*Its All about Pj Problem Strings (S _{i}P_{j}A_{jk}) -
7 Spaces Of Interest (S_{i}) and their associated Basic Sequences; 7 Pj Problems of Interest (PPI) and their Alleles (A_{jk})*

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, “*Don’t 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

"I have reduced physics to mathematics" - **Descartes**.

Strings (S_{i}P_{j}A_{jk}) reduce all knowledge to mathematics.

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.

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 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 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 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.

Figure 123.2 illustrates *an enhancement-mode NMOS*. The threshhold voltage, V_{T} = 2 V, The *source* terminal of the nmos is grounded and a voltage source V_{GG} = 3 V DC is connected to the *gate*.

Determine the operating state of the transistor for each of the following values of v_{D}:

(a) v_{D} = 0.5 V

(b) v_{D} = 1 V

(c) v_{D} = 5 V

→ Solution: the strings and the math.

*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.

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 V_{CC} = 5 V? Assume a *transistor-transistor logic* (**TTL**) as follows:

For V_{in}, logic low (0) = 0 V; logic high (1) = 5 V

For V_{out}, logic low (0) = 0 V; logic high (1) = 5 V

→ Solution: the strings and the math.

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):

V_{s} = 1 cos(6.28 x 10^{3}t) mV; V_{CC} =15 V

Current gain, β = I_{C}/I_{B} = 100 .

R_{1} = 68 kΩ; R_{2} = 11.7 kΩ;

R_{C} = 200 Ω; R_{E} = 200 Ω;

R_{L} = 1.5 kΩ; R_{s} = 0.9 kΩ;

Determine V_{CEQ} and the region of operation.

→ Solution: the strings and the math.

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 = R_{B} = 1 kΩ; V_{on} = 5 V; V_{off} = 0 V; current, I = 5 mA.
**Transistor**: V_{CC} = 5 V; offset voltage, V_{γ} = V_{BE} = 0.7;

Current gain, β = I_{C}/I_{B} = 95; V_{CEsaturation} =0.2 V.
**LED**: offset voltage, V_{γLED} = 1.4 V; I_{LED} > 15 mA; P_{max} = 100 mW.

(a)Determine collector resistance R_{C} 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.

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:

R_{1} = 100 kΩ; R_{2} = 50 kΩ; R_{C} = 5 kΩ; R_{E} = 3 kΩ;

V_{CC} = 15 V; offset voltage, V_{γ} = V_{BE} = 0.7;

Current gain, β = I_{C}/I_{B} = 100.

→ Solution: the strings and the math.

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.

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.

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.

Suppose the components of the Zener diode voltage regulator circuit of figure 121.11(b) have the following values:

V_{s} = 24 V; V_{Z} = 12 V; R_{s} = 50 Ω; R_{L} = 250 Ω;
**(a)** Determine the minimum acceptable power rating of the Zener diode.

Now suppose the values are changed as follows:

V_{s} = 50 V; V_{Z} = 14 V; R_{s} = 30 Ω; P_{Z} = 5 W.
**(b)** Determine the range of load resistances, R_{L} such that the diode power rating is not exceeded.

→ Solution: the strings and the math.

*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, V_{Z} = 8.2 V.

Allowable range of operational Zener current: 75 mA≤ *i _{Z}* ≤1 A.

Load resistance, R

Size R

→ Solution: the strings and the math.

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.

*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, v_{s} = 52cosωt V. Assume ideal diode.

→ Solution: the strings and the math.

The circuit of figure 121.7 (a) consists of an ideal diode (the black triangle), voltages and resistors (R_{1}, R_{2}, R_{3}). The values of the voltages are as indicated in the diagram. R_{1} = 5 Ω, R_{2} = 10 Ω, R_{3} = 10 Ω.

(a) Is the diode conducting current?

(b) If resistor R_{2} is removed from the circuit, will the diode conduct current?

→ Solution: the strings and the math.

*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.

Time In Between

Just Suppose

Why I Love You

Love Me

Step By Step - Now Available

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.

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.

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 σ_{x} = 12,000 lb/in^{2}

Vertical compressive stress σ_{y} = 15,000 lb/in^{2}

Lateral compressive stress σ_{x} = 9,000 lb/in^{2}

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.

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 in^{2}; modulus of elasticity = 30,000,000; allowable stress = 15,000 lb/in^{2}.

Sectional area of aluminum tube = 2 in^{2}; modulus of elasticity = 10,000,000; allowable stress = 10,000 lb/in^{2}

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.

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/in^{2}.

Coefficient of thermal expansion = 0.0000065/^{o}F.

Suppose bar is subjected to a drop in temperature of 50^{o}F. Determine maximum tensile stress in bar.

→ Solution: the strings and the math.

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.

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.

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.

(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.

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.

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.

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 n^{2} 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 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, ^{n}Σ_{k=1} = [n(n+1)]/2

(a) Prove, by induction, that, for every integer ≥ 5, 2^{n} > n^{2}.

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

→ Solution: the strings and the math.

