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Details: Published: Monday, 14 August 2023 22:58; Hits: 923

Force:

There are 4 fundamental forces or interactions in nature. All the motion that you see around us are due to forces. These forces may be friction, gravitation, push, etc. They ultimately belong to one of these 4 forces. It's easier to visualize these forces by considering a field, and then this field exerting forces. Force is always a vector as the effective force due to multiple forces is a vector sum of these forces. We are omitting the arrow sign on top of force for simplicity, but it should always be there to indicate it's a vector. The 4 fundamental forces or fields are:

Gravitational force: This is attraction force due to mass of an object, and it's range is infinite. Newton's formula for gravitational force (published in 1686) gives us the formula for this:
1. Newton's Gravitational Law: Force between 2 bodies of mass m1 and m2 as F = G*m1*m2/R^2, where G=Gravitational Constant of 6.674*10^-11 m^3/(kg*s^2). Here both F and R are vectors.
2. Gravitation Field is defined as field due to presence of mass M and is given as g = F/m = G*M/R^2. The other mass is removed in eqn here, as F =m*g where g is the Gravitation Field due to mass M,
3. The value of G wasn't determined at Newton's time. It was used to calculate forces relative to one another. It was in 1798 (>100 years later) that value of G was found out experimentally. It can be seen that this gravitational force is very weak (as G is a very small number), and is only relevant when at least one of the bodies is very large as moons, planets, stars, etc. What in the mass causes gravity or gravitational field is unknown. Even Newton hypothesized that cause of this power is unknown and contrary to sound science.
Electromagnetic Force: This is the attractive or repulsive force due to charge. Look under "ElectroMagnetism" section.

Newtonian Laws:

3 of Newton's Laws shaped human advancement, as his 3 laws were revolutionary and considered one of the greatest scientific advancements. His book "Philosophiæ Naturalis Principia Mathematica", published in 1987 in Latin, contained his 3 laws. An excerpt of his laws from the original Latin and translated to English is here => https://www.gsjournal.net/Science-Journals/Communications-Mechanics%20/%20Electrodynamics/Download/4537

Another link for the 3 Newronion Laws and their translation => https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/66%3A_Appendices/66.18%3A__Newtons_Laws_of_Motion_(Original)

Below are Newton's 3 laws, directly translated from his original book.

First law: Law of Inertia => Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.
Second law: Force and momentum => The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. We write it as F=m*a, but in reality it's F=d/dt(m*v) or Force is equal to the rate of change of momentum.
Third law: Action and Reaction: To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. Most important thing to note is that the 2 forces of action and reaction are acting on 2 different bodies.

Ex of force/work => To get an idea of how much 1N of force is, consider a 100g weight that you carry on your hand. That is 1N of force that you are applying to counteract earth's gravitational force. Now if you lift it up by 1m, that's 1J of work that you did. So, 1N of force and 1J of force look like very small amounts, as your body doesn't feel any tiredness doing this.

Pressure: Pressure is Force per unit area. i.e P = F/A. SI unit of P is Pascal (Pa) which is 1N/m^2. 1 N is the force equiv to holding 100g of weight. This doesn't sound like too much weight. Now if we spread this weight over 1m^2, that's almost negligible Pressure. 100g of weight over 100cm*100cm = 0.01g/cm^2 of weight is 1 Pa. It's more convenient to use 1kPa (1 kilo pascal) or 1 bar (=100 kPa). 1kPa is 10g/cm^2, while 1 bar is 1kg/cm^2. Atmospheric pressure due to air is measured in atm (atmospheric pressure), where pressure at sea level is defined as 1 atm=101,325 Pa, which is ~1 bar (To be precise, 1 atm is slightly greater than 1 bar). Pressure is also measured in inches/mm of Mercury (Hg) where pressure at sea level = 760mm = 29.9 inches (approximated to 30 inches) = 1 atm pressure.

In USA, pressure is measured in pounds of force per square inch (psi)—for example, in car tires. 1 lb=453.6g => 1 lb force = 453.6g*9.8N/kg = 4.445N. So 1 psi = 4.45N/(0.0254)^2=6897.5N/m^2 = 6.9kPa. => 1 bar = 100/6.9=14.5 psi. So, 1 atm=14.7 psi. Pressure readings in excess of atmospheric pressure are usually more relevant in measurements, so we use psig (pounds per square inch gauge) in such cases. PSIG is measured relative to atm pressure, i.e 1 PSIG = 1 PSI - 14.7, so at sea level PSIG =0. PSI measures pressure relative to vacuum.

