Its All about Pj Problem Strings (SiPjAjk) - 7 Spaces Of Interest (Si) and their associated Basic Sequences; 7 Pj Problems of Interest (PPI) and their Alleles (Ajk)

WiseBites - Chew And Swallow

My Brain Is Stringed - Iremisan Adegiga : I became a TECian (a person or any other being that sees the Universe through the TECTechnics Prism) about two years ago after ... more

God And The Homo-Sapiens Series: The word God is a mapping onto the belief in the existence of an Ultimate Being. The philosophical presentation... more

Onset Of Homo (Sapiens)3 : The homo sapiens series was introduced in Homo (Sapiens)n And The End Of Time. The gist ... more

Gender Math Based On Non-Autosomal Chromosome Pairings: Language is a mapping of symbols into space in order to establish generally accepted meanings for space and its contents. ... more

The Greatest Black History: History is an account of past events. Often, the account is chronological. The veracity of ... more

Suspension of Chief Justice Of Nigeria: Two days ago, Adegiga's Why President Buhari Will Be Re-elected was featured here at tectechnics.com. Since then,... more

Why Buhari Will Be Re-elected - I. Adegiga: The single most important current problem facing Africa is political stealing. Political stealing is a polity cancer... more

Immigration - Migration Conflicting Rights (2)

Two natural rights were highlighted with respect to human migration and immigration in ... more

By Their Fruits We Know Them - O. A. Asemota: Happy New Year! Recently, some of our visitors wanted to know if I am still here at tectechnics.com since I have been literarily quiet for some time ... more

Homo (Sapiens)n And The End Of Time: Homo sapiens (wise man) emerged as a new human species on earth about 400,000 years ago... more

Kings And Queens Who Are No Longer Monarchs - I. Adegiga: King, Queen (non-consort) and monarch used to be one and the same in my mind until ... more

Immigration - Conflicting Rights Early humans were nomads. They moved from place to place in search of food and sometimes because of disequilibriums in their spaces .. more

Crack In The American Democracy: Complex structures are always made up of parts. Humans copied this design template from Nature and have used it extensively to construct various structures (e.g. social systems, political systems, economic systems, machine systems, etc) ... more

Palace Intrigues: A palace is the residence of a king or queen. An intrigue is a secret plot. Palace intrigues are secret plots that originate in palaces. This meaning has been generalized in this presentation to mean ... more

Thy Kingdom Come - O. A. Asemota : Jesus Christ, The Messiah, Son of God, King of kings, Lord of lords. These are some of the names Christians use to refer to Jesus of Nazareth ... more

The Meaning Of Great: The concepts words describe predate the words that describe them. Consequently, if a word is to be contextually invariant (i.e. absolute in meaning) in all human spaces, the spaces must... more

The Risk View Of Loss Risk, the probability or chance of loss is an existential reality. Risk lurks in the existence of all entities. Consequently, its absolute elimination is ... more

Rest - I. Adegiga: Rest is simply a break from work and a time to rejuvenate the body, mind and spirit. The duration of rest may... more

While Africa Slumbered - O. A. Asemota: The African Story to date, is bittersweet. Volumes have been written about the first continent. Also, there have been many movies (fictions and nonfictions) about Africa. The Black Panther is an example of a recent fictional movie about Africa. I saw the movie, The Black Panther. The cinematography, costumes and cast are excellent. However, after all the pageantry, I was reminded that... more

Jealousy The Green-Eyed Monster - Iremisan Adegiga : Human imperfection with respect to existential living is an existential reality. As humans live day by day, they fall short of the marks they set for themselves, or that societies set for them; or that their religion set for them; etc. I am definitely a member of the group of imperfect humans. However, there is .. more

The Simplicity Of Nature's Infinite Intelligence: Nature has dual meanings in this presentation: (a) the totality of the being of the physical Universe ... more

Why Greatness Eludes Nigeria - O.A. Asemota: While contemplating the title of this article, two thoughts came to mind ... more

The Animal In Humans: the domestication of humans began centuries ago as humans became smarter than their fellow animals in the jungle. Civilization after... more

Great Concepts From Africa: Africans established human existence on Earth. There is no human accomplishment greater than this. In addition to pioneering human existence on Earth, Africans gave humanity the following great concepts ... more

Hi-story - Iremisan Adegiga: I called my parents at the end of my freshman year in college to inform them about my decision to major in history instead of economics as... more

Political Thieves Within Nigeria: The Implication - O. A. Asemota: Stealing is the taking of property without permission from the owner of the property. A thief is one who steals... more

A Grain Of Faith In The Scientific Method...more

Genesis Chapter One (KJV) -Reconciling Creation With Facts: Belief in the creation account in Genesis Chapter one, is mostly faith-based. Nonetheless, there is...more

War. Lessons Unlearned: Cognitive beings defend being and space when faced with existential threats ...more

Denatured Conquest: The ability to conquer (defeat or overwhelm) was incorporated into the being of cognitive beings at their creation because ...more

Many Kingdoms Within Nigeria's Democracy - O. A. Asemota: There were sophisticated political systems in the space now called Nigeria prior to the coming of the white men... more

Bloom - Kimberlee June Benart: My momma said to me, ďDonít hide your light under ... more

Leaders And Leaders-Makers: Human political systems did not fall from the sky. They are consequences of gradual political evolutions that... more

Policing The Pursuit Of Knowledge : Policing is the enforcement of the system of laws of a space. The police are ... more

Spiritual But Not Religious - Kimberlee J. Benart : I saw the title of the blog and took the time to read it, but how it saddened me to see it full of harsh unkindness ... more

Mis-Information As A Weapon - The Larger Issue: Information is shared knowledge. The knowledge shared does not have to be accurate...more

I Charlie - A Farmer At Heart: From growing up on a farm in America to pioneering and working in Africa...more

One Nation Under What?: A nation is a group of individuals with different identities... more

Selective Freedom: Freedom is the condition of not being controlled by another. The implication here is not that a person is ... more

Ultimate Reality: the awareness of being establishes reality ... more

All is Mathematics: Nature only speaks mathematics within the context of 7 universal concepts (Pj problems). This language is uniform everywhere in the Universe and is ... more

Good Walls Bad Walls: A wall is a barrier that encloses a space. A wall does not have to be visible to the naked eye. The structure of a wall... more

Expressions Of Pj Problems

"I have reduced physics to mathematics" - Descartes.
Strings (SiPjAjk) reduce all knowledge to mathematics.

Cams - The PjProblemStrings

Cams The PjProblemStrings

Cams are used to change rotary motion to linear or up-and-down motion

(a) Figure 134.1 is an illustration of a cam-driven valve. Explain how the cam help to open and close the valve.
(b) Write the PjProblemStrings of interest in the cam-driven valve.

                → Solution: the strings and the math.

Gears - The PjProblemStrings

Gears The PjProblemStrings

Gears are simple machines. They are used to change the direction of motion; decrease or increase the speed of motion; and magnify or reduce applied force.

(a) Figure 133.1(a) is an illustration of an eggbeater. Gears A, B and C enable the eggbeater to do its work. Physicists say, the gear is a type of a lever. Classify the eggbeater as a lever.
(b) Give a range for the theoretical mechanical advantage of the eggbeater of 133.1(a).
(c) Explain how the gears of the eggbeater change the speed of motion.
(d) Figure 133.1(b) is a rack-and-pinion arrangement of gears. What is its usefulness.
(e) Imagine a gear train consisting of four gears: A (10 teeth), B (40 teeth) C (20 teeth) and D (10 teeth). A in mesh with B, C is rigidly fixed on the same shaft as B; C in mesh with D. What is the overall speed ratio of the gear train?
(f) What is an idler gear?
(g) Write the PjProblemStrings at play with respect to figures 133.1(a) and 133.1(b).

                → Solution: the strings and the math.

The Screw - The PjProblemStrings

The Screw The PjProblemStrings

Figure 132.1 illustrates a jack screw. This machine belongs to the class of simple machines called the screw. Some common members of this group are the micrometer, the foodprocessor used to grind meat, the rigger's vice and the friction brake.

(a) Physicists say, the screw is a type of an inclined plane. What simple experiment shows that the screw is an adaptation of the inclined plane?
(b) What is the theoretical mechanical advantage of the jack screw of 132.1 if R = 24 and p = 1/4?
(c) Why is it that most of the theoretical mechanical advantage of the jack screw is lost to friction?
(d) Write the PjProblemStrings at play with respect to the jack screw of figure 132.1

                → Solution: the strings and the math.

Wheel And Axle - The PjProblemStrings

Wheel And Axle The PjProblemStrings

Figure 131.1 illustrates a brace with a screwdriver bit driving a screw. This machine belongs to the class of simple machines called the wheel and axle. Some common members of this group are the doorknob and the steering wheel of automobiles.

(a) Physicists say, the wheel and axle is a type of lever. What type of lever is the brace and bit of figure 131.1 ?
(b) What is the theoretical mechanical advantage of the brace and bit of figure 131.1?
(c) Write the PjProblemStrings at play with respect to the brace and bit of figure 131.1?

                → Solution: the strings and the math.

Block And Tackle - The PjProblemStrings

Block And Tackle The PjProblemStrings

Figure 130.1 is a basic combination of a type of simple machines.

(a) What type of simple machine is illustrated in figure 130.1?
(b) What is the theoretical mechanical advantage of the machine illustrated in figure 130.1
(c) Write the PjProblemStrings at play with respect to figure 130.1?

                → Solution: the strings and the math.

Inclined Planes - The PjProblemStrings

Inclined Planes The PjProblemStrings

Inclined planes and levers are the two main groups of simple machines. Figures 129.1(a) and (b) illustrate a simple inclined plane and a wedge respectively.

(a) Explain why the length NQ is longer than the length NM.
(b) Write an expression for the effort that pushes the solid ball into the cart if the weight of the ball is W and the angle of inclination of the plane is θ:
(i) If friction is not at play.
(ii) If friction is taken into account.
(c) The wedge with sides EF = 1" and GH = 3" is used to split a log (figure 129.1(b). How wide does the force of the effort force the log apart?
(d) What is the mechanical advantage of the wedge?
(e) Write the PjProblemstrings of the work done in figures 129.1(a) and 129.1(b).

                → Solution: the strings and the math.

Levers - The PjProblemStrings

Engines The PjProblemStrings

Levers are simple machines. Figure 128.1 illustrates a 6-inch file scraper (brown) being used to pry up the lid of a can.

(a) Name the other types of simple machines.
(b) What type of lever is the file scraper illustrated in figure 128.1?
(c) The blue arrows associated with the file scraper are
implicitly PjProblemStrings. Make them explicit.
(d) What is the mechanical advantage of the file scraper,
given that MN is 1 inch and NQ is 5 inches?