The right triangle XYZ of figure 121.1 has sides of lengths x and y, and hypotenuse of length z. Its area is z^{2}/4. Using the Forward-Backward proof techniques, proof that triangle XYZ is isosceles.

→ Solution: the strings and the math.

(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.

(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.

Suppose the execution time function for an algorithm is:

f(x) = 3 + 8x + x^{2}

Determine the order of f(x).

→ Solution: the strings and the math.

(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.

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.

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.

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.

Figure 118.1 illustrates the conceptual essence of statistic.

(a) What is statistics?

(b) Describe its triadic components.

→ Solution: the strings and the math.

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.

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 10^{6}. Determine for tension and compression:

(a) The proportional limit.

(b) The modulus of resilience.

(c) percent elongation.

→ Solution: the strings and the math.

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, u_{r}.

(b) Calculate the probable maximum relative error in u_{r}.

→ Solution: the strings and the math.

(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.

The proportional limit of a member made of a type of steel is 30,000 psi. Modulus of elasticity of member is 30 x 10^{6} 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.

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 10^{6} psi and tensile stress σ_{y} = 52, 0000 psi

(b) Magnesium Alloy, for which modulus of Elasticity, E = 6.5 x 10^{6} psi and tensile stress σ_{y} = 28,500 psi

(c) Molded Nylon, for which modulus of Elasticity, E = 410,000 psi and tensile stress σ_{y} = 8000 psi.

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

→ Solution: the strings and the math.

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 10^{6}

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.

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 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.

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:
**u _{tt} = α^{2}u_{xx}** ---------------(1)

where u(x,t) is the displacement of a point on the vibrating substance from its equilibrium position.

u

u

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.

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) = 60x^{3/4}y^{1/4}. 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.

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.

(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 v_{a} and v_{o} respectively?

(b) If ambulance and observer are moving in opposite directions at velocities v_{a} and v_{o} respectively?

→ Solution: the strings and the math.

Figure 114.4 is an illustration of the rays from an object being refracted by a convex lens with left radius of curvature R_{1}, right radius of curvature R_{2} 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.

*Mathematical statements* used by mathematicians are usually provable. In other words, the mathematical statement: "volume of a sphere = (4/3)πr^{3}" (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 80^{o} F), Maybelline goes braless." What scenario in the illustrated truth table presents X as a liar?

→ Solution: the strings and the math.

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.

(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.

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.

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.

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.

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.

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.

(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.

(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.

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 10^{14} 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.

(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.

"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.

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 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.

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 23^{o} C to 60^{o} C?

→ Solution: the strings and the math.

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.}

Determine the amount of energy in *one quantum of light energy* if its wavelength is 650 nm (nanometer).

→ Solution: the strings and the math.

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.

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.

"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.

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.

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.

*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.

(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.

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.

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.

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

3Fe(s) + 2Au^{3+} --------> 3Fe^{2+} + 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.

Given the following Daniell cell (a typical voltaic or galvanic cell):

Zn|Zn^{2+}||Cu^{2+}|Cu

Determine the cell voltage (potential) at 25^{o}C If the concentrations of the zinc ions and the copper ions are 0.50m and 0.20m respectively.

→ Solution: the strings and the math.

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.

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.

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.

Figure 14.42 presents three chemical reactions. Which of the reactions are homogeneous and which are heterogeneous?

→ Solution: the strings and the math.

The activation energy for a process is 55,000 cal/mole. The rate of this process is known at 400^{o}C.

What is the incremental temperature needed to double the rate?

→ Solution: the strings and the math.

(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 35^{o}C.

→ Solution: the strings and the math.

What is the ΔG (change in Gibbs free energy) when liquid water changes phase to vapor at 100^{o}C and 1 atm. Molar enthalpy of vaporization is 9720 cal.

→ Solution: the strings and the math.

What is ΔE (change in internal energy) when liquid chlorine changes phase to chlorine gas at standard boiling point, 284^{o}K (vapor pressure 1 atm)? The enthalpy of vaporization of chlorine, Cl_{2}, is 20.41 KJ/mole.

→ Solution: the strings and the math.

Given the information in figure 14.21, determine:

ΔS^{o}, ΔH^{o}, ΔG^{o} for the following reaction at 25^{o}C:

CO(g) + Cl_{2}(g) -------> COCl_{2}(g)

40 grams of ice at 0^{o}C is mixed with 100 grams of water at 60^{o}C.

Determine the final temperature of the water after equilibrium has been established.

Heat of fusion of water (H_{2}O) = 80 cal/gram

Heat capacity/Specific Heat of water = 1 cal/gram degrees C.

→ Solution: the strings and the math.

Determine the weight of ice melted at 0^{0}C by the heat liberated when 100 grams of steam at 100^{0}C condenses to liquid.

Heat of vaporization = 540 cal/g

Heat of fusion = 80 cal/g.

→ Solution: the strings and the math.

The molar entropy of ice at 0^{o}C is given as 51.84 J deg^{-1} mole^{-1}.

