling water and the other end in ice. Calculate the quantity of
heat transferred through the rod within 1 minute. (Given, k = 205 W m-1 K-1).
(1.45 J)

4. A glass window of cross-sectional area 1.50 m2 and thickness 0.20 cm is closed in winter.
The temperatures of the inner and outer surfaces of the window are 15˚C and 0˚C.
(i) Calculate the rate of heat flow through the window.
(9.45 k Watt)
(ii) Suggest how you would reduce the amount of heat loss to the surroundings
through this window.

(Thermal conductivity of glass = 0.84 W m-1 K-1)

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13.2 THERMAL EXPANSION

1. A strip is made of two metals P and Q of the same length and cross-sectional area as shown
in FIGURE 14.1. The linear expansion of P is twice that of Q. What will happen to this strip
when it is heated?

PQ

FIGURE 14.1

A. The strip expands vertically.
B. The strip bends to the left side.
C. The strips bends to the right side
D. The area of the strip increase but its length does not increase.

2. The coefficient of area expansion is
A. half the coefficient of linear expansion.
B. double the coefficient of linear expansion.
C. double the coefficient of volume expansion.
D. triple the coefficient of volume expansion.

3. a) (i) Define the coefficient of linear expansion.
(ii) Write the equation of linear expansion and state the meaning of all the

symbols used.

b) A 100 cm rod A expands by 8.0 mm when heated from 0oC to 100oC. Calculate
the coefficient of linear expansion for rod A.
(80 µoC-1)

4. A technician cuts a hole of area 2.0 cm2 through a copper sheet. The temperature of the sheet
rises to 150oC. Find the area of the hole when the sheet is cooled down to room temperature

30oC. (the coefficient of linear expansion of copper is 1.7 x 10-5 K-1)
(1.992 cm2)

5. A copper ball with a radius of 1.6 cm is heated to 353 C. The diameter of the ball has
increased by 0.18 mm. If the coefficient of volume expansion for the copper is 51  106
C1, calculate the initial temperature of the ball.
(22.1oC)

6. A 60 liter steel tank is full of petrol at a temperature of 30oC. If the cover of the tank is not
tightened, how much petrol will spill out at 40oC?. (coefficient of linear expansion of steel =
1.2 x 10-5 K-1, coefficient of volume expansion of petrol = 9.5 x 10-4K-1)

(548.4 cm3)

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Chapter 14

GAS LAWS AND KINETICS THEORY OF GASES

By the end of this chapter students
should be able to
14.1 IDEAL GAS EQUATIONS
a) Solve problems involving ideal gas
equation
b) Discuss the following graph of an
ideal gas :

i) p-V graph at constant
temperature.

ii) V-T graph at constant pressure.
iii) p-T graph at constant volume.

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14.2 KINETIC THEORY OF GASES
a) Discuss root mean square (rms)
speed of gas molecules

b) Solves problem involving root
mean square (rms) speed of gas
molecules

176

14.3 Molecular kinetic energy and
internal energy
a) Discuss translational kinetic energy
of a molecule

b) Discuss internal energy of gas.
c) Solve problems related to internal
energy

177

178

Molar Specific Heat

14.4 Molar Specific Heats
a) Define molar specific heat at
constant pressure Cp and volume Cv
b) Use equation

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CHAPTER 14
GAS LAWS AND KINETIC THEORY OF GASES

LEARNING OUTCOMES

14.1 Ideal Gas Equations
a) Solve problems related to ideal gas equation, =

b) Discuss the following graphs of an ideal gas :

i) p-V graph at constant temperature.
ii) V-T graph at constant pressure.
iii) p-T graph at constant volume.