                → Solution: the strings and the math.

Engines - The PjProblemStrings

Engines The PjProblemStrings

Matter's adventure in apriori Space, is the Story of the Universe. This Story is a mesh of infinitely many PjProblemStrings. As a result of their sophisticated mind and physical weakness, humans have developed myriad machines (simple and complex) to aid them in their adventures.

(a) What is the difference between a machine and an engine?
(b) PjProblemStrings are everywhere. Review this
fundamentally universal concept with a brief description.
(c) All 7 Pj Problems are featured in the construction of an engine. However, only two PjProblemStrings represent
the Pj Problems of Interests (PPI). List them and explain.

                → Solution: the strings and the math.

Market Triad

Market Triad

A market is basically a space (may be virtual) where goods or/and services are sold. A medium of exchange (aka money) is implied in the market-triad of figure 127.1.
(a) Describe the components of the market-triad of figure 127.1.
(b) What is the relationship between market and economy?
(c) What is the relationship between investment and market
(d) How does inflation in an economic space reflect market-activities in the space.
(e) How does unemployment in an economic space reflect market-activities in the space.
(f) Why has advertising become big business?
(g) Which space invented paper money?

                → Solution: the strings and the math.

Communication Triad

Communication Triad

Communication is basically the transmission and reception of knowledge. Data and Information are implied in the communication-triad of figure 126.1.
(a) Define knowledge, data and information.
(b) Describe the components of the communication-triad of figure 126.1.
(c) How have the components of the communication-triad
been conceptually advanced over the years?

                → Solution: the strings and the math.

Triadic Unit Mesh - The Mathematics

Triadic Unit Mesh - The Mathematics

Human technological advancement is impressive and very much needed in modern civilization. However, The human mind, a pen/pencil and paper remain the best tools for concepts-sketching. The Triadic Unit Mesh (TUM) is an effective intellectual abstraction for concepts-sketching. Its effectiveness is inherent in its conceptual simplicity.

Explain the meaning of the mathematics of TUM.

                → Solution: the strings and the math.

Triadic Unit Mesh - Constructing With Triangles

Triadic Unit Mesh - Constructing With Triangles

A structure is a composite of grouped components. The Triadic Unit Mesh (TUM) (figure 125.1a) is a trapezoid consisting of three triangles and seven sides. Essentially, it groups components in order to realize the static or dynamic equilibrium of an entity. The primary groupings of the human-body (a very well engineered structure) has been mapped onto the Triadic Unit Mesh as reference.

(a) Using the Triadic Unit Mesh (TUM), determine the cell component, the tissue component and the organ component of a n-storey block building.
(b) A painter is painting on a canvas. Using TUM, what is the cell component of the painting.
(c) A sculptor begins work on an original stone by chipping the stone. Adapt TUM to his or her sculpturing.

                → Solution: the strings and the math.

Electric/Electronic Circuit Triad

Electric/Electronic Circuit Triad

The circuit-triad of Figure 124.1 illustrates the three basic conceptual components of an electric/electronic circuit.
(a) Describe the three basic conceptual components of the circuit-triad.
(b) What roles do amplification and switching play in the circuit-triad?
(c) What roles do active and passive elements play in the circuit-triad?
(d) The electrical/electronic circuits of a given automobile consists of an automotive battery connected to the parallel connections of the following components: headlights, taillights, starter motor, fan, power locks, and dashboard panel.
d(i) Illustrate the circuits. How many meshes in the illustration?
d(ii) How many loops in the illustration?

                → Solution: the strings and the math.

EnhanceMent-Mode MOSFETS AS Switches - The NAND Gate

Enhancement Mode MOSFETS As Switches - The NAND Gate

Enhancement-mode MOSFETS are primary components in amplification and switching circuits. They are combined in various ways as gates to realize complex switching configurations.

Figure 123.6(a) is an arrangement of two pMOS connected in parallel and two nMOS connected in series to form a NAND gate. Figure 123.6(b) is the symbol and Figure 123.6(c) is the truth table for the configuration.
Show how the circuit of figure 123.6(a) functions as a NAND gate.

                → Solution: the strings and the math.

EnhanceMent-Mode MOSFETS AS Switches - The NOR Gate

Enhancement Mode MOSFETS As Switches - The NOR Gate

Enhancement-mode MOSFETS are primary components in amplification and switching circuits. They are combined in various ways as gates to realize complex switching configurations.

Figure 123.5(a) is an arrangement of two pMOS connected in series and two nMOS connected in parallel to form a NOR gate. Figure 123.5(b) is the symbol and Figure 123.5(c) is the truth table for the configuration.
Show how the circuit of figure 123.5(a) functions as a NOR gate.

                → Solution: the strings and the math.

EnhanceMent-Mode MOSFETS AS Switches - The NOT Gate

Enhancement Mode MOSFETS As Switches - The NOT Gate

Enhancement-mode MOSFETS are primary components in amplification and switching circuits. They are combined in various ways as gates to realize complex switching configurations.

Figure 123.4(a) is an arrangement of a pMOS and an nMOS to form a NOT gate (an Inverter). Figure 123.4(b) is the symbol and Figure 123.4(c) is the truth table for the configuration.
Show how the circuit of figure 123.4(a) functions as an inverter.

                → Solution: the strings and the math.

Operating Region Of An Enhancement-Mode NMOS Transistor

Operating Regions Of An Enhancement Mode NMOS Transistor

Figure 123.2 illustrates an enhancement-mode NMOS. The threshhold voltage, VT = 2 V, The source terminal of the nmos is grounded and a voltage source VGG = 3 V DC is connected to the gate.

Determine the operating state of the transistor for each of the following values of vD:
(a) vD = 0.5 V
(b) vD = 1 V
(c) vD = 5 V

                → Solution: the strings and the math.

Enhancement-Mode Metal Oxide Semiconductor Field-EFFect Transistors

Enhancement Mode Metal-Oxide Semiconductor Field-Effect Transistors

Field-Effect Transistors (FETs) are another main group of transistors. There are two primary types of FETs: metal-oxide semiconductor field-effect transistors (MOSFETs) and Junction Field Effect Transistors (JFETs). MOSFETs are grouped into enhancement-mode MOSFETs and depletion-mode MOSFETs. Each of these transistors can be either an n-channel device or a p-channel device depending on the nature of the doping. Figure 123.1 illustrates both the n-channel enhancement MOSFET and the p-channel enhancement MOSFETs.

(a) Explain the conduction state of the NMOS (figure 123.1c).
(b) Define the following terms: Threshold Voltage, Conductance Parameter, Early Voltage.
(c) Name and characterize the three operating regions of the NMOS transistor.

                → Solution: the strings and the math.

Bipolar Junction Transistor Switching Characteristic

Bipolar Junction Transistor Switching Characteristic

Figure 122.7 illustrates a simple BJT switch (122.7a) and its collector charateristic (122.7b).

(a) Base on the switch-triad model, identify the signal, The on state and the off state of the BJT.

(b) How is the switching characteristic of the BJT implemented given that VCC = 5 V? Assume a transistor-transistor logic (TTL) as follows:
For Vin, logic low (0) = 0 V; logic high (1) = 5 V
For Vout, logic low (0) = 0 V; logic high (1) = 5 V

                → Solution: the strings and the math.

Bipolar Junction Transistor Configurations

Bipolar Junction Transistor Configuration

There are three primary types of BJT configurations: common-emitter (emitter is common to both input and output circuits), commom-collector (collector is common to both input and output circuits) and common-base (base is common to both input and output circuits). Figure 122.6 illustrates the common-emitter and common-collector .

(a) Indicate the common application scenario for each BJT configuration

(b) The following are circuit values for the common-emmitter of figure 122.6(a):
Vs = 1 cos(6.28 x 103t) mV; VCC =15 V
Current gain, β = IC/IB = 100 .
R1 = 68 kΩ; R2 = 11.7 kΩ;
RC = 200 Ω; RE = 200 Ω;
RL = 1.5 kΩ; Rs = 0.9 kΩ;

Determine VCEQ and the region of operation.

                → Solution: the strings and the math.

Transistor Amplifier Supply For A LED

Transistor Amplifier Supply For A LED

Figure 122.5 is a transistor amplifier circuit designed to supply a Light Emitting Diode (LED). The output signal from the microcomputer acts as the switch for the LED (symbol for LED is circled diode with emanating arrows). The circuit values are as follows:
Microcomputer: output resistance = RB = 1 kΩ; Von = 5 V; Voff = 0 V; current, I = 5 mA.
Transistor: VCC = 5 V; offset voltage, Vγ = VBE = 0.7;
Current gain, β = IC/IB = 95; VCEsaturation =0.2 V.
LED: offset voltage, VγLED = 1.4 V; ILED > 15 mA; Pmax = 100 mW.

(a)Determine collector resistance RC such that the transistor is in the saturation region when the microcomputer outputs 5 V.
(b) Determine the power dissipated by the LED.

                → Solution: the strings and the math.

Operating Point Of A Bipolar Junction Transistor

Operating Point Of A Bipolar Junction Transistor

Figure 122.4 illustrates a Bipolar Junction Transistor (BJT) self-bias DC Circuit. Determine the operating point of the transistor in the circuit given the following circuit values:
R1 = 100 kΩ; R2 = 50 kΩ; RC = 5 kΩ; RE = 3 kΩ;
VCC = 15 V; offset voltage, Vγ = VBE = 0.7;
Current gain, β = IC/IB = 100.

                → Solution: the strings and the math.

Operation Of The Bipolar Junction (BJT) Transistor (2)

Operation Of The Bipolar Junction Transistor

Using figure 122.3(a) as reference, determine the operating region of each of the Bipolar Junction Transistor illustrated in 122.3(b), 122.3(c) and 122.3(d) by determining whether the BE and BC junctions are forward-biased or reverse-biased.

                → Solution: the strings and the math.

Operation Of The Bipolar Junction (BJT) Transistor

Operation Of The Bipolar Junction Transistor

Figure 122.2(a) and 122.2(b) illustrate the two primary types of the Bipolar Junction Transistor (BJT): pnp BJT transistor and the npn BJT transistor. Figure 122.2(c) and 122.2(d) illustrate its operation. The + superscript indicates additional doping (the addition of impurities to semiconductors).

(a) The operation of the BJT is defined in terms of two currents and two voltages. Identify these currents and voltages and incated the relationship between them.

(b) There are four basic regions that define the operating states of a BJT. Name the regions.

                → Solution: the strings and the math.

Amplification Triad

Amplification Triad

A signal is amplified when it is strengthened by the application of energy. Humans often amplify electrical and electronic signals for various reasons. Sound and light signals are some signals in nature often amplified by humans. The concept of amplification is ubiquitious in Nature. In fact growth is amplification.