Barometer: Barometers are used to measure pressure. We take a tube closed at end end, and fill it with liquid as Mercury or water. Now, if we invert and immerse it in container of same liquid, then liquid inside the tube will rise until the point where pressure due to weight of liquid (+ any air/vapor pressure inside the tube in the empty space above the liquid) inside is same as pressure due to air outside. Mercury rises to 760 mm (or 2.5 ft), while water rises to 34 ft (since Hg is ~13,6 times denser than water). P = h*ρ*g.

In pressure gauges (usually in HVAC applications), pressure is usually shown as PSIG. In a barometer, if the air inside the tube is at atm pressure, then the mercury level will be at 0 inside the tube. But if the air inside the tube is in perfect vacuum, then the mercury will rise to 30 inches. We take these readings inverted to indicate relative pressure and indicate it as inch Hg Pressure, So a pressure reading of "0 in Hg" means the pressure is same as atmospheric pressure, while a "-30 in Hg" pressure means absolute 0 pressure. These can also be shown in bar as "0 bar" for 1 atm pressure and "-1 bar" for absolute 0 pressure. However, showing these on linear scale is hard, because we can never go to 0 vacuum. We keep on going down by a factor of 10 or something infinitely. So, a log scale is preferred in many gauges, where very precise measurement is required.

Micron unit: Since pressure becomes very low close to vacuum, we use another unit called micron in pressure gauges to indicate very low level of pressures. 1 mm Hg of pressure is divided into 1000 parts called microns. So, 1 micron pressure = 0.001 mm of Hg pressure. At 1kPa (or 0.01 bar), water boils at 10C. We usually try to get to this vacuum level, when we want to get rid of moisture in any system (as at this pressure, liquid water will get converted to steam, and get sucked out via vacuum pump). 100 kPa = 760 mm of Hg, so 1kPa=7.6mm Hg or 7600 micron. As a thumb rule, going to 1% of atm pressure starts the process of water to boil at room temp. People in HVAC industry accept a pressure of 1000 micron or lower in order to guarantee that no water is present in the evacuated system. Usually 500 micros or lower is what people shoot for when vacuuming an HVAC system. These kind of low pressures can only be read by "micron gauge", which are very expensive ($200 for branded ones)

1 Pascal = 7 micron of Hg (implying 1 micron is extremely low pressure, since 1 Pascal is already too low of pressure)

1000 micron = 1 mm of Hg pressure = 0.02 psi

100K micron = 100 mm of Hg pressure = 2 psi or 0.1 atm (this is the highest pressure that a micron gauge will read, since it's designed to read vacuum pressure and NOT high pressure)

Link => https://vacaero.com/information-resources/vacuum-pump-technology-education-and-training/633-understanding-vacuum-measurement-units.html

Manometer: These are similar to Barometer, and are used to measure pressure of gases. Manometers are used in measuring blood pressure using device called a sphygmomanometer (Greek sphygmos = “pulse”).

Details: Published: Friday, 11 August 2023 13:21; Hits: 837

Passive Elements = Resistors, Capacitors, Inductors

Almost all the matter around us has the 3 passive elements built as intrinsic part of that matter. These 3 elements are Resistors, Capacitors, Inductors. They arise due to physical properties of resistance, capacitance and Inductance that gives rise to these 3 passive elements respectively.

Resistance:

Mobility of electron:

What gives rise to resistance in any material? Well, it's the cloud of electrons present in any material that starts drifting in presence of an applied voltage. These electrons are the ones present in outer orbit around each atom. For conducting materials, there is a sea of electrons (instead of each electron tied to a specific atom), moving in random direction, which is relatively free to drift in presence of an electric field. When we apply a n Electric field, these electrons experience F = q*E and start accelerating as per Newton's law. They hit the atom lattice, and scatter off and lose some of their energy. Then it again starts accelerating in same direction, and this process continues. The final result is that the electron moves with a finite average velocity, called the drift velocity V_d . This causes an electric current in the direction of the Electric Field. This motion is lot smaller than normally occurring random motion. If these electrons went uninterrupted, they would keep on accelerating for ever, but in a material, they can't.

