Circuits Complex Analysis Equation Simplification

You want a common denominator to simplify things.

I replaced jw x 10^-5 with the variable X, so I didn’t have to write it as many times.

You can factor out a term later on that cancels.

10^5 * 10^-5 = 1
complex_simplification

Tutoring Physics, some basics about circuits

We looked at circuits with resistors. Saw current, voltage, and resistance. The circuits included branches.

Current is the flow of electrons. Electrons have a negative charge.

Used Ohm’s law, V = IR and moved the variables around.

Current flows in an optimal combination when it comes to a branch. Electrons tend to flow where there is less resistance but the presence of more electrons creates more resistance.

Current flow is much like water flow through pipes. Though different in a few ways.

Used 1/R1 + 1/R2 = 1/R for two resistors in parallel.

Circuit Branch with Resistors, Crocodiles and Spock

Made this video as an analogy for circuit branches with resistors.

“How cold would a sphere of silver with a radius of 5 meters be in order to freeze the ocean upon contact?”

Short answer is “no”.

There is a lower limit to how cold something can be, absolute zero.

It looks like the specific heat of silver is 0.233 J/gm K ( at 20 C). Let’s say it stays that value throughout this process, that should be fine.

The density of silver is about 10490 kg/m^3

A sphere of that size would be 523.598775598 m^3

Multiply the density by the volume to get a mass of 5492551.15602 kg

Multiply that by the specific heat (careful with the units) and you get

Energy per degree for the silver:

1279764419.35 J/K

The ocean across the world varies a bit in temperature.

Salt water freezes at about -2° C

Absolute Zero is −273.15° C

A change in temperature of about 271° for the silver sphere to absorb the energy needed to reach equilibrium with freezing salt water.

Energy to bring the silver to -2°

1279764419.35 J/K * 271

346816157644 J

(that could be absorbed by the silver)

How much water could the sphere freeze?

Let’s say some ocean water is 10° C. It would decrease by 12° to freeze.

Specific heat of water, 4179 J/kg

We will take the energy the silver could absorb and figure out much water at this temperature would transfer that energy to change its temperature to -2 C

346816157644 J = 4179 J/kg X

X = 82990226.7633 kg

About 8,000,000 kg, 8000 m^3 (20 m x 20 m x 20m)

The Great Lakes have about 22,700 cubic kilometers of water.

One cubic kilometer of water is 1000000000000 kg

You could lower the temperature of 8000 m^3 water with the silver sphere to -2° C. But at that point its still not frozen, you would use the latent heat of fusion to figure that out.

But no where near being able to freeze the ocean. Maybe a small pond.

“What are some tips on how to solve Integration problems using U-Substitution?”

Question from Quora

u-sub-integration-1

There are basically two terms here, the 5x and the 1- x^2 in parentheses.

The general idea for u-substitution is to make u something that when you take the derivative of it can be substituted for what is left over (with a little manipulation).

So if you chose 5x as your u, then du would be 5dx which you could not easily substitute.

If what you choose seems to not be working, either you made a mistake or you should try a different option.

For this problem choosing u = 1 -x^2 seems to work better. You can probably even take the derivative of that in your head to figure out in advance if it will work well.

u-sub-integration-2

At that point it becomes much easier to solve.

“Why is sin(70°) the same as sin(110°)?

From Quora,

Here’s a diagram.

Each is 20°away from 90°. If you think in terms of the ratios of the sides of the triangles for the angles, SOHCAHTOA, then the opposite sides are the same and the hypotenuse is the same. So the ratio is the same for each.

sin_70_sin_110

“Imagine you are a layperson with a basic liberal arts education who wants to learn quantum mechanics on his/her own: What steps should s/he take?”

From Quora,

The first book in college (as a physics major) I used for Quantum Mechanics was the book by Griffiths.