(a) What is the molar entropy of water at 0^{o}C?

(b) What is the molar entropy of water at 25^{o}C ?

→ Solution: the strings and the math.

Why is heat of vaporization larger than heat of fusion?

→ Solution: the strings and the math.

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

Consider compounds M_{a} and M_{b}.

The properties of M_{a} are as follows:

Melting pt. = 0^{0}C at normal conditions, 1 atm (101325Pa)

Boiling pt. = 100^{0}C at normal conditions, 1 atm(101325Pa)

Critical Temperature = 374^{0}C, at 218 atm

Triple pt. =0.0098^{0}C, at 0.006 atm (611Pa)

The properties of M_{b} are as follows:

Melting pt. = -78.5^{0}C at normal conditions, 1 atm (101,325Pa)

Boiling pt. = -57^{0}C at normal conditions, 1 atm (101,325Pa)

Critical Temperature = 31.1^{0}C, at 7.39MPa

Triple pt. =-56.6^{0}C, at 518kPa

(a) Determine the identity of M_{a} and M_{b} and sketch their respective phase diagrams

(b) What is the implication of the triple point pressure of M_{b} being above 1 atm?

→ Solution: the strings and the math.

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: u _{t} = α^{2} u_{xx} + f(x,t)** 0 < x < 1; 0 < t < ∞

Determine the function

→ Solution: the strings and the math.

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: u _{t} = α^{2} u_{xx}** 0 < x < L; 0 < t < ∞

Transform the non-homogeneous BCs to homogeneous BCs.

→ Solution: the strings and the math.

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.

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: u_{t} = α^{2} u_{xx} 0 < x < 1; 0 < t < ∞

BCs: u(0,1) = 0; u_{x}(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.

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 *g _{1}(t)* and

→ Solution: the strings and the math.

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.

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.

*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.

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 C_{1} when in a vertical position. Suppose the prism is pushed slightly such that it tilts and its center of gravity changes to C_{2} and thereafter, falls on its own to a horizontal position where its center of gravity is C_{3}. The vertical distance between C_{1} and C_{2} is y_{1}. The vertical distance between C_{1} and C_{3} is y_{2}.

(c) What is the activation energy required by the prism inorder to fall to its horizontal position?

→ Solution: the strings and the math.

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 25^{o}C.

(b) Calculate the total energy in one mole of the crystalline solid at 25^{o}C.

→ Solution: the strings and the math.

The bonding force F, between atoms may be expressed approximately as follows:
** F(r) = A/r ^{M} - B/r^{N}** (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.

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

Hence,

(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, r

(c) What is the significance of U(r

(d) Explain the meaning of the area under the curve U(r) from r

→ Solution: the strings and the math.

The bonding force F, between atoms may be expressed approximately as follows:
** F(r) = A/r ^{M} - B/r^{N}** (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) Express the equilibrium spacing r

(b) Derive another form for F(r) in which the only constants are r

(c) From the equation for F(r) derived in (b) calculate the following:

(i) The spacing r

(ii) The value of the maximum force F

→ Solution: the strings and the math.

The above photo is of a sample of Bismuth. The atomic number of Bismuth is 83 and its electron configuration is as follows:

1s^{2}2s^{2}2p^{6}3s^{2}3p^{6}
4s^{2}3d^{10}4p^{6}5s^{2}4d^{10}5p^{6}6s^{2}4f^{14}5d^{10}6p^{3}

Determine in Coulombs the maximum charge on a bismuth ion.

→ Solution: the strings and the math.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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**.

(d) State Faraday's Law that relate magnetic flux φ to eletromotive force (emf), *e*.

→ Solution: the strings and the math.

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

→ Solution: the strings and the math.

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.

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.

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:

V_{2} = 120 V; V_{3} = 120 V.

Three resistive loads (R_{1}, R_{2}, R_{3}) are connected to the transformer (connection not shown in figure 8.42).

R_{1} is connected to the 240 V line.

R_{2} and R_{3} are connected to each of the 120 V lines.

Determine the power absorbed by each of the loads, if:

Power absorbed by R_{2} = P_{2}

Power absorbed by R_{1} = 5P_{2}

Power absorbed by R_{3} = 1.5P_{2}

Ccurrent through primary coil, I_{1} = 1.5625 A.

→ Solution: the strings and the math.

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.

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.

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.

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.

Figure 8.25 shows the sum V_{s}(t) of the sinusoidal voltage signals V_{1}(t) and V_{2}(t).

Determine the phasor form of V_{s} given the following information:

V_{1}(t): amplitude = 15; frequency = 377; phase angle = 45^{o}.

V_{1}(t): amplitude = 15; frequency = 377; phase angle = 30^{o}.

→ Solution: the strings and the math.

Figure 6.11 shows a circuit of the given elements. Determine the behavior of the frequency response of the voltage V_{0} at extremely high and low frequencies.

→ Solution: the strings and the math.

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.

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)

(ii)

(iii)

→ Solution: the strings and the math.

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* "**.**" 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

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