F= m*a=e*E (where m is mass of electron, e is it's charge)

mdv/dt = e*E => Δv = eEΔt/m => Avg drift velocity (V_d) = a*τ = (e*E*τ)/m where τ=mean free time (i.e avg time between collisions).

So, V_d = -μ*E, where μ=e*τ/m and is known as mobility of electron. mobility is avg_velocity/electric_field. The concept of mobility is only relevant for moving charges inside materials, as in a open space, electrons will just keep on accelerating for ever in an electric field. But in a material, we see that drift velocity is directly proportional to E, and that constant is called mobility. SI unit of mobility is m^2/(V.sec) but is almost always expressed in cm^2/(V.sec)

I = -n*(A*L)*e/t = -n*(A*L)*e/(L/V_d) = -n*(A*L)*e*V_d (n=electron density per unit volume, Length L of distance covered in time t)

J=I/A = -n*e*V_d = -(-n*e*μ*E) = n*e*μ*E

Conductivity (σ) / Resistivity (ρ)

Conductivity of a material is defined by ohm's law as: J = σ*E where J=current density=I/A (I=current, A=Area, Unit of J is A/cm^2), σ = conductivity of material and E=Electric field thru that material. Current is defined as # of electrons flowing thru an area per unit time

J = -q*n*<v> = -q*n*µ*E = σ*E where σ = -q*n*µ = n*q²*t/m, where n=number of electrons crossing per unit area

Resistivity (ρ) = 1/σ = inverse of conductivity.

For a material with uniform cross section A and length L, E=V/L (In a medium with no source/sink point, electric field is constant. It it weren't, charges would start accumulating at the point where electric fields are different)

=> I/A = σ*V/L

=> Resistance R = V/I = ρ*L/A => This is the classic Ohm's law that is stated in books, but it applies only to special case of homogeneous material with uniform cross section.

Units of ρ are Ω-m. The value of ρ comes by experimentally measuring it for different materials. Metals are good conductors and have lower ρ.

Cu=1.7*10^-8 ohm-m (Cu less resistive than Al and used in most conductors now, including wires on chip)

Al=2.6*10^-8 ohm-m

Resistivity is also written in units of ohm-mm^2/m, so Cu resistivity is 17 mΩ-mm^2/m => For a wire with area=1mm^2, Resistance is 17mΩ/m. Or for 1km of 1mm thick wire, Resistance is 17Ω. Gauge (AWG) is used in transmission wires to specify the thickness and current carrying capacity of wire (see in Power Transmission section).

Ohm's law: J = σ*E => σ = n*e*μ

Semiconductors: For metallic materials, electrons are the only carriers. But for semiconductors, we have another carrier which is opposite of electron (i.e a +ve charge). But where do these +ve charges come from?

So, conductivity of any material is related to the mobility of electron in that conductor as well as the concentration of free electrons. That is why different metals with same number of free electrons may have different resistivity as electron mobility might be different between them, depending on the lattice scattering. Thus electron mobility depends on following:

Lattice structure: Depending on the lattice structure, scattering (or collision) may occur sooner, which will cause lower electron mobility. That is why mobility varies as you cut the material along <110> or <111>. Basically, electrons can travel not only perpendicular to lattice, but also at an angle to the lattice structure resulting in more or less hinderance to their movement. In semiconductors, this is even more true, and materials are crystallized differently to give different mobility thru them.
Velocity saturation for electrons may occur. Electrons don't keep going faster and faster with increasing Electric field. At some point, sufficiently high drift velocity is reached, and velocity maxes out. This is called Velocity Saturation (v_sat). v_sat is on the order of 1×10⁷ cm/s for both electrons and holes in Si. It is on the order of 6×10⁶ cm/s for Ge. So, basically μ starts decreasing as velocity sat approaches, and finally goes to very low values.
Temperature affects electron mobility => With increasing temperature, phonon concentration increases and causes increased scattering, thus lowering the carrier mobility. In semi Conductors (Si, Ge), μ ∝ 1/T^2. The power to T isn't exactly but varies based on material and whether it's electrons or holes. So, conductivity is supposed to decrease with increasing Temperature (as mobility decreases), but at the same time, carrier concentration may increase due to higher Temperature (true for semiconductors)

Typical electron mobility at room temperature (300 K) in metals like gold, Cu, Silver is 30–50 cm²/ (V⋅s). Carrier mobility in semiconductors is doping dependent. In Silicon (Si) the electron mobility is of the order of 1,000, while in germanium it's around 4,000.