(a) A seed is planted and eventually grows into a tree. Identify the components of the amplification-triad for the planting: the signal, the applied energy and the amplified signal.

(b) A nation is founded and eventually grows into a great nation. Identify the components of the amplification-triad for the nation: the signal, the applied energy and the amplified signal.

                → Solution: the strings and the math.

Maximum Zener Diode Power Dissipation

Maximum Zener Diode Power Dissipation

Suppose the components of the Zener diode voltage regulator circuit of figure 121.11(b) have the following values:
Vs = 24 V; VZ = 12 V; Rs = 50 Ω; RL = 250 Ω;

(a) Determine the minimum acceptable power rating of the Zener diode.

Now suppose the values are changed as follows:
Vs = 50 V; VZ = 14 V; Rs = 30 Ω; PZ = 5 W.

(b) Determine the range of load resistances, RL such that the diode power rating is not exceeded.

                → Solution: the strings and the math.

Zener Diode In A Voltage Regulator Circuit

Zener Diode In A Voltage Regulator Circuit

Zener diodes (symbol in 121.10(a)) are voltage regulator devices. They help to keep the rectifier DC voltage output constant. They are deliberately designed to operate in the reverse-breakdown region.

The voltage regulator circuit of figure 121.10(b) has the following characteristics:
Constant reverse breakdown Zener voltage, VZ = 8.2 V.
Allowable range of operational Zener current: 75 mA≤ iZ ≤1 A.
Load resistance, RL = 9 Ω

Size Rs so that vL = Vz = 8.2 V is maintained while Vs varies by + 10% from its nominal value of 12 V.

                → Solution: the strings and the math.

The Bridge Rectifier

The Bridge Rectifier

The half-wave rectifier converts only the positive half-cycle of an AC input signal. So, the negative half-cycle is unused. The bridge rectifier is a full-wave rectifier that coverts both cycles of the AC signal.

(a) Circuit 121.9 (a) is a bridge rectifier. Which diodes are conducting during the positive AC cycle and which diodes are conducting during the negative AC cycle?

The box of 121.9(b) and its AC input and DC output represents an Integrated Circuit (IC) Rectifier. An integrated circuit is a collection of electronic circuits on a silicon chip.
(b) Why are the resistor and capacitor added in the circuit of figure 121.9(b)?

(c) Suppose the bridge rectifier of circuit 121.9(a) is used to provide a 50 v, 5 A DC supply. Determine the resistance of the load that will draw exactly 5 A and the rms source voltage that realized the desired DC voltage. Assume an ideal diode.

                → Solution: the strings and the math.

Load Voltage In A Half-Wave Rectifier

Load Voltage In A Half-Wave Rectifier

AC retification (change of AC signal to DC signal) is one of the important diode applications.

Circuit 121.8 is a half-wave rectifier. Determine the DC value of the rectified waveform if the AC voltage source, vs = 52cosωt V. Assume ideal diode.

                → Solution: the strings and the math.

Conduction State Of An Ideal Diode

Conduction State Of An Ideal Diode

The circuit of figure 121.7 (a) consists of an ideal diode (the black triangle), voltages and resistors (R1, R2, R3). The values of the voltages are as indicated in the diagram. R1 = 5 Ω, R2 = 10 Ω, R3 = 10 Ω.

(a) Is the diode conducting current?

(b) If resistor R2 is removed from the circuit, will the diode conduct current?

                → Solution: the strings and the math.

Switch Triad

Switch Triad

Switches are everywhere. The electrical and electronic push-buttons and clicks are familiar switches. However, there are many other switches that do important work but are usually not perceived as switches.

1. Switches in codes: identify the nature of the signal and the binary states (on and off) of the switche(s) in the following simple code:


<?php
session_start();
set_include_path('/somedir/');
include('/somedir/somefile');
somefunction();
.
.
.
echo "someoutput";
?>

2. Doors and gates as switches: identify the signal and the binary states (on and off) when doors and gates are considered as switches.

3. Biological switches: name at least one biological switch. Identify the signal and the binary states (on and off) of the biological switch.

                → Solution: the strings and the math.

Music Of Peter Oye Sagay

Time In Between
Just Suppose
Why I Love You
Love Me
Step By Step - Now Available

Torsional Shear Stress In A section Of A Shaft

Torsional Shear Stress In A Section Of A Shaft

Determine the horse power (hp) transmitted at 1800 rpm by a shaft with diameter, d =1.306 if the torsional shear stress is limited to 8000 psi.

                → Solution: the strings and the math.

Maximum Tensile Flexure Stress In A Section Of A Beam

Maximum Tensile Flexure Stress In A Section Of A Beam

Determine maximum flexure stress at section A-B of the beam resting on simple supports as illustrated in figure 125.6(a).

                → Solution: the strings and the math.

Total Longitudinal Strain Of A Rectangular Steel Block Under Combined Stress

Total Longitudinal Strain Of A Rectangular Steel Block Under Combined Stress

The dimensions of a rectangular steel block are:
Length = 12 in, height = 4 in, thickness = 2 in.
Member is subjected to the following stresses:
Longitudinal tensile stress σx = 12,000 lb/in2
Vertical compressive stress σy = 15,000 lb/in2
Lateral compressive stress σx = 9,000 lb/in2
Poisson ratio μ, = 0.30.
Modulus of Elasticity of Steel E, = 30, 000,000.

Determine the steel block's total longitudinal strain (i.e, total change in length).

                → Solution: the strings and the math.

Initial Stress In A Composite Member Under Tensile Loading Given Allowable Stresses

Initial Stress in A Composite Member Under Tensile Loading Given Allowable Stresses

A composite member consists of a steel rod shaft in an aluminum tube. The members are fastened together by adjustable nuts. Dimensions are:
Sectional area of steel rod = 1.5 in2; modulus of elasticity = 30,000,000; allowable stress = 15,000 lb/in2.
Sectional area of aluminum tube = 2 in2; modulus of elasticity = 10,000,000; allowable stress = 10,000 lb/in2

Determine initial stresses (prestressed) in the members so that under tensile loading both members will attain their allowable stresses simultaneously.

                → Solution: the strings and the math.

Maximum Tensile Stress In Relation To Temperature Drop In A Steel Bar

Maximum Tensile Stress In Relation To Temperature Drop In A Steel Bar

A steel bar in the form of a frustum (truncated cone) is rigidly fixed at both of its ends (figure 125.2). Its dimensions are:
length = 24 inches; diameter of circular section at one end = 1 inch.
diameter of circular section at the other end = 3 inches.
Modulus of elasticity, E = 30,000,000 lb/in2.
Coefficient of thermal expansion = 0.0000065/oF.

Suppose bar is subjected to a drop in temperature of 50oF. Determine maximum tensile stress in bar.

                → Solution: the strings and the math.

Total Force In Portions Of A Bar Under Axial Loading

Total Force In Portions Of A Bar Under Axial Loading

Figure 125.1 illustrates an axially loaded uniform bar that is rigidly fixed at its ends A and B. Axial loading is at points B and C.

(a) Determine the total force in portions AB, BC and CD of the bar. Assume consistent deformation and superposition.
(b) Interprete the result of problem (a) if the bar is not rigidly held at its ends A and D, but instead, prestressed by wedging it between rigid walls under an initial compression of 10,000 lb.
(c) What happens if the initial compression in problem (b) is less than 8000 lb?

                → Solution: the strings and the math.

Properties Of A Plane Area

Properties Of A Plane Area

Consider figure 124.1:
(a) Where is the centroid of the plane area?
(b) Write the general expression for the moment of inertia about the x axis
(c) Write the general expression for the moment of inertia about the y axis
(d) Write the general expression for the polar moment of inertia
(e) Write the general expression for the product of inertia
(f) Write the general expression for the radius of gyration with respect to the x axis
(g) Write the genaral expression for the radius of gyration with respect to the y axis
(h) Show that the polar moment of inertia is the sum of the moment of inertia about the x axis and the moment of inertia about the y axis.

                → Solution: the strings and the math.

Bernoulli On Pressure In A Moving Fluid

Bernoulli On Pressure In A Moving Fluid

Nature has established many codes in the Universe (perhaps infinitely many). Birds were able to decode a critical aspect of the flight code long before the emergence of humans. Then came Daniel Bernoulli, son of John Bernoulli (1667-1748), with the Bernoulli's Principle. John Bernoulli and his elder brother Jacob Bernoulli contributed greatly to mathematics. Daniel Bernoulli and his brother Nicholas Bernoulli also contributed greatly to mathematics.

(a) What is the Bernoulli's Principle?
(b) Can one relate the principle of suction to the Bernoulli's Principle?
(c) Isaac Newtons third law established the action -reaction relationship. Identify the action and the reaction in the Bernoulli's Principle.
(d) What other scenarios does one find the Bernoulli's Principle in action?

                → Solution: the strings and the math.

Pascal On Pressure In Liquids

Pascal On Pressure In Liquids

(a) One pushes the brake pedal in a car with a hydraulic brake system and the car comes to a halt. How does figure 123.1 explain this observation?
(b) Name other scenarios where one observes Pascal's Principle in action.

                → Solution: the strings and the math.

Kepler's Laws Of Planetary Motions

Kepler's Laws Of Planetary Motions

planet A revolves around the Sun, S (figure 122.1):
(a) Describe its path around the sun.
(b) Is the velocity of A constant throughout its revolution around the sun?
(c) Relate the period of A's revolution to its mean distance from the sun.

                → Solution: the strings and the math.

Uniqueness Proof Technique

Uniqueness Proof Technique


Suppose If A then B is a proposition involving statements A and B. The existence of B does not necessarily establish the uniqueness of B. The Uniqueness Proof Technique can be used to establish the uniqueness of B.

Prove, by the direct uniqueness method, that if a, b, c, d, e, and f are real numbers such that (ad - bc) ≠ 0, then there are unique real numbers x and y such that (ax + by) = e and (cx + dy) = f

                → Solution: the strings and the math.

Contradiction Proof Technique

Contradiction Proof Technique

The Contradiction Proof Technique is one of the common proof techniques used when the Forward-Backward Proof Technique is not suitable for the given proof problem.
Suppose If A then B is a proposition involving statements A and B. The contradiction proof technique (Figure 121.3) begins by assuming that B is false (i.e, not B). The problem is proved if at the end the assumption is contradicted.

(a) Prove, by contradiction, that if n is an integer and n2 is even, then n is even.
(b) Prove, by contradiction, that at a party of x people, where x ≥ 2, there are at least two people who have the same number of friends at the party.
(c) Prove, by contradiction, that there are an infinite number of primes.

                → Solution: the strings and the math.