14.2 Kinetic Theory of Gases
a) Discuss use root mean square (rms) speed, of gas molecules

b) Solve problems related to use root mean square (rms) speed, of gas molecules

14.3 Molecular Kinetic Energy and Internal Energy

a) Discuss translational kinetic energy of a molecule, Ktr= 3  R  T= 3 kT.
2 NA 2

b) Discus internal energy of gas

c) Solve problems related to internal energy, U= 1 fNkT
2

14.4 Molar Specific Heats
a) Define molar specific heat at constant pressure Cp and volume, Cv
b) Use equation Cp-Cv=R and   Cp/Cv

Physical Constant
NA  6.021023 molecules mol-1
k  1.38 1023 JK 1
Molar gas constant = 8.31J mol-1 K-1

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14.1 Ideal Gas Equations
14.2 Kinetic Theory of Gases
14.3 Molecular Kinetic Energy and Internal Energy
14.4 Molar Specific Heats

1. A diatomic gas is at temperature of 600K. What is the mean kinetic energy of each of these

molecules?

A. 4.14x1020 J C. 4.14x1020 J

B. 2.07x1020 J D. 2.07x1020 J

2. Which of the following statements is incorrect about the kinetic theory of gases?
A. Each degree of freedom is associated with an amount of energy given by ½kT.
B. The temperature of a gas is proportional to the average kinetic energy of the
molecules.
C. For a gas of diatomic molecules, the average translational kinetic energy per
molecule is 5kT / 2.
D. The pressure of a gas is depends on the average value of the square of the speeds
of the molecules.

3. Which of the following statements is correct about the internal energy of a gas?
A. It is the total amount of work which has been done on the gas.
B. It is equals to the maximum amount of work that can be done by the gas.
C. It is the sum of the kinetic and potential energies of the molecules in the gas.
D. It is the thermal energy needed to raise the temperature of the gas by one Kelvin.

4. Which of the following properties of gas molecule the one that is same for all ideal gases
at a particular temperature?
A. Mass
B. Velocity
C. Momentum
D. Kinetic energy

5. A vessel of volume 30 liters contains 2.5 moles of gas at 30ºC.

a) Calculate the gas pressure.
(2.1 x 105 Pa)

b) If the gas is neon, calculate the mass of the gas in the vessel.
Molar gas constant = 8.31J mol-1 K-1; relative atomic mass of neon = 40

(0.10 kg)

6. a) i. State Boyle’s Law.

ii. Sketch a graph P against V.

b) Define the ideal gas equation and write the unit for each term.

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7. A cylinder is filled with 80 x 10-3 kg of gas at 3 atm and 0° C. If the gas is compressed to

quarter of the initial volume and the temperature is recorded 500°C, find
Given the molecular weight is 5.32 x 10-26 kg mol-1.

a) the initial volume (1.12 x 1027 m3)
b) final pressure for the gas. (34 x 105 atm)

8. A balloon of diameter 30 cm is filled with helium gas at 200C and 1 atm.
a) How many atoms of helium gas fill a balloon having a diameter of 30 cm at 20ºC
at 1 atm?
()
b) Calculate the average kinetic energy of the helium atoms.
()

9. A chamber is filled with 25.0 moles of helium gas at 200oC.
a) Calculate the internal energy of the system
(1.47 × 105 J)
b) If the helium gas is replaced with 10 moles of nitrogen (N2) gas, determine the
difference between the internal energy of helium and nitrogen gases.
(9.83x104 J)

10. Three cylinders A, B and C are containing different gasses and stay at temperature 27oC.
Cylinder A contains 1 mole of argon gas, cylinder B contains 1 mole of oxygen gas and
cylinder C contains ozone gas. For each type of gas:
a) the degree of freedom
b) the percentage of translational energy contributing the total energy.
(100%, 60%, 50%)
c) the total energy (internal energy) possessed.
(3741.1 J, 6235.6 J, 7482.75 J)

11. The initial volume of one mole of hydrogen gas is at 0.05 m3 at 400 K. The gas is
adiabatically compressed to half of its original volume. Calculate

a) The initial pressure of the gas

b) The final pressure of the gas.

Given hydrogen behaves as an ideal diatomic gas with ratio of heat capacities = 1.40

183

12. A cubic ice cooler has inside dimension of 25cm. The wall of the cooler is 3.0 cm thick
and made of plastic with thermal conductivity of k = 0.050 Wm-1 K-1. If the outside
temperature is 270C, calculate the rate of heat transfer into the cooler.