Determine ρ of Cu using mobility:

1 mole of Cu has weight = 64g => 1 atom of Cu has weight = 64g/(6.02*10^23 ) = 10^-22 g.

Density(Cu)=9g/cm^3 => # of atoms in 1cm^3 = 9g/(10^-22) = 10^23 atoms. Assume 1 free electron per Cu atoms

σ = n*e*μ = 10^23*1.6*10^-19*40 cm²/ (V⋅s) = 0.6*10^6 => ρ = 1/σ = 1.6 * 10^-6 ohm-cm = 1.6*10^-8 ohm-m, which matches closely to the ρ shown above (1.7*10^-8 ohm-m)

Capacitance:

Capacitance of a material is defined by gauss's law.

We saw insulators above which don't conduct electrons that well (i.e electrons drift very slowly in response to Electric field). Within insulators, we have a subset of materials which will get polarized when an Electric field is applied. This is called permittivity of the material. The electrons in such materials don't drift, but instead just shift slightly causing an internal Electric field. Such materials are called dielectric. A perfect dielectric is a material with zero electrical conductivity.

Dielectrics: https://en.wikipedia.org/wiki/Dielectric

Atomic level: Every material is made up of atoms. Atoms contain a =ve proton and -ve electrons around it. In presence of electric field, the charge is distorted slightly with electrons moving in the opposite direction to the direction of electric field. In insulator type materials (called as dielectric), these electrons don't flow from one end to other, but instead shift just slightly. This creates a dipole for each atom (think of it as separated +ve and -ve charges at 2 ends with some distance b/w them). When the electric field is removed, the atom returns to its original state. This shift of electrons from their average equilibrium positions, causes dielectric polarisation and creates an internal electric field that reduces the overall field within the dielectric itself (the applied electric field minus the dielectric polarisation induced Electric field). This tendency to polarize is referred to as the permittivity of the material, an is expressed as:

D (displacement field) = ε . E (electric field) where ε is the permittivity of material. Vacuum has a permittivity of ε₀ =

A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material.

Details: Published: Wednesday, 02 August 2023 15:24; Hits: 1053

Car Buying: Electric Vehicle (EV) vs Internal Combustion Engine (ICE)

Which Car should you buy? One that runs on Battery (Called an Electric Vehicle or EV) or one that is traditional, still running on Internal Combustion Engine (ICE) which burns gasoline (petrol/diesel). Then there are hybrid cars. These are the cars that have an Electric battery as well as gasoline tank (a mix of EV and ICE as they have both). Hybrids are divided into Conventional Hybrids and Plug in Hybrid Electric Vehicles (PHEV). The conventional Hybrid car runs 100% on gasoline, but uses the battery pack to provide to provide extra boost where needed. Thus, the range of car in miles per gallon improves, as some of the energy is provided by Battery. The battery doesn't need to be charged separately, as Car's alternator keeps charging this battery. On the contrary, in PHEV, the car is mostly run by the battery, and gasoline is used only when battery is almost out of Juice. However the range of car on Battery is not too large. It ranges from 10 miles to 50 miles, before switching to gasoline. They are on purpose designed for smaller range on battery, so that battery pack won't be too large.

Cost of Car:

Let's look at the cost side. Pure EV sell for around $40K, while similar ICE can be had for $20K.

Cost of upkeep

Depreciation

Cost of gas vs electricy

Also, it's not cheaper to get electricity compared to gasoline. Gasoline costs $4/gallon, while electricity cost is $0.15/KwHr (Avg for residential pricing).

Link on msn comparing price of gas vs electricity => https://www.msn.com/en-us/autos/news/most-cars-still-cost-more-to-charge-than-to-fill-up-with-gas/ar-AA1eDQ0R

Details: Published: Sunday, 30 July 2023 14:56; Hits: 1225

Home maintenance and Repairs

As a homeowner, you will soon find that there are a lot of things that will breaking every year. If you go out to hire someone for each and every repair, it will easily cost you 1000's of $, as any service is very expensive in USA. You will probably give the same hourly rate as what you make in hiring someone,and that is if you lucky. So, we'll save that money and try to learn to do it ourselves. You can think of the money saved as extra non taxable pay. The learning that you gain as DIY is un measurable.