Mathematical Induction Proof Technique

Mathematical Induction Proof Tecnique

Mathematical induction proof technique is well suited for proof problems of the type:
For a given population of integers, some event occurs. An example of this type of proof problem is as follows:

For all integers n ≥ 1, nΣk=1 = [n(n+1)]/2

(a) Prove, by induction, that, for every integer ≥ 5, 2n > n2.
(b) Prove, by induction, that any integer n ≥ 2 can be expressed as a finite product of primes.

                → Solution: the strings and the math.

Forward-Backward Proof Techniques

Proof - Backward Process - Forward Process

The right triangle XYZ of figure 121.1 has sides of lengths x and y, and hypotenuse of length z. Its area is z2/4. Using the Forward-Backward proof techniques, proof that triangle XYZ is isosceles.

                → Solution: the strings and the math.

Law Triad - The Basic Concepts Of Law

Law Triad Basic Concepts of Law

(a) Describe the components of the Law Triad illustrated in figure 119.1.
(b) Does the Law Triad apply to natural laws?
(c) On the basis of the Law Triad, Timothy posits that in general, animals in the jungle are better behaved than their human cousins. Speculate on the correctness of Timothy's assertion.
(d) State in one sentence, why Timothy's assertion in (c) may not be conclusive.
(e) Contexualize the rule of law within the Law Triad.

                → Solution: the strings and the math.

The Triadic Unit Mesh - Nature's Construction Template

Triadic Unit Mesh

(a) All of medicine has one singular objective. Do you know what it is?
(b) Can you string the vertices of the triadic unit mesh in the context of the human body system?

                → Solution: the strings and the math.

Order Of Execution Time Function For An Algorithm

Order Of Execution Time Function For An Algorithm

Suppose the execution time function for an algorithm is:
f(x) = 3 + 8x + x2
Determine the order of f(x).

                → Solution: the strings and the math.

The Normal Probability Curve

The Normal Probability Curve

(a) Suppose that the frequencies of some data is normally distributed and figure 118.5 represents the probability curve. What is the probability of a value occurring between a and b?
(b) The weight of a large number of grapefruits were found to be normally distributed with a mean of 1 lb and a standard deviation of 3 oz. What is the probability that any one grapefruit has a weight between 1 lb 3 oz and 1 lb 6 oz?
(c) The average number of persons joining a certain queue in one minute is 2. What is the probability that 5 persons will join the queue in one minute?

                → Solution: the strings and the math.

Conditional Probabilities, Independent Events And Odds

Conditional Probabilities

A bag contains 12 balls (4 reds, 3 blues, 3 blacks, 2 greens) as indicated in Figure 118.4.
(a) Determine the probability of choosing a second blue ball from the bag given that the first ball chosen from the same bag is blue.
(b) Suppose there is a second bag containing 12 balls (4 reds, 3 blues, 3 blacks, 2 greens). Determine the probability of choosing a second blue ball from the second bag given that the first ball chosen from the first bag is blue.
(c) Determine the probability of chosing at least one blue ball, given both bags.
(d) What are the odds in favor of throwing a head in a single throw of a coin?
(e) What are the odds in favor of throwing at least one head on a single throw of two coins?

                → Solution: the strings and the math.

Probabilities - The Likelihood Of Events

Probabilities - The Likelihood Of Events

Figure 118.3 is a Venn Diagram for events A and B. Given the following information:
Probability of A union B = P[A ∪ B] = 0.7
Probability of A union B' = P[A ∪ B'] = 0.9. B' is the complement of B.
Probability of A = P[A].
Probability of B = P[B].
Determine P[A].

                → Solution: the strings and the math.

The Normal Frequency Curve

The Normal Frequency Curve

Figure 118.2 is the frequency distribution of a certain variable x. Is this frequency distribution a normal frequency curve if 78.9 % of the data lie within one standard deviation of the mean?

                → Solution: the strings and the math.

Statistical Triad - Grouping, Data Collection And Analysis, Inference

Statistical Triad - Grouping, Data Collection And Inference

Figure 118.1 illustrates the conceptual essence of statistic.
(a) What is statistics?
(b) Describe its triadic components.

                → Solution: the strings and the math.

Algorithmic Triad - The Basic Concepts Of Algorithms

Algorithmic Triad - The Basic Concepts Of Algorithms

Figure 117.1 illustrates the conceptual essence of algorithms. Assuming a problem has been properly defined, What is meant by Algorithmic Triad.

                → Solution: the strings and the math.

Stress-Strain Diagram For Gray Cast Iron

Stress-Strain Diagram For Gray Cast Iron

Figure 116.1 is a sketch of the stress-strain diagram (tension and compression) for gray cast iron. Modulus of elasticity, E = 12.5 x 106. Determine for tension and compression:
(a) The proportional limit.
(b) The modulus of resilience.
(c) percent elongation.

                → Solution: the strings and the math.

Modulus Of Resilience Of A Molded Phenolic Plastic

Modulus Of Resilience Of A Molded Phenolic Plastic

Data from a tension test to determine the elastic properties of a molded phenolic (sythetic resin) plastic are as follows:
Specimen diameter, d = 0.400 + 0.001 in.
Gage length, l = 1 + 0.01 in.
Load at the proportional limit, P = 500 + 20 lb
Elongation due to P, δ = 0.0030 + 0.0001 in.

(a) Determine the modulus of resilience, ur.
(b) Calculate the probable maximum relative error in ur.

                → Solution: the strings and the math.

True Stress - True Strain

True Stress True Strain

(a) The conventional strain in a member subjected to a tensile stress of 14,815 psi is 0.350. Calculate the true stress and the true strain. Assuming constant volume.

(b) The original diameter of a tension specimen is 0.505 inches (figure 115.4). At a certain load, the diameter is found to be 0.388 inches. Calculate the true and conventional strain at this point. Assuming constant volume.

What is the ratio of elastic strain to plastic strain?

                → Solution: the strings and the math.

Elastic Strain - Plastic Strain

Elastic Strain - Plastic Strain

The proportional limit of a member made of a type of steel is 30,000 psi. Modulus of elasticity of member is 30 x 106 psi. When this member is subjected to a tensile load of 45,000 psi, the strain is 0.0615 in./in. When this member is subjected to a tensile load of 60,000 psi, the strain is 0.2020 in./in.

What is the ratio of elastic strain to plastic strain?

                → Solution: the strings and the math.

Strain Energy Stored In An Elastic Material

Strain Energy Stored In An Elastic Material

A 15 in member is to be designed using a safety factor of 1.50, to withstand a tensile load of 6000 lb. The three choices of material available are:
(a) Aluminum Alloy, for which modulus of Elasticity, E = 10 x 106 psi and tensile stress σy = 52, 0000 psi
(b) Magnesium Alloy, for which modulus of Elasticity, E = 6.5 x 106 psi and tensile stress σy = 28,500 psi
(c) Molded Nylon, for which modulus of Elasticity, E = 410,000 psi and tensile stress σy = 8000 psi.

Calculate the total amount of strain energy stored by each member at the 6000 lb load.

                → Solution: the strings and the math.

ΔL, ΔV In A Columbium Member Under Tensile Load

Change In Dimensions Of A Cold-Worked-Columbium-Member-Under-Tensile-Load

A member made of cold-worked columbium is 15 inches long and has a rectangular cross-section 1/4 in by 3/4 in. This member is under a tensile load of 5,000 lb.
Modulus of elasticity, E = 22.7 x 106
poisson's ratio μ = 0.28.
Assuming elastic behavior, determine:
(a) total change in Length
(b) lateral strain
(c) total change in Volume

                → Solution: the strings and the math.

The Unidirectional Flow Of Genetic Information

The Unidirectional Flow Of Genetic Information

There is a Primiordial Equilibrium associated with every organism. Consequently, there exist in every organism the information necessary to maintain its primordial equilibrium.

How is this information used to maintain primordial equilibrium?
                → Solution: the strings and the math.

The Genetic Code - Quadric Alphabet, Triadic Words

The Genetic Code

The language of an organism's characteristics is composed from three-letter words derived from a four-letter alphabet.
(a) Determine from the illustrated table, the three-letter words (code) for the principal amino acids in proteins.
(b) Which of the three-letter words in (a) means house in the Yoruba Language?
                → Solution: the strings and the math.

Derivation Of The One Dimensional Wave Equation

Derivation Of The One Dimensional Wave Equation

There are three basic types of Linear Partial Differential Equations (PDEs). Parobolic PDEs, Hyperbolic PDEs and Elliptic PDES. The one dimensional wave equation is a hyperbolic PDE and is of the form:
utt = α2uxx ---------------(1)
where u(x,t) is the displacement of a point on the vibrating substance from its equilibrium position.
utt is the second partial derivative of u(x,t) with respect ot t
uxx (concavity) is the second partial derivative of u(x,t) with respect to x
α is the proportinality constant.

Show that the transverse vibrations of a string of length L (figure 114.8a) fastened at each end can be described mathemathecally by equation (1).
                → Solution: the strings and the math.

Geometric Interpretation Of Partial Derivatives

Geometric Interpretation Of Partial Derivatives

z = f(x,y) is the three dimensional surface illustrated in figure 114.7.
(a) What is the meaning of ∂f(x,y)/∂x?
(b) Supppose the variables x, y and z represent the length, width and height of a building respectively and the heat loss function for the building is:
f(x,y,z) = 11xy + 14yz + 15xz
Interprete ∂f(10,7,7)/∂x
(c) Suppose the production function of a manufacturer is:
f(x,y) = 60x3/4y1/4. Where x and y are units of labor and capital respectively
(i) What is the marginal productivity of labor for f(81,16)?
(ii) What is the marginal productivity of capital for f(81,16)?
                → Solution: the strings and the math.

Sound Navigation And Ranging - SONAR

Sonar

A sonar device sends sound waves into ocean water in order to determine the distance of a reflecting object in the ocean. If it takes 3 secs for the sound wave to make a round trip from the sonar device, how far down in the ocean is the object if the speed of sound in ocean water is 1530 m/sec?
                → Solution: the strings and the math.

Derivation Of Volume Of A Torus

Derivation Of Volume Of A Torus
                → Solution: the strings and the math.

The Doppler Effect

Doppler Effect

(a) What is the relationship between the frequency of the sound from the siren of the futuristic ambulance illustrated above and the frequency heard by an observer if both ambulance and observer are approaching each other at velocities va and vo respectively?
(b) If ambulance and observer are moving in opposite directions at velocities va and vo respectively?
                → Solution: the strings and the math.

Derivation Of Volume Of A Paraboloid

Derivation Of Volume Of A Sphere
                → Solution: the strings and the math.

Gaussian Lens Equation

Gaussian Lens Equation

Figure 114.4 is an illustration of the rays from an object being refracted by a convex lens with left radius of curvature R1, right radius of curvature R2 and focal length f.

