It got me thinking about the formulas I use on a daily basis at work. Funny enough as I wrote them down I found there were 10 main formulas I use all the time. 10? That sounds like it calls for a top 10 list... the nerdiest top 10 list ever!
Granted these are mainly mechanical formulas but since we all know that mechanical engineers are the best kind I figured no would would begrudge me. :)
On to the geekiness...
1. F = ma
Force equals mass times acceleration. The granddaddy of all physics formulas. From this one equation you can derive all the rest.
2. V = √(2ad)
This describes the final velocity of an object falling from height d. This of course ignores wind resistance and terminal velocity but works fine for say an anvil dropping on a cyote's head.
3. k=1/2mv2
So now you need to know the kinetic energy with which said anvil hits said coyote's head. This equation gives you that energy. Also useful for computing the force with which two object collide. This equations also illustrates why a 70mph crash is so much more devastating than a 35 mph crash as the velocity components is squared.
4. S α 1/T
While this is not really a formulas it states that speed is inversely proportional to torque. For example with a 30:1 gearbox you'll decrease the speed of the motor by 30 times but increase the motor's torque by 30 times. This equations is what makes your 10 speed bicycle work. When going up a hill yo have to pedal faster but you create more torque to get up that hill.
5. Pv=nRT
This is the Ideal Gas Law. It states that the Pressure times the volume of an ideal gas is equal to the number of moles of the gas times the gas constant time ther temperature of the gas. A lot of gases are close enough to "ideal" for this equation to work. While most rarely use this form of the equaiton can can easily derive:
P1V1=P2V2
Which shows that the pressure times the volume of a gas is constant. As pressure increases, the volume must decrease or
P α 1/V.
Also it shows that
PV/T=C
This illustrates that as the temperature of a gas increases the pressure and/or volume must increase. This is useful for back computing the temperature change in a gas as it is rapidly compressed or expanded. Think about that can of compressed air you use for cleaning out your computer. This explains why it's so cold coming out of the can.
6. Combinations & Permutations:
There are actually 4 equations for combinations and permutations, but I'm only going to focus on two. The important things to remember is that in combinations order is not important (think a lottery) where in permutations, order IS important.
For combination with no repetition in the objects (for example a lotteries):
C=n!/(r!*(n-r)!)
For those unfamiliar with the factorial ! notation it simple means n! = n * (n-1) * n-2) * (n-3)...etc.. You'll find a factorial button on most scientific calculators.
When you select "r" objects (or numbers) from a possible set of "n" objects (or numbers). Take for example a lottery with 53 possible numbers where you select 6 numbers. There are then
(53!)/(6!*(53-6)!)) = 22,957,480
possibilities. Your odds are 1:22,957,480 of winning. Which is why lotteries are taxes on the mathematically challenged ;)
For permutation with repetitions (for example a combination lock on a safe) the formula is:
P=nr
So a standard Master Lock combination lock (really should be called a permutation lock) has 40 positions and you have to choose 3 of them to unlock it. It then has:
40^3 = 64,000 possible solutions. Assuming each ones takes you 5 seconds it would take you 3.7 days working 24 hours a day to try them all. Forgot your PIN number to your ATM card? Only 10,000 possible solutions.
7. F=kx
This is known as Hookes law and describes the forces generated by stretching an elastic material (in engineering terms a LOT of things are considered elastic including steel). Hooke states that the force required to stretch an object distance x is that distance times the spring constant of the material. This can be used in the simplest form to determine the force that is exerted by a spring that has bee compressed some distance. Hooke's law is the basis for determining stresses, strains and deflections in beam members hence is the forumlas upon whihc all FEA software is built. So elegant..
8. V=IR
Ohm's Law. The only non-mechanical equation on the list. This states that the voltage of a circuit is equal to the current times the resistance. As resistance goes up, current goes down for a constant voltage. The cool things about this equation is that it's analogous to fluid flow in pipes. If you change out voltage for pressure and current for flow you end up with a working equation that states that as the pipe size is reduced (increasing R) that you need to increase the flow to achieve a constant pressure. Who says electrical and mechanical engineers can't get along.
9. a2+b2=c2
Can you name the famous movie in whihc this equation is incorrectly cited? (answer at the bottom of the post). Pythagoras, the Greek mathematician derived that the sum of the squares of two sides of a right triangle are equal to the square of the hypotenuse. It is the basis of trigonometry and geometry.
10. y=mx+b
While this equation is probably the simplest of the bunch it's also one of the most used. This is the equations of a straight line. the value of Y on the Cartesian plane is equal to the value of x times the slope constant (m) plus the Y intercept (b). The y intercept is the value at which x=0.
The cool things about this equation is how it can be used. For example recently I received a price for a length of conveyor 50" long and another price for the same conveyor $100 long. Using these two equation and two unknowns I was able to determine the base price of the conveyor and the amount per inch. So then I was able to determine how much ANY length of conveyor would cost.
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So that wraps up the top 10 list. Let me know if you see any mistakes (I'm not the best typist in the world).
Oh yeah and the movie with mangled Pythagorean theorem? The Wizard of Oz. When the scarecrow gets his brain he states
The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.
Of course we all know it's a RIGHT triangle, not an isosceles. Yeah, yeah.. nerdy!
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