Garage:

The one thing that goes bad in your garage is your garage door. Installing a garage door by yourself is NOT a DIY. I would be surprised if your house didn't come with a garage door already installed. Most of the times, there will be issues with the automatic garage door opener (Chamberlain Brand for ex). At that point, you might have to replace it or fix it, which is certainly a DIY. Or fix the sensors on the side, or issue with the wireless button on the side of the garage outside your house. These are all DIY, and don't cost much.

Plumbing:

Plumbing is the one of the highest cost repair after the AC. Plumbing and AC are 2 things, that you can't do much about. With an AC, may be you can live for a couple of days/weeks, while you shop around, but plumbing issues need to be taken care of immediately.

Cracked Pipes: There are both copper and PVC (plastic) pipes carrying water in and around the house. The drainage pipes taking waste water out of your house are all PVC pipes. Most of the pipes going under the ground are also PVC pipes. It's easy to get cracks in these PVC pipes 9usually the ones in your yard). Number 1 reason is someone just hit the pipe while shovellingor digging. Calling a plumber isn't required, as this can be fixed for couple of dollars.

How to fix PVC pipes in your yard => https://www.youtube.com/watch?v=x7NWmdlvVIQ

Electric:

Installing lights/Fans:

Very easy to install a fan, if the box is already there in place. It will take good 2 hours to do it, but is well worth the effort. You can get a very good fan with lights for about $50-$100.They regularly go on clearance for 75% off at HomeDepot and Lowes, usually after summer. So, no reason to pay full price.

Details: Published: Sunday, 30 July 2023 02:04; Hits: 752

Questions and interrogative words:

Interrogative words are very essential in any language to form a question. Even if you don't know any other word, you can just use an interrogative word, and get the message across (i.e Where Bathroom?). In spanish, we add question mark ¿ ..... ? around a question. In English, the verb and pronoun placement is interchanged. i.e "You are tall" becomes Are you tall? when phrased as a question. In Spanish, there's lot of flexibility around the order.

There are multiple ways to form a question. Let's explore 3 of them

No change in order (keep SVO order): This is the most common way for questions in Spanish. It's different order than what we use in English for asking Questions. Here, there's no difference b/w question and answer, the order in both of them is the same. By keeping SVO order, the emphasis is on the 1st word in sentence, which is the subject. Does Juan read books => Here we translate "Juan reads books", and just raise our inclination at the end, to pose it as a question. So, translation is ¿Juan lee libros? => We are placing emphasis on Juan.
Change to English order (VSO order): By changing to VSO order, the emphasis is still on the 1st word in sentence, but here it is the subject. This is how we ask questions in English. Only subject and verb are interchanged. Object remain unchanged no matter where it appears in sentence (it may not necessarily be at the end, it may may be in beginning too). Does Juan read books? => Does read juan, the books? => So, translation is ¿lee Juan libros? => We are placing emphasis on Read (or lee).
- ex: Which book would you want? => Here "you want" is changed in order to "want you". So, ¿Qué libro quiere usted? (Here object "libro" is in front, remains unchanged)
- ex: What did the man order? => Here "man ordered" is changed to "ordered the man". ¿Qué ordenó el hombre? (Here there is no object)
Informal way: Add "right?" at end: Juan read books, right? OR Juan read books, didn't he? Here we translate the answer as in case 1, but add "no/right" at the end to make it a question.
- No?: ¿no? => Juan read books, didn't he? => translation is Juan lee libros ¿no?
- Right/true? => ¿verdad? (i.e Your name is Jose, right? => ¿tu es nombre Jose, verdad? ). es verdad => It is true (or right).

In Spanish, most of the times pronoun is not explicitly used. In such cases, both "You are good" and "Are you good?" translate to same thing, except for question mark. ex: You are good => Eres Bueno. But Are you good => ¿Eres Bueno? is also translated the same, as we omitted the word "tu" for you in both of them. With "tu", it would be ¿Tu Eres Bueno?

Link => https://www.thoughtco.com/asking-questions-spanish-3079427

Common Interrogative words: Shown below are common words used for questions. One thing to note that all such words shown below have accent on the vowel. Without the accent, the meaning changes.