(a) Express the focal length f, in terms of the radii of curvature and the refractive indices of the lens and the medium through which the refracted light travels.
(b) Calculate the distance of the image from the lens in terms of the distance of the object from the lens and the focal length of the lens
(c) Calculate the height of the image in terms of the height of the object, the distance of the object from the lens and the distance of the image from the lens.
                → Solution: the strings and the math.

Derivation Of Volume Of A Frustum

Derivation Of Volume Of A Frustum
                → Solution: the strings and the math.

Truth Table - The Truth Of A Implies B

The Truth Table

Mathematical statements used by mathematicians are usually provable. In other words, the mathematical statement: "volume of a sphere = (4/3)πr3" (where r is the radius of the sphere), has a mathematical proof.

(a)What is a mathematical statement?
(b) What is a mathematical proof?
(c) Explain the truth of A implies B
(d) What is a truth table?
(e) Suppose X makes the following statement to Y: "If it is hot (above 80o F), Maybelline goes braless." What scenario in the illustrated truth table presents X as a liar?
                → Solution: the strings and the math.

Derivation Of Volume Of A Sphere

Derivation Of Volume Of A Sphere
                → Solution: the strings and the math.

Volume Obtained By Revolving y = x2 About The X Axis

Derivation Of Volume Of A Cone
                → Solution: the strings and the math.

Harmonic Wavelengths Of A String Fixed At Two Points Distance L Apart

Harmonic Wavelengths Of A Fixed String Of Distance L

Sinusoids 1, 2 and 3 represent the first, second and third harmonics respectively, of a string fixed at two points distance L apart.

(a) Compare the octaves of 2 and 3 relative to 1.
(b)What is the harmonic wavelength of the fourth harmonic?
(c) What is the general equation that relates the harmonic wavelength of the string to the distance L, between the two fixed points?
                → Solution: the strings and the math.

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Cone
                → Solution: the strings and the math.

Derivation Of Volume Of A Cone

Derivation Of Volume Of A Cone
                → Solution: the strings and the math.

Frequencies Of Simple Sound And Complex Sound

Frequencies Of Simple Sound And Complex Sound

(a) The sound produced by a tuning fork (figure 113.4) is considered a simple sound. In general, the sound wave of a simple sound can be represented by the following simple sinusoid:
y = asin2πft --------------(1)
Where a is the amplitude of the sound wave, f is the frequency and t is time.
What is the amplitude and frequency of the simple sound represented by 10sin(π/2)16t?

(b) Sounds from musical instruments and the human voice are complex sounds so they are not representable by only the simple sinusoid of equation (1). However, intelligible sounds (simple or complex) are periodic even if they are not sinusoids. Consider the following complex sound:
y = 0.07sin480πt + 0.05sin760πt + ... -------------(2)
(i) What are the frequencies of the fundamental, first harmonic and first partial?
(ii) What is the frequency of the second harmonic?
(iii) Why is the frequency of a complex sound always that of the first harmonic?
(iv) What is the difference between natural frequency (resonant frequency) and fundamental frequency?
(v) Can humans hear an infrasonic or an ultrasonic sound?
                → Solution: the strings and the math.

Derivation Of Area Of An Ellipse

Derivation Of Area Of A Circle, Sector Of A Circle And Circular Ring
                → Solution: the strings and the math.

Derivation Of Areas Of A Circle, Sector Of A Circle And A Circular Ring

Derivation Of Area Of A Circle, Sector Of A Circle And Circular Ring
                → Solution: the strings and the math.

Derivation Of Areas Of A Trapezoid, A Rectangle And A Triangle

Derivation Of Area Of Trapezoid, Rectangle And Triangle
                → Solution: the strings and the math.

Light Amplification By Stimulated Emission Of Radiation - LASER

Light Amplification By Stimulated Emission Of Radiation

Its Tee again, a sharp 7 year old. "Come here Tee". Called Tee's mom. "Explain to me what is meant by monochromatic light, coherent light and LASER". Said Tee's mom. "Just excitation mom, just excitation". Said Tee.

Describe Tee's explanation.
                → Solution: the strings and the math.

Werner Heisenberg Uncertainty In The Location Of A Mosquito

Werner Heisenberg Uncertainty In The Location Of A Mosquito

The mass of the mosquito illustrated above is 150 mg. It is moving at a speed of 1.40 m/s. The certainty of its speed is within + or - 0.01. Calculate the uncertainty in the location of the mosquito.
                → Solution: the strings and the math.

de Broglie Wavelength As Determinant Of Velocity Of Electron

de Broglie Wavelength As Determinant Of Velocity Of An Electron

Assume the particle illustrated in Figure 22.13 is an electron moving with velocity v:
(a) How fast would the electron be moving in order to have a wavelength of 0.711 Å?
(b) Under certain conditions, the element molybdenum emits light with characteristic wavelength of 0.711 Å. What region of the electromagnetic spectrum do the emitted light belong?
(c) Name an important use for the light emitted by molybdenum.
                → Solution: the strings and the math.

Energy Of An Electron In A Quantum Energy Level

Energy Of An Electron In A Quantum Energy Level

Figure 22.12 presents four quantum energy levels of an arbitrary quantum mechanical system:
(a) Indicate the quantum energy levels associated with the transition that requires the most energy.
(b) Indicate the quantum energy levels associated with the transition that requires the least energy.
(c) Which transition will absorb or emit light with the longest wavelength?
(d) Which transition will absorb or emit light with the shortest wavelength?
(e) Suppose the electron of this quantum system is from a nucleus with atomic number 3. Compare the energy of its n = 4 to n = 3 transition with the energy of a n = 2 to n = 1 transition of an electron from a nucleus with atomic number 2.
                → Solution: the strings and the math.

Absorption Emission Spectra - The Fingerprints Of Elements

Absorption And Emission Spectra - The Fingerprints Of Elements

Figure 22.11 illustrates the spectral lines of five elements (A, B, C, D) and a celestial star E.
(a) Identify the elements with the spectral lines of A, B, C and D.
(b) What elements in the star produced the spectral lines of E?
(c) A gaseous substance is more likely to exibit a line spectrum while a liquid or solid is more likely to exhibit a continuous spectrum. Why?
                → Solution: the strings and the math.

Energy Levels Of Electrons As Determinants Of Emission Lines Of Electrons

Energy Levels Of Electrons As Determinants Of Emission Lines Of Electrons

(a) What are emission lines of electrons?
(b) Figure 22.10 shows the six energy levels contained in a given molecule. Determine the maximum number of emission lines one would expect to see in this molecule
                → Solution: the strings and the math.

Electron Behaviors

Electron Behaviors

(a) Table 22.1 highlights important concepts associated with an electron. Associate each concept with a behavior of an electron.
(b) Derive the equation for the De Broglie wavelength.
                → Solution: the strings and the math.

Wavelength Of Light Determines Color Of Light

Wavelength Of Light Determines Color Of Light

Figure 22.8 illustrates the different colors of the visible spectrum of light and their wavelength ranges. Suppose the frequency of light a potassium compound emits is 7.41 x 1014 Hz. What color in the visible spectrum would you expect to see when the potassium compound is heated in a Bunsen burner flame?
                → Solution: the strings and the math.

Dispersion Of White Light

Dispersion Of White Light

(a)Why did the white light that went through the prism in figure 22.7a disperse?
(b) List the dispersed lights in decreasing order of wavelengths.
(c) Show that the product of the frequency and wavelength of each of the dispersed light are the same.
(d) What color of light will be formed if the various colors are combined again?
(e) Figure 22.7b is a rainbow. The colors that constitute the rainbow also result from the dispersion of light. What replaced the prism in this dispersion.
                → Solution: the strings and the math.

Pigments Of An Object Determine Its Color

Pigments Of An Object Determine Its Color

"Come here Tee. Since you know light, explain to me in simple terms, how pigments determine the color of an object". Said Tee's mom. "Ok mom". Said Tee, a sharp 7 year old.

Describe Tee's explanation.
                → Solution: the strings and the math.

Light Polarization By Polarizing Filters

Light-polarization-by-polarizing-filter

It's Tee (a sharp 7 year old) again. "Mom, you're always squinting because of the sun's glare. Just use your polarizing sunglasses and you'll squint no more". said Tee. "You know I don't like wearing sunglasses. By the way how does the polarizing sunglasses stop me from squinting?" asked Tee's mom. "Light polarization mom, just light polarization". replied Tee.

Explain what Tee means by light polarization.
                → Solution: the strings and the math.

RADAR - Bistatic And Monostatic Range Calculations

RADAR Range Calculations

RADAR is an acronym for Radio Detecting And Ranging. Short- wavelength microwaves are used in radar to locate objects and monitor speed.

(a) Explain what is meant by bistatic, monostatic and quasi-monostatic radar.

(b) The basic quantity measured by most radars, is target range. Write the formula for calculating the target range for: bistatic and monostatic radar.
                → Solution: the strings and the math.

Photons Required To Heat Coffee In A Microwave Oven

Photons Required To Heat Coffee In A Microwave Oven

Microwave ovens use microwave radiation to heat food. Moisture in the food absorbs microwaves. Food becomes hotter as moisture in food becomes hotter.

How many photons must a microwave radiation with a wavelength of 11.2 cm produce, in order to heat 200 mL of coffee in a microwave oven, from 23o C to 60o C?
                → Solution: the strings and the math.

Photoelectric Effect And Einstein Photoelectric Equation

Photoelectric Effect And Einstein Photoelectric Equation

The photoelectric effect explained by the Einstein photoelectric equation established the particle nature of light.

(a) Sodium metal A is illuminated by a light source producing light of wavelength 650 nm. Sodium metal B is illuminated by a light source producing light of wavelength 325 nm (nanometer). Which light source imparted significant kinetic energy to the emitted electrons if the photoelectric threshold of sodium is 650 nm?

(b) A 0.01 Watts beam of light with wavelength 6500 Å (angstrom) which strikes a photoelectric cell is completely used in the production of photoelectrons. Determine the magnitude of the current that flows in the circuit of the photoelectric cell.

(c) What retarding potential would be required to stop the flow of photoelectrons in a photoelectric cell with sodium metal illuminated by light with wavelength of 325 nm, if 3.06 x 10-19 J is used to remove the electron from the metal?
                → Solution: the strings and the math.

Energy In One Quantum Of Light

Energy In One Quantum Of Light

Determine the amount of energy in one quantum of light energy if its wavelength is 650 nm (nanometer).
                → Solution: the strings and the math.

Electromagnetic Spectrum - Scaling Visible And Invisible Light

Electromagnetic Spectrum - Scaling Visible And Invisible Light

The wave nature of light presents the electromagnetic spectrum.