13. An ideal gas absorbs 2500 J of heat and expands isobarically at 18 kPa from volume of
50 x 10-3 m3 to a volume of 80 x 10-3 m3. Calculate the change of internal energy of the
gas.

14. The pressure of a 2 mole gas changes from 2.2 x 105 Pa to 1.4 x 105 Pa at a constant
temperature of 300K. Calculate the work done by the gas.

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CHAPTER 15

THERMODYNAMICS

LEARNING OUTCOMES

15.1 First law of thermodynamics (1 hour)

a) State the first law of thermodynamics.
b) Solve problem related to first law of thermodynamics
15.2 Thermodynamics processes

a) Define the following thermodynamics processes

i) Isothermal

ii) Isochoric

iii) Isobaric

iv) Adiabatic

b) Discus the P-V graph for all the thermodynamics processes.

c) Determine the initial and final state for adiabatic process :
=
−1 =

15.3 Thermodynamics work

a) Discuss work done in isothermal, isochoric and isobaric processes

b) Solve problem related to work done in

i. Isothermal process,

= ln 2 = ln 1
1 2

ii. Isobaric process,

= ∫ = ( 2 − 1)

iii. Isochoric process,

= ∫ = 0

192

1. The first law of thermodynamics is expressed as Q = ΔU + W. When a real gas
undergoes a change at constant pressure,
A. Q is zero.
B. ΔU is zero.
C. W is zero.
D. None Q, ΔU or W is zero.

2. In adiabatic process, the internal energy of a system of gas is decreases by 800J.
Which of the following statements is CORRECT?
A The system loses 800 J by heat transfer to its surrounding.
B The system gains 800 J by heat transfer from its surrounding.
C The system does 800 J of work on its surrounding.
D The surrounding does 800 J of work on the system.

3. The work done in the expansion from an initial to a final state
A. Is the area under the curve of a pV diagram.
B. Depends only on the end point
C. Is independent of the path
D. Is the slope of a pV curve

4. In an isobaric process
A. The volume remains constant
B. The temperature remains constant
C. The pressure remains constant
D. No heat is transferred between a system and its surroundings

15.1 FIRST LAW OF THERMODYNAMICS

1. a) Write an expression of
i) first law of thermodynamics and state the meaning of all the symbols.

ii) work done by an ideal gas at variable pressure.
b) 2500 J heat is added to a system and 1800 J work is done on the system.

Calculate the change in internal energy of the system.
(4300 J)

2. A gas in container is heated with 10 J of energy causing the lid of container to rise 2
m with 3N of force. What is the total change in energy of the system.
(4J)

15.2 THERMODYNAMICS PROCESSES

1. The gas in a cloud chamber at a temperature of 292 K undergoes a rapid expansion.
Assuming the process is adiabatic, calculate the final temperature if γ = 1.40 and the
volume expansion ratio is 1.28.
(265 K)

193

2. In a steam locomotive, steam at boiler pressure of 16.0 atm enters the cylinder, is
expanded adiabatically to 5.60 times its original volume, and then exhausted to the
atmosphere. Calculate the steam pressure after expansion
(1.62 x 105 Pa)

3. A sample containing 1.00 mol of the ideal gas helium undergoes the cycle of
operations as shown in FIGURE 15.2. BC is an isothermal process. Pressure at A is
at stp (standard temperature and pressure) and pressure at B is 2.00 atm. The standard
temperature is 273 K (0° Celsius) and the standard pressure is 1 atm pressure. At
STP, one mole of gas occupies 22.4 L of volume (molar volume). Calculate
a) temperature at A.
(0K)

b) temperature at B.