A verb usually follows the interrogative word, which makes it easy to understand. Noun/pronoun follows the verb. ex: What is the capital of Peru? => ¿Qué es la capital del Peru? => Here, es verb comes right after interrogative word.
When the verb needs a subject other than the interrogative, the subject follows the verb (VSO order). In short, verb always follows the interrogative word. ex: Why did he go to the city? => ¿Por qué fue él a la ciudad? Here él follows the verb. With multiple nouns, some nouns may come before verb, but the noun associated with the verb comes after the verb , i.e How many dollars does he have? => ¿Cuántos dólares tiene el?
ex: Where does your grandma go on Fridays? => ¿Dónde/ADónde va tu abuela los viernes? (It's translated as => where go, your grandma, on Fridays?)
ex: What do you do on Mondays? => Que haces (tú) los lunes? (What do you, on Mondays?)

english	spanish	ex
what	¿Qué? => pronounced "kay" NOTE: que (w/o accemt on e) is used as "to" in verb "have to".	What is that? => ¿Qué es eso?
which or which one	¿Cuál / Cuáles? => pronouned kual / kuales, (add es for plural) NOTE: In spanish, "which" is used in a lot of places where "what" should be used, there's no hard rule about it. Just remember this when interpreting.	Which is your car? – ¿Cuál es tu carro? What is your profession? => ¿Cuál es tu profesión? (NOTE: Cuál used even though it's what?) What is the price? => ¿Cuál es el precio?
when	¿Cuándo? => pronounced kuaando ("when=Cuándo" and "how many=Cuánto" are similar translations)	When is your birthday? – ¿Cuándo es tu cumpleaños? When is Independence day? – ¿Cuándo es el día de la independencia?
how many or how much	¿Cuánto / Cuántos? =>for masculine noun (singular/plural) ¿Cuánta / Cuántas? =>for feminine noun (singular/plural) These act like adjective, and need to match gender and number of the noun. Sounds like count.	How much coffee do you have? - ¿Cuánto café tienes? (café is singular maculine)
how	¿Cómo? => NOTE the accent on o. NOTE: ¿cómo? is also a common way to say “huh?” in Spanish	How are you? – ¿Cómo estás? =>This is the most common way to greet someone. See in "general greetings" below.
where	¿Dónde?	Where are you? – ¿Dónde estás? (temporary condition, so estar). I'm at home => estoy en casa Where is the store? – ¿Dónde esta la tienda? (where is something => this is temporary condition)
who whose = of who	Who => ¿Quién/ Quienes? => pronounced kee-en / kee-enes. (add es for plural). Remember as hindi "kaun" which means who (Quién sounds like kaun) Whose => ¿De quién/ De quienes? (of who)	Who is she? – ¿Quién es ella? Who are those people? – ¿Quiénes son esas personas? Whose TV is that? – ¿De quién es el televisor? Whose shoes are those? – ¿ De quiénes son esos zapatos?
why / because	¿Por qué? / Porque => pronouned por-kay "why" in spanish is 2 separate words (por+que), while "because" is just single word (porque). Basically, if you combine Por+que as 1 word (no accent on e), it becomes "because". Por translates to "because of", while qué translates to what => Porque (accent left out when combining) => because of what => why? Por also translates to "for" and many other words, but it also means "because of".	Why are you sad? – ¿Por qué estás triste? Because I’m sick – Porque estoy enfermo Why not? => ¿Por qué no?

NOTE: Several of these interrogatives can be preceded by prepositions: a quién (to whom), de quién (of whom), de dónde (from where), de qué (of what), etc.

Personal "a" in questions: "a" is also added in front of a question. This happens when a question will require an answer with a personal A in response. Then we need to add it in front of the interrogative word. personal "a" can be thought of as English "to" in a loose sense.

Examples:

¿A Dónde vamos? => Where are we going? "vamos" means "we go", so ¿Dónde vamos? means "where we go?". Response will be "we go to Mexico" or "vamos a mexico", which requires "a" in response (since verb "go" is being done to mexico)
¿A quién crees que engañas? – Who do you think you are fooling? A nadie. – Nobody. Here response is "someone" which is a person.
¿A cuántas clases faltaste? – How many classes did you miss? Falté a cinco clases. – I missed five classes. Here "verb" miss is being done on classes, which is inanimate, so not sure why "a" used in response.

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