(a) What is a wave?
(b) Why are light waves electromagnetic?
(c) How is the electromagnetic spectrum scaled?
                → Solution: the strings and the math.

Refraction Of Light


Refractive Prism And Lens
Refraction And Total Reflection
Refraction Cone Refraction Air To Water

Its Tee again! "Mom, check out my periscope", said Tee a sharp 7 year old. "Interesting! So what concept did you use?" Asked her mom. "Total reflection, just total reflection". Tee replied.

Explain Tee's total reflection in the context of the refraction of light.
                → Solution: the strings and the math.

Reflecting Property Of A Paraboloidal Mirror

Reflecting Property Of A Paraboloidal Mirror

"Mom, check out my headlight", said Tee, a sharp 7 yr old. "cool! how did you do that?", asked his mom. "Conics, just conics", replied Tee.

Explain the conic concept Tee is referring to.
                → Solution: the strings and the math.

Law Of Reflection Of Light

Law Of Reflection Of Light

The ray model of light is an established simple assumption about the motion of light in the macro-realm (non-quantum realm). The ray model of light assumes that light travels in straight-line paths; light rays from luminous objects spread out in all directions and an image is formed when the light rays leaving the object from the same point meet. The ray model of light does not disrupt the particle-wave nature of light.

(a) Prove the law of reflection of light which states that the angle of incidence of a ray of light is equal to the angle of reflection.
(b) Indicate the position of the mirror image of a point O (figure 21.1) which is in front of a plane mirror.
                → Solution: the strings and the math.

1895-1911. 16 Years Of Open Brains

1895 - 1911 Sixteen Years Of Great Discoveries

The human story is primarily about survival and discoveries. Many persons young and old from various parts of the world have been contributors to the important story of great human discoveries. The discoveries indicated in 1895-1911. 16 years of open brains are important snapshots from the human Knowledge Continuum.

(a) Indicate the discoverers and primary underlying concepts of the discoveries highlighted in 1895-1911. 16 years of open brains.
(b) What is meant by open brain and stringed brain?
                → Solution: the strings and the math.

Quantum Electrodynamics (QED)

Photon Electron Interaction

Feynman's QED Summary was posited by physicist Richard P. Feynman (1918-1988).

(a) What is a photon?
(b) How are photons emitted or absorbed?
(c) Give examples of photons moving from place to place.
(d) Give examples of electrons moving from place to place.
                → Solution: the strings and the math.

Acid - Base Neutralization

Aicd-Base Neutralization

(a) 12.5 ml of a 50 ml solution of sulphuric acid containing 0.490 g of sulphuric acid completely neutralized 20 ml of a sodium hydroxide solution during titration. Determine the concentration of the sodium hydroxide solution.

(b) The hydrochloric acid (HCl) concentration in the gastric juice of a patient with duodenal ulcer is 80 x 10-3 M. The patient produces 3 liters of gastric juice per day and his doctor has prescribed a medication containing 2.6 g Al(OH)3 per 100 ml of solution, for the relief of excess stomach acidity (Al(OH)3 and Mg(OH)2 are common ingredients in medications designed to neutralize stomach acid). Determine the patient's daily dose of the prescribed medication that will neutralize the acid.
                → Solution: the strings and the math.

Acidity-Basicity Of An Aqueous Solution

Acidity-Basicity Of An Aqueous Solution

Assuming complete ionization, determine:
(a) the pH of 0.0001 N HCl (Hydrocloric acid)
(b) the pOH of KOH (potassium hydroxide)
                → Solution: the strings and the math.

Acid - Base Definitions

Acid-Base Definition
Tika is a bright chemistry student. She posits that the formation of a complex ion is an acid-base reaction in which the complex ion is the acid and the ligand is the base. What acid-base definition did Tika used to arrive at this conclusion?
                → Solution: the strings and the math.

Electrodes And Electrolytes Determination Given A Redox Reaction

Electrode And Electrolyte Determination Given A redox Reaction

Figure 15.7 is a conceptual sketch of the construction of a galvanic cell based on the following spontaneous reaction:

3Fe(s) + 2Au3+ --------> 3Fe2+ + 2Au(s)

(a) Determine electrodes A and B, and electrolytes A and B in figure 15.7.
(b) What is the sign of the change in enthalpy for this reaction?
                → Solution: the strings and the math.

Nernst Cell Voltage Equation

Nernst Cell Voltage Equation
Given the following Daniell cell (a typical voltaic or galvanic cell):

Zn|Zn2+||Cu2+|Cu

Determine the cell voltage (potential) at 25oC If the concentrations of the zinc ions and the copper ions are 0.50m and 0.20m respectively.
                → Solution: the strings and the math.

Oxidation-Reduction Reaction In A Voltaic Cell

Oxidation-Reduction Reaction In A Voltaic Cell

Figure 14.63 is a simple illustration of a voltaic cell called the Daniell cell. Container 1 contains zinc sulphate solution in which a zinc electrode is immersed; container 2 contains copper(II) sulphate solution in which a copper electrode is immersed. A conducting wire connects the electrodes through a volmeter. The ends of an inverted U-shape salt bridge are immersed in the solutions respectively.

(a) Describe the half-reactions of the chemical reaction in the cell then derive arithmetically, the whole oxidation-reduction reaction from the half-reactions.

(b) Determine the approximate reading of the voltmeter.
                → Solution: the strings and the math.

Electrolysis As Oxidation-Reduction Reaction

Electrolysis As Oxidation-Reduction Reaction

Figure 14.56 is a simple conceptual illustration of the electrolytic decomposition of molten sodium chloride.
(a) Describe the half-reactions of the chemical reaction then derive arithmetically, the whole oxidation-reduction reaction from the half-reactions.

(b) Using Faraday's Law of electrolysis, calculate the amount of Chlorine (in grams), a chemist could produced from molten sodium chloride (NaCl) if she uses a current of 1 ampere for 5 minutes?
                → Solution: the strings and the math.

Oxidation Numbers As Determinants Of Oxidation-Reduction Reactions

Oxidation Numbers As Determinants Of Oxidation-Reduction Reactions

Figure 14.49 presents a heterogeneous synthesis reaction of sodium and chlorine.
This chemical reaction is also an oxidation-reduction reaction. Why?
                → Solution: the strings and the math.

Homogeneous And Heterogeneous Reactions

Homogeneous And Heterogeneous Reactions

Figure 14.42 presents three chemical reactions. Which of the reactions are homogeneous and which are heterogeneous?
                → Solution: the strings and the math.

Arrhenius Rate Equation

Arrhenius Rate Equation

The activation energy for a process is 55,000 cal/mole. The rate of this process is known at 400oC.
What is the incremental temperature needed to double the rate?
                → Solution: the strings and the math.

Carnot Cycle And The Efficiency Of A Perfect Heat Engine

The Carnot Cycle

(a) Derive the equation for the efficiency of a perfect heat engine in terms of the temperatures of the primary heat reservoir and the secondary heat reservoir using the Carnot cycle.

(b) Determine the efficiency of a steam engine (heat engine) operated reversibly between a primary reservoir and a secondary reservoir at 35oC.
                → Solution: the strings and the math.

ΔG When Liquid Water Changes Phase To Vapor At SBP



What is the ΔG (change in Gibbs free energy) when liquid water changes phase to vapor at 100oC and 1 atm. Molar enthalpy of vaporization is 9720 cal.
                → Solution: the strings and the math.

ΔE when One Mole Of Liquid Chlorine Changes Phase To Chlorine Gas At SBP

What is ΔE (change in internal energy) when liquid chlorine changes phase to chlorine gas at standard boiling point, 284oK (vapor pressure 1 atm)? The enthalpy of vaporization of chlorine, Cl2, is 20.41 KJ/mole.
                → Solution: the strings and the math.

ΔSo, ΔHo, ΔGo In CO + Cl2 Reaction

Entropy, Enthalpy, Free Enegy Changes In CO+Cl<sub>2</sub> Reaction

Given the information in figure 14.21, determine:
ΔSo, ΔHo, ΔGo for the following reaction at 25oC:

CO(g) + Cl2(g) -------> COCl2(g)

                → Solution: the strings and the math.

Equilibriun Temperature Of Water Mixed With Ice

40 grams of ice at 0oC is mixed with 100 grams of water at 60oC.
Determine the final temperature of the water after equilibrium has been established.
Heat of fusion of water (H2O) = 80 cal/gram
Heat capacity/Specific Heat of water = 1 cal/gram degrees C.
                → Solution: the strings and the math.

Weight Of Ice Melted By Heat Liberated From Steam Condensation

Determine the weight of ice melted at 00C by the heat liberated when 100 grams of steam at 1000C condenses to liquid.
Heat of vaporization = 540 cal/g
Heat of fusion = 80 cal/g.
                → Solution: the strings and the math.

Molar Entropy Of Water

Phase Diagram Of Water

The molar entropy of ice at 0oC is given as 51.84 J deg-1 mole-1.

(a) What is the molar entropy of water at 0oC?
(b) What is the molar entropy of water at 25oC ?
                → Solution: the strings and the math.

Why Heat Of Vaporization Is Larger Than Heat Of Fusion

Why is heat of vaporization larger than heat of fusion?
                → Solution: the strings and the math.

Phase Diagram

Phase Diagram

A phase diagram is a graphical representation of the change of matter from one phase (solid, liquid, gas) to another.

Consider compounds Ma and Mb.

The properties of Ma are as follows:
Melting pt. = 00C at normal conditions, 1 atm (101325Pa)
Boiling pt. = 1000C at normal conditions, 1 atm(101325Pa)
Critical Temperature = 3740C, at 218 atm
Triple pt. =0.00980C, at 0.006 atm (611Pa)

The properties of Mb are as follows:
Melting pt. = -78.50C at normal conditions, 1 atm (101,325Pa)
Boiling pt. = -570C at normal conditions, 1 atm (101,325Pa)
Critical Temperature = 31.10C, at 7.39MPa
Triple pt. =-56.60C, at 518kPa

(a) Determine the identity of Ma and Mb and sketch their respective phase diagrams
(b) What is the implication of the triple point pressure of Mb being above 1 atm?
                → Solution: the strings and the math.

The Spectrum Of A Function

Spectrum Of A Function

                → Solution: the strings and the math.

Eigenfunction Expansions As A Solution Method For Non-Homogeneous PDEs



The Eigenfunction Expansion Method is one of the methods used to solve non-homogeneous PDEs. Consider the following IBVP for a one-dimensional heat flow in a laterally insulated rod of unit length:

PDE: ut = α2 uxx + f(x,t)    0 < x < 1;    0 < t < ∞

BCs: u(0,t) = 0;   u(1,t) = 0

IC: u(x,0) = g(x)     0 ≤ x ≤1

Determine the function u(x,t) by the eigenfunction expansion method.
                → Solution: the strings and the math.