Introduction to Quantum Mechanics (2nd Edition)

David J. Griffiths: 9780131118928: Amazon.com: Books

Seems like you can get a paperback copy for under $20, hardcover is a bit more.

You say you have a college-level overview of math/sciences. I’m not sure exactly what that entails. Does it include calculus?

I took the class with Griffiths book in my third year as an undergraduate.

A book that will help you understand the necessary mathematics is,

Mathematical Methods in the Physical Sciences

Mary L. Boas: 9780471198260: Amazon.com: Books

This is assuming you know some calculus. If you do not know calculus, you should also get a calculus book. There are quite a few options. Also some lessons about calculus on Khan Academy.

You should also know some chemistry. I had only taken a single quarter of chemistry (10 weeks) in college before I took Quantum Mechanics. Later I took a second quarter.

I read through the Griffiths Quantum Mechanics book the summer before taking the class. It’s interesting and you can understand a little by just reading it. But you need to work the problems to understand it better. Having that book could be a first step since then you can know more of what is involved in learning the subject.

Tutoring Physics Sound Waves

Some notes from tutoring today,

His class is studying sound waves right now.

We looked at questions related to a lab. One question had to do with the accuracy of finding the speed of sound using either standing waves in a tube or an echo. We reviewed the equation for velocity and how to graph it which was relevant for two of the problems.

Fundamentally, sound waves are expansions and compressions of air or another medium. We often graph them like sine waves, but those waves represent density.

The sounds of instruments are really combinations of different levels of frequencies. In a similar way, the tone of voices is made of a combination of frequencies.

Flutes and clarinets change the tube length by covering or uncovering holes in the tube.

Even though clarinets and flutes are about the same length, the flute functions like an open-open tube and the clarinet functions like a closed-open tube.

“My 8 year old grandson asked, “Could all the water on earth put out the sun?”

Saw this question on Quora and thought it was interesting.

The sun is hot enough to have nuclear fusion.

But let’s pretend that it was just a fire like we would have burning wood on Earth and compare the sizes of the Sun and the Earth.

The radius of Mercury is 2,440 km

The radius of Venus is 6,052 km

The radius of the Earth is 6,371 km

The radius of Mars is 3,390 km

The radius of Jupiter is 69,911 km

The radius of Saturn is 58,232 km

The radius of Uranus 25,362 km

The radius of Neptune is 24,622 km

The radius of the Sun is 696,300 km

The radius of Pluto is 1,186 km

This drawing is fairly accurate for comparing the sizes of the planets and the Sun. Pluto is the hardest to see since it’s the smallest so I labeled it. I went ahead and labeled Jupiter, the largest planet, as well in addition to the Earth and the Sun.

Sun_planets

If you imagined that you had a fire that was an ordinary size and you had some water that was the same ratio compared to the fire as the Earth compared to the Sun, you would think that the water would simply evaporate when it got near the fire but not extinguish it or really take away much energy at all from the fire.

And you could say that the sun is much hotter than ordinary fires.

“What is the square root of 96 in radical form?”

Like Terry Moore said, it actually is pretty easy to work with in the original form,

√96.

Sometimes it is better to ‘simplify’ by getting a coefficient multiplied by a smaller number within the radical. Other times, you’re really not simplifying, but making it more complicated. This is probably one of those times where leaving it as √96 is better.

If you had something like √196, you would want to change that form since it turns out that 196 is a perfect square.

Regardless, I’m assuming this is a problem for a class and your teacher/text book wants you to do the following……

You could start by factoring it in small steps within the radical sign.

If you find factors that are perfect squares, you can take them out of the radical.

You can find large factors if you want, but you can also start with small factors.

Probably the easiest number to factor out is 2 since 96 is an even number.

96/2 = 48.

square root 96 radical form

48 is also an even number, so you can factor another 2.

Now you have √[(2*2)*24]

If you can factor out larger numbers, it can save you a little time.

So from here let’s factor out a 4.

√[(2*2)*(4*6)]

= 4 √7