(540 K)

P

B

C V
A

FIGURE15.2

4. A fixed mass of ideal monatomic gas is contained in a cylinder. The cylinder volume
can be varied by moving a piston in or out. The gas has an initial volume 0.010 m3 at
100 kPa pressure and its temperature is initially 300 K. The gas is cooled at constant
pressure until its volume is 0.006 m3.
a) Sketch a graph of pressure against volume for the above process.
b) Calculate the
i. final temperature of the gas.
(180 K)
ii. work done on the gas.
(-400 J)
iii. number of moles of gas.
(0.40 mol)
iv. change of internal energy of the gas.
(597.6 J)
v. heat transfer from the gas.
(997.6 J)

194

5. Many “empty” aerosol cans contain remnant propellant gases under approximately 1
atm of pressure ( assume 1.00 atm ) at 20˚C. They display the warning

“Do not dispose of this can in an incinerator or open fire”.
a) Explain why it is dangerous to throw such a can into fire.
b) What is the change in internal energy of such a gas if 500J of heat is added

to it, raising its temperature to 2000˚F.
(+500J)

c) What is the final pressure of the gas?
(4.66 atm)

d) Suppose the can were designed to withstand pressure up to 3.5 atm. What
would be the highest temperature it could reach without exploding?
(1025 C)

15.3 THERMODYNAMICS WORK

p/ × 105 Pa B

A
4.0

2.0 C
D

1.5 4.0 V /cm3

Figure 15.1

1. The p-V diagram in Figure 15.1 applies to a gas undergoing a cyclic change in a
piston-cylinder arrangement. Calculate the work done by the gas in
a) AB path.
(1J)

b) ABCDA path.

(0.5 J)

2. A sample of gas expands isothermally and does 4.00 x 103 J of work in the process.
Find
a) the change in the internal energy
(zero)

b) the heat absorbed by the gas

(4 x 103 J)

195

9. FIGURE 16.3 shows a system undergoing a change from A to B following a path of
ACB 90 J heat flows into the system. 70 J of work is done by the system.

FIGURE 16.3
a) Calculate the heat flows into the system that follows the path of ADB when

the work done by the system has a value of 15J.
(35 J)

b) When a system reversing its path to A by a curve path, the amount of work
done by the system is 45 J, decide whether the system gaining or losing heat.
Determine its value.
(-65 J)

c) When UA = 0 and UD = 8 J, evaluate the amount of heat gains during the
process A → D and D → B
(23 J, 12 J)

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197

EXPERIMENT 1: MEASUREMENT AND UNCERTAINTY

Course Learning Outcome:
Solve problems related to Physics of motion, force and energy, waves, matter and
thermodynamics
(C4, PLO 4, CTPS 3, MQF LOD 6)

Learning Outcomes:
At the end of this lesson, students will able to describe technique of measurement and
determine uncertainty of length of various objects.

Student Learning Time:

Face-to-face Non face-to-face

1 hour 1 hour

Direction: Read over the lab manual and then answer the following question.

Introduction

1. Complete Table 1

Basic Quantity Symbol SI Unit Measuring Instrument
(with symbol)
Length l
Table 1
Mass m

Time t

Electric Current I

Temperature T

2. …………………………….. is used to measure the diameter of a coin.

3. Micrometer screw gauge is usually used to measure the …………………… of a thin wire
or the ………………………………. of paper.

4. Complete Table 2 Sensitivity Uncertainty
0.1 cm 0.1cm
Measuring Apparatus 0.01 cm
Meter rule 0.01mm
Vernier calipers 0.1oC 0.01mm
Micrometer screw gauge 198
Travelling microscope
Thermometer

Measuring Apparatus Sensitivity Uncertainty
Voltmeter 0.1 V 0.1A

Ammeter 0.01 g
Table 2
Electronic Balance

5. State TWO types of reading;
i. ………………………………………………………
ii. ………………………………………………………

6. The repeated reading for a measurement is given as a, b, c, d, e, and f. Write the equation
of Average Value and Uncertainty.

EQUATION

Average Value, x

Uncertainty, x

Experiment Measuring Uncertainty/ Type of reading
Instrument Smallest (single point/two
7. Complete Table 3 scale
point/Vernier
Measurement scale)

Length of a metal Two points
rod
Length and width of Two points
a laboratory book
Mass of a ball Single Point
bearing
Diameter of a ball Vernier scale
bearing
Diameter of a coin Vernier scale
External diameter Vernier scale
of a glass rod

Table 3

199