Homogenizing Non-Homogeneous Time Varying IBVP Boundary Conditions



The solution of PDEs with the separation of variables method is only possible when the IBVP is linearly homogeneous. When the boundary conditions (BCs) are non-homogeneous, it is often desirable to transform them to homogeneous BCs. Consider the following IBVP for a one-dimensional heat flow in a laterally insulated rod of length L:

PDE: ut = α2 uxx    0 < x < L;    0 < t < ∞

BCs: u(0,t) = g1(t);  ux(L,t) + hu(L,t) = g2(t)

IC: u(x,0) = p(x)     0 ≤ x ≤L

Transform the non-homogeneous BCs to homogeneous BCs.
                → Solution: the strings and the math.

Boundary-Value Problems As Sturm-Liouville Problems



The following is a Sturm-Liouville problem

ODE: X"(x) + λX(x) = 0    0 < x < 1;

BCs: X(0) = 0;   X'(x) = 0
where ' implies first derivative and " implies second derivative

(a) What is a Sturm-Liouville problem?

(b) What are the eigenvalues and eigenfunctions of the given Sturm-Liouville problem?
                → Solution: the strings and the math.

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Figure 14.14 shows a one-dimensional heat flow problem. The bottom end of a laterally insulated unit rod is immersed in a water solution at a fixed reference temperature. The top end is also at the same fixed reference temperature. The initial boundary value problem (IBVP) of the heat flow problem is as follows:

PDE: ut = α2 uxx    0 < x < 1;    0 < t < ∞

BCs: u(0,1) = 0;  ux(1,t) + hu(1,t) = 0

IC: u(x,0) = x     0 ≤ x ≤1

Determine the function u(x,t) by the separation of variables method
                → Solution: the strings and the math.

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Many important physical phenomena can be modeled as problems of systems of partial differential equations (PDEs) or ordinary differential equations (ODEs). Usually, the mathematical expressions of the initial conditions (IC) and boundary conditions associated with a particular problem are stated with the PDEs or ODEs. The PDE, BC and IC, together constitute an Initial-Boundary-Value-Problem (IBVP).

Consider the laterally insulated one-dimensional copper rod with length L (figure 14.12(a)), the ends of which are enclosed in containers of liquids at temperatures described by the functions g1(t) and g2(t) respectively.
                → Solution: the strings and the math.

Fourier Series

Fourier Series
                → Solution: the strings and the math.

Derivation Of Heat Equation For A One-Dimensional Heat Flow

Derivation Of Heat Equation For A One-Dimensional Heat Flow

The flow of heat is a consequence of temperature gradient. Consider the one-dimensional rod of length L in figure 8.105. The following assumptions apply to the rod:
(1) The rod is made of a single homogeneous conducting material
(2) The rod is laterally insulated, that is, heat flows only in the x-direction.
(3) The rod is thin, that is, the temperature at all points of a cross section is constant.
(4) The principle of the conservation of energy can be applied to the heat flow in the rod.

(a) Derive the heat equation for a one-dimensional heat flow.
(b) How does the heat equation change if the rod is not laterally insulated, the surrounding is kept at zero, and the heat flow in and out across the lateral boundary, is at a rate proportional to the temperature gradient between the temperature u(x,t) in the rod and its surrounding.
                → Solution: the strings and the math.

Gibbs Free Energy As Predictor Of Chemical Reaction Spontaneity

Gibb's Free Energy As Predictor Of Chemical Reaction Spontaneity

When deviations from generally accepted rules occur, smart humans want to know why.
Generally, an exothermic reaction involves an increase in disorder. So, an exothermic reaction that involves a decrease in disorder (increase in order) is a deviation from expectation.
Generally an endothermic reaction involves a decrease in disorder. So, an endothermic reaction that involves an increase in disorder (decrease in order) is a deviation from expectation.
Theoretical physicist and chemist J. Willard Gibbs (A.D. 1839 - 1903) was one of the people who wanted to know why the deviations stated above exist. The answer he developed, introduced a quantity called free energy (now called Gibbs Free Energy in his honor).

(a) State Rudolf Clausius' mathematical definition of entropy
(b)Relate change in Gibbs Free Energy to Change in enthalpy, temperature and change in entropy
(c) Relate change in Gibbs free energy to chemical equilibrium constant
(d) Relate change in Gibbs free energy to standard cell potentials through the Nernst equation
)e) Explain the spontaneity of the reaction of the combustion of 2 moles of hydrogen gas despite the decrease in entropy.
(f) State Ludwig Boltzmann's mathematical definition of entropy.
                → Solution: the strings and the math.

Thermodynamic Stability As A Function Of Enthalpy Of Formation

Thermodynamic Stability As A Function Of Enthalpy Of Formation

Synthesis and decomposition reactions are two important chemical reactions. For example, the combustion of carbon (as coal) and the decomposition of the carbon dioxide that is the product of the combustion are synthesis reaction and decomposition reaction respectively (figure 11.2).
In general, a systhesis reaction has the following form:
element or compound + element or compound -------> compound
In general, a decomposition reaction has the following form:
compound -------> two or more elements or compounds Thermodynamic stability is the non-spontaneity of the decomposition of the product of a synthesis chemical reaction.

(a)What is enthalpy of formation?
(b) Show that the carbon dioxide that is the product of the combustion of carbon (as coal) is thermodynamically stable.
                → Solution: the strings and the math.

Activation Energy As Prerequisite For Activated Complex

Activated Energy As Prerequisite For Activated Complex
There is an energy hill all chemical reactions must climb inorder for the reactants in the chemical reaction to produce the desired products. The activation energy is the energy required to climb to the peak of this hill. The peak is the activated complex. It is a short-lived high-energy (excitations due to absorption of activation energy) zone where energetic collisions cause changes in the electron cloud of the colliding molecules and allow bonding rearrangement. Consequently, the reaction is able to slide down the hill as the products are being formed.

(a)Figures 11.1(a) and 11.1(b) are energy diagrams. Which diagram represents an endothermic reaction and which diagram represents an exothermic reaction?

Compare the activation energy required for exothermic reaction with that required for endothermic reaction.

Mechanical processes also need activation energy. Consider figure 11.1(c). A is a rectangular prism in a vertical position. Its weight is W and its center of gravity is C1 when in a vertical position. Suppose the prism is pushed slightly such that it tilts and its center of gravity changes to C2 and thereafter, falls on its own to a horizontal position where its center of gravity is C3. The vertical distance between C1 and C2 is y1. The vertical distance between C1 and C3 is y2.

(c) What is the activation energy required by the prism inorder to fall to its horizontal position?
                → Solution: the strings and the math.

Energy In A Mole Of Crystalline Solid

Crystalline Scandium
The above photo is of a sample of crystalline scandium (Sc, atomic number 21).
The atoms and molecules of a solid are in constant motion at ordinary temperatures eventhough there is equilibrium spacing between them. Thermal agitation due to heat causes the atoms and molecules to oscillate about their equilibrium positions oftentimes at very high frequency. The resulting kinetic energy acquired by the atoms and molecules constitute the thermal energy of the substance.
Consequently, both kinetic energy (thermal) and potential energy (structure) consitute the total energy in solids and liquids (this potential energy is not in gases).

(a) Suppose an individual atom of a crystalline solid behaves as a point mass. Calculate the total energy of the atom at 25oC.

(b) Calculate the total energy in one mole of the crystalline solid at 25oC.
                → Solution: the strings and the math.

Potential Energy Between Atoms As A Function Of Nucleic Spacing

Potential Energy Between Atoms
The bonding force F, between atoms may be expressed approximately as follows:

F(r) = A/rM - B/rN (N > M) -----------(1)
Where r, is the center-to-center spacing between atoms and A, B, M, and N are constants that vary according to the type of bond.
A/rM represents the attractive force while B/rN represents the repulsive force.

In general, the potential energy, U(r) between atoms is defined as the work capacity of interatomic forces for a given reference frame.

Hence, U(r) = ∫F(r)dr = ∫(A/rM - B/rN)dr -----------(2)

(a) Determine the expression for U(r) by integrating equation (2).
(b) Show that the curve U(r) in figure 10.8 has a minimum at equilibrium spacing, r0.
(c) What is the significance of U(ro)?
(d) Explain the meaning of the area under the curve U(r) from r0 to infinity.
                → Solution: the strings and the math.

Interatomic Force As A Function Of Nucleic Spacing

Interatomic Force
The bonding force F, between atoms may be expressed approximately as follows:

F(r) = A/rM - B/rN (N > M) -----------(1)
Where r, is the center-to-center spacing between atoms and A, B, M, and N are constants that vary according to the type of bond.
A/rM represents the attractive force while B/rN represents the repulsive force.

(a) Express the equilibrium spacing r0 in terms of A, B, M, and N.
(b) Derive another form for F(r) in which the only constants are r0, A, M, and N.
(c) From the equation for F(r) derived in (b) calculate the following:
(i) The spacing r1 for which F is maximum
(ii) The value of the maximum force Fmax.
                → Solution: the strings and the math.

Maximum Charge In Coulombs On A Bismuth Ion

Maximum Charge Of Bismuth Ion In Coulombs
The above photo is of a sample of Bismuth. The atomic number of Bismuth is 83 and its electron configuration is as follows:
1s22s22p63s23p6 4s23d104p65s24d105p66s24f145d106p3
Determine in Coulombs the maximum charge on a bismuth ion.
                → Solution: the strings and the math.

Properties Of Elements Due to Valence-Shell Electrons

Similar Valence Shell Electrons Similar Chemical properties
The electron configurations of seven elements of the Periodic Table are shown in Table 10.1. Which of the elements should have similar physical and chemical properties?
                → Solution: the strings and the math.

Electron Residency In Atomic Orbitals

Periodic Table Showing Sub-Shells That Are Filling
Figure 9.16 shows the order in which the electrons of atoms fill their orbitals. Use this information to determine the electron configuration of the following elements:
(a) Iron (b) Aluminum (c) Bismuth
                → Solution: the strings and the math.

Volume Of A Barrel

The Volume Of A Barrel
Determine the approximate volume of the barrel in figure 5.1, if the sides are bent to the arc of a parabola and D = 60 inches, d= 50 inches, and h = 120 inches
                → Solution: the strings and the math.

The Grocer And His Fruits

The Grocer And His Fruits
Suppose a grocer sells 2 apples for 5 cents and 3 oranges for 5 cents. He surmises that the simple arithmetic of selling 5 fruits for 10 cents results in the same average price. In other words, according to him, a sale of 2 apples and 3 oranges have the same average price as a sale of any 5 pieces of fruits for 10 cents.
Is the grocer right? What is the correct average price ?
                → Solution: the strings and the math.

Time Population A Equals Population B

The population of town A is 10,0000 and is increasing by 600 each year. The population of town B is 20,000 and is increasing by 400 each year. After how many years will the two towns have the same population?
                → Solution: the strings and the math.

Break-Even Point Of A Publisher

Love Royal And True
A publisher finds that the cost of preparing a book for printing and of making the plates is $5000. Each set of 1000 printed copies costs $1000. He can sell the books at $5 per copy. How many copies must he sell to at least recover his costs?
                → Solution: the strings and the math.

Circle Test For Deductive Reasoning

Circle Test For Deductive Reasoning
Radeen examined figure 1.1 and posited the following premises and deductive conclusion:
All college students are clever. All young boys are clever. Therefore All college students are young boys.
Evadin examined figure 1.1 and posited the following premises and deductive conclusion:
All college students are clever. All young boys are clever. Some clever college students are young boys.
Who has the correct deductive conclusion base on figure 1.1. Radeen or Evadin?
                → Solution: the strings and the math.

Conditional River Crossing With Wolf, Goat And Cabbage

Wolf, Goat, Cabbage
A wolf, a goat, and a cabbage are to be rowed across a river in a boat holding only one of these three objects besides the oarsman. How should he carry them across so that the goat should not eat the cabbage or the wolf devour the goat?
                → Solution: the strings and the math.

Select A Fruit

Select A Fruit
Container A, contains 7 apples. Container B, contains 7 peaches. You are asked to select a fruit from each of the following spaces formed by containers A and B:
(a) A ∪ (A ∩ B)
(b) A ∩ (A ∪ B)
(c) (A - B) ∪ B
What is the maximum number of apples you can select? What is the maximum number of peaches you can select?
Hint: simplify expressions before selecting. A ∪ B (fig. 7.87a) means the set of elements in A or in B or in both (if in both, counted only once). A ∩ B ( fig. 7.87b) means the set of elements in A and in B (elements common to both A and B). A - B (fig. 7.87c) means the set of elements in A but not in B.
                → Solution: the strings and the math.

Average Rate Of Work - Two Ditch Diggers

Average Rate Of Work - Two Ditch Diggers
A ditch-digger can dig a ditch in 2 days and another ditch-digger can dig the same ditch in 3 days. What is their average rate of ditch-digging per day?
                → Solution: the strings and the math.

Energy Stored In A Magnetic Field And Incremental Inductance

Energy Stored In A Magnetic Field And Incremental Inductance
Figure 16.2 shows the i - λ characteristics of an iron-core inductor:
(a) Calculate the energy and incremental inductance for i = 1 A.
(b) Given that the sinusoidal current i(t) = 0.5sin2πt and coil resistance is 2Ω, calculate the voltage across the terminals of the inductor.
                → Solution: the strings and the math.

Induced Voltage Of Coil In A Magnetic Field

Induced Voltage Of Coil In A Magnetic Field
A coil having 100 turns is immersed in a magnetic field that is varing uniformly from 80 mWb (milliWeber) to 30 mWb in 2 seconds. Determine the induced voltage in the coil.
                → Solution: the strings and the math.

Charge Moving In A Constant Magnetic Flux

Charge Moving In A Constant Magnetic Field
Figure 1.2 shows a charge q moving with velocity u (a vector) in a magnetic field with magnetic flux density B (a vector). Assuming that the field is a scalar field (i.e, it is spatially unidirectional).
(a) Express the vector force f in terms of the charge q, and the vectors u and B.
(b) What is the magnitude of f If u makes an angle θ with the magnetic field?
(c) Suppose the magnetic flux lines are perpendicular to a cross sectional area A (fig1.3). Express the magnetic flux ψ, of the field in terms of the flux density B.
Magnetic Flux Perpendicular To A Cross Sectional Area
(d) State Faraday's Law that relate magnetic flux φ to eletromotive force (emf), e.
                → Solution: the strings and the math.

3 Phase Power

3 Phase Power
Figure 8.56 is a Wye (or Y) configuration of a Balanced three phase AC circuit. Show that:
(a) The magnitude of the line voltages is equal to √3 times the magnitude of the phase voltages.
(b) No conducting wire is needed to connect nodes n and n'.
(c) If the 3 balanced load impedances are replaced with 3 equal resistances R, the total instantaneous power delivered to the balanced load by the 3-phase generator is constant.
                → Solution: the strings and the math.

AC Power Triangle

AC Power Triangle Figure 8.14 shows a simple AC circuit. Figure 8.14(a) is the time domain circuit while figure 8.14(b) is its power triangle. Given that v(t) = 16cosωt; i(t) = 4cos(ωt - π/6); and ω 377 rad/sec. Determine:
(a) The power factor, pf.
(b) The values of the real power P, the reactive power Q and the apparent power S of the power triangle.
                → Solution: the strings and the math.

Average AC Power

Average AC Power
Figure 8.7 shows a simple AC circuit. Figure 8.7(a) is the time domain circuit while figure 8.7(b) is its phasor form.
Given that the sinusoidal voltage and current of the circuit are as follows:
v(t) = Vcos(ωt); i(t) = Icos(ωt - θ); Determine:
(a) The average power of the circuit in the time domain
(b) The average power of the circuit in the frequency domain.
                → Solution: the strings and the math.

Center Tapped Transformer

Center Tapped Transformer
Figure 8.42 shows a center-tapped Transformer. The following information is given about the transformer:
Voltages and current are rms values.
Primary voltage = 4,800 V
Secondary voltage of 240 V is split (because transformer is center tap) into two voltages:
V2 = 120 V; V3 = 120 V.
Three resistive loads (R1, R2, R3) are connected to the transformer (connection not shown in figure 8.42).
R1 is connected to the 240 V line.
R2 and R3 are connected to each of the 120 V lines.
Determine the power absorbed by each of the loads, if:
Power absorbed by R2 = P2
Power absorbed by R1 = 5P2
Power absorbed by R3 = 1.5P2
Ccurrent through primary coil, I1 = 1.5625 A.
                → Solution: the strings and the math.

Ideal Transformer

Ideal Transformer
Figure 8.35 shows an Ideal Transformer. Show that: The apparent power of the primary coil equals the apparent power of the secondary coil.
                → Solution: the strings and the math.

Band Pass Filter

Frequency Response Of Band Pass Filter
Figure 7.56 shows a circuit of a simple RLC filter. Determine:
(a) The frequency response of the RLC filter in terms of the natural or resonant frequency.
(b) The bandwidth in terms of the natural or resonant frequency and the quality factor.
                → Solution: the strings and the math.

High Pass Filter

Frequency Response Of High Pass Filter
Figure 7.50 shows a circuit of a simple RC filter. Determine:
(a) The phasor form of the frequency response, H(jω) in terms of ω, R and C.
(b) The cutoff frequency of the RC filter.
                → Solution: the strings and the math.

Low Pass Filter

Frequency Response Of Low Pass Filter
Figure 7.49 shows a circuit of a simple RC filter. Determine:
(a) The phasor form of the frequency response, H(jω) in terms of ω, R and C.
(b) The cutoff frequency of the RC filter.
                → Solution: the strings and the math.

Phasor Form Of Periodic Signals

Phasor Form Of A Periodic Signal
Figure 8.25 shows the sum Vs(t) of the sinusoidal voltage signals V1(t) and V2(t).
Determine the phasor form of Vs given the following information:
V1(t): amplitude = 15; frequency = 377; phase angle = 45o.
V1(t): amplitude = 15; frequency = 377; phase angle = 30o.
                → Solution: the strings and the math.

Frequency Response Of Output Voltage At High And Low Frequency

Frequency Response
Figure 6.11 shows a circuit of the given elements. Determine the behavior of the frequency response of the voltage V0 at extremely high and low frequencies.
                → Solution: the strings and the math.

Random Signals

The signal v(t) from a voltage source is a binary waveform: it is either 0.5 V or -0.5 V.
The sign change has a 50-50 chance of occurrence within the interval of 1 μs. In other words, v(t) has an equal chance for positive or negative values within this interval.
What is the average and effective values of v(t) over a period of 5 secs?
                → Solution: the strings and the math.

Periodic Signals

Signaling is a ubiquitious phenomenon in the societies of cognitive beings.
(a) What is a signal?
(b) What is a periodic signal?
(c) Identify the following periodic waveforms:
(i) Sinusoids
(ii) Periodic Pulse
(iii) Sawtooth Waveform
                → Solution: the strings and the math.

Even Functions - Odd Functions

Even And Odd Functions
Figure 7.13 shows two periodic signals (blue and red). Which of the signals is an even function and which is an odd function.
                → Solution: the strings and the math.

Mind Warm Ups

The Universe is composed of matter and radiant energy. Matter is any mass-energy that travels with velocities less than the velocity of light. Radiant energy is any mass-energy that travels with the velocity of light.

The Point

The point "." is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
Functions
Conics
Ordinary Differential Equations (ODEs)
Vector Spaces
Real Numbers

More Strings (SiPjAjk)


Rocket Work Take That Load Up
Cable Work Bring That Load Up
Friction Work Bring That Moving Object To Stop
Escape Velocity
Signals From Mars
Speed Of A Satellite In An Orbit
Acceleration Of Object In Circular Motion With Constant Speed
Object Thrown Up From The Roof
Maximum Height Of A Projectile
Maximum Range Of A Projectile
Direction Of A Projectile
Difference Between Matter And Radiant Energy
Difference Between Mass And Weight
Einsteins Energy Formula Applied To Uranium-235 Detonation
Water At Bottom Of A Water Fall Warmer Than Water At Top
Air Plane In Flight-Head Wind Tail Wind
Couples In Equilibrium
Sound Travel In An Iron
Shell Fired At An Angle
Range Of A Projectile
How Deep Is The Well
Slow And Steady Average Speed Of Auto
Kings An Aces
Inclined Plane
Car Choices
Free Falling Pebble-2
Free Falling Pebble
Meridian Travel
Row Your Boat
Composition Structure Of Matter
How Matter Gets Composed
How Matter Gets Composed (2)
Structure Of Matter
Bond Length Bond Angle
Valence Shell Electron Pair Repulsion
Molecular Shape Orbital Hybridization
Sigma Bonds Pi Bonds
Non ABn Molecules
Molecular Orbital Theory

What is Time?
St Augustine On Time
Bergson On Time
Heidegger On Time
Kant On Time
Sagay On Time
What is Space?
Newton On Space
Space Governance
Leaders
Imperfect Leaders
Essence Of Mathematics
Toolness Of Mathematics
The Number Line
Variables
Equations
Functions
The Windflower Saga
Who Am I?
Primordial Equilibrium
Primordial Care
Force Of Being
Forgiveness

Blessed are they that have not seen, and yet have believed. John 20:29

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