Problem #2

One of the most important pieces of equipment in biomedical labs is the ultracentrifuge. It’s used for separation of compounds consisting of particles of different masses through the process of sedimentation (more massive particles sink to the bottom). Instead of relying on the acceleration of gravity to accomplish sedimentation, an ultracentrifuge utilizes the centripetal acceleration from rapid rotation to speed up the process. Some ultracentrifuges can reach centripetal acceleration values of up to 106g (g=9.81 m/s2)

Ultracentrifuge L-100 XP

Problem:

If you want to generate 840,000g of centripetal acceleration in a sample rotating a distance of 23.5 cm from the ultracentrifuge’s rotation axis, what is the frequency you have to enter into the controls? What is the linear speed with which the sample is then moving?

Post your answers and solutions as a comment.

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Problem #1

Think of a track on a compact disc (CD); The track is a spiral, originating at an inner radius of r1=25 mm and terminating at an outer radius of r2=58 mm. The spacing between successive loops of the track is a constant, Δr=1.6 μm.

Problem: What is the total length of this track?

Solution:

Notice that I haven’t used calculus for solving this problem.

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An Introduction to the Quantum Mechanics (an article by Milad Ebrahimpour)

One of my articles titled “An Introduction to the Quantum Mechanics” Seems to be unachievable by the website it was uploaded to. Anyways, here’s another copy of it uploaded on Google Docs:

PDF

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Auguries of Innocence

by: William Blake

To see a World in a Grain of Sand

And a Heaven in a Wild Flower,

Hold Infinity in the palm of your hand

And Eternity in an hour.

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Uniform motion on a straight line

When we’re speaking about a uniform motion on a straight line, we could get to three equations that are usable for such physics problems.

Since it’s a uniform motion, we can say:

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Velocity-Time equation:

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Acceleration equation:

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Darkness on the Edge of the Universe

A New York Times article about Dark Energy By BRIAN GREENE

The New York Times

Brian Greene

IN a great many fields, researchers would give their eyeteeth to have a direct glimpse of the past. Instead, they generally have to piece together remote conditions using remnants like weathered fossils, decaying parchments or mummified remains. Cosmology, the study of the origin and evolution of the universe, is different. It is the one arena in which we can actually witness history.

The pinpoints of starlight we see with the naked eye are photons that have been streaming toward us for a few years or a few thousand. The light from more distant objects, captured by powerful telescopes, has been traveling toward us far longer than that, sometimes for billions of years. When we look at such ancient light, we are seeing — literally — ancient times.

During the past decade, as observations of such ancient starlight have provided deep insight into the universe’s past, they have also, surprisingly, provided deep insight into the nature of the future. And the future that the data suggest is particularly disquieting — because of something called dark energy.

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f(x)=x-2[x], find: f(f(x)).

Problem: f(x)=x-[2x], find: f(f(x)).

I posted this problem on Facebook but non of my friends seemed interested to answer or at least think about the problem. Today I worked it out myself. It was not hard at all!

Solution:

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Properties of Integer Part

1.

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2.

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3.

     Using the second property: clip_image006

4.

     Using the third property: clip_image008

5.

     Using the fourth property: clip_image010

            How to get there? Here’s how: clip_image012

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10.

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11.

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12.

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©2011, All rights reserved.

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From the birth to the death of the universe

This article is the intellectual property of Milad Ebrahimpour. You may not reproduce, edit, translate, distribute, publish or host this document in any way without the permission of Milad Ebrahimpour.

In this article, I want to take you to a journey which will tell you how the universe is the way it is and how it came to be like this.

Everything, according to our present physics, commenced from a massive explosion. What do I mean using the term everything? Time and space are merely considerable after the bang. What was before that? Before the big bang, all of the physical laws we know fall apart. Was there a god before then? You can say that, because we have no clue of rejection! I believe that we are capable of understanding what was before then, if we concentrate more on black holes. (Just a simple conception for now!). According to math, it’s believed that at the beginning, everything that you can see (e.g. galaxies, stars, planets, you and me)  and everything that you cannot see (e.g. dark energy and dark matter) were kept inside a super tiny dot, which is named the singularity. Shame on us that we don’t even know what made the singularity to bang! Of course, superstring theory has got some notions to explain the explosion moment of the big bang. It suggests that you can think of the big bang as the collision of two parallel universes; two parallel universes hit each other, a lot of energy comes inside our universe, gets transferred into matter and makes up all the visible and invisible matter in our universe. But still black hole and having nothing before that seems more understandable for most of the people!

At the moment of the big bang, entropy was so high (it was highly ordered). But as the time was passing, the entropy was going higher and higher. Entropy says that we can define time as the increase in entropy (disorder). The most simplex exemplar is to think about the reason of this question: Why does an egg splatter and why doesn’t it unsplatter?! By the burst of the singularity, time and space get meaning. Besides, small particles of matter and anti-matter get released and as they hit each other in pairs, they get disappeared. Were matter particles more than anti-matter particles or vice versa? To answer that question, first answer this question: Are you reading these words or not? Yes you are, then matter particles were more! We’ve got to thank the preponderance of matter particles, cause in the other situation, there would be no us.

Because of gravity, early particles began attracting each other and forming the early child stars. The more massive the stars were getting, the more pressure was forced to the core of the child stars. Let’s talk about some entropy again: The formation of stars makes us think of entropy going down. But actually it’s not. As it’s lowering the entropy by gathering particles from here and there, it’s also increasing entropy by the energy it gives out in form of heat. Energy gets out, E=mc^2, then matter gets out and entropy becomes high. But how does it give out energy? Well, as I mentioned earlier, the more massive the child gaseous star was getting, the higher the pressure and temperature was getting in the core of it. At the time that the temperature reached to 15(10^6) degrees Celsius, the nuclear fusion reaction would start working in the core. In nuclear physics, nuclear fusion is the process by which two or more atomic nuclei join together, or “fuse”, to form a single heavier nucleus. This is usually accompanied by the release large quantities of energy. Normal stars, create their energy by the process of hydrogen fusion – the process of fusing two hydrogen atoms to create one helium atom. Energy is created because a helium atom weighs slightly less than the two hydrogen atoms, and the excess mass is converted into energy, as related by, again, Einstein’s famous equation E=m*c2. After about ten billion years, a normal star has converted approximately 10% of its hydrogen to helium. Although this might seem as though it could still undergo hydrogen fusion for another 90 billion years, this is not the case. Remember that there are immense pressures at the core of stars, and it is only because of these pressures that the fusion can occur — in a fixed volume, increased pressure leads to increased heat. Outside of the range of pressures there is still mostly hydrogen, but it cannot be used because the pressures are not high enough to initiate fusion.The helium core begins to contract, and the outer layers expand and cool, glowing redder. The star is now called a red giant. At this point, helium fusion begins. The star was previously unable to fuse the helium; however, now that the core has contracted, the added pressure is enough to fuse helium into heavier elements. Simultaneously, hydrogen fusion also occurs at this point in a shell around the helium core, for pressures there have also increased enough for hydrogen to fuse. Life expectancy from here on is about 100,000,000 years. After this time, the red giant is made of mostly carbon. The next fusion process would be to fuse the carbon into iron. The problem in this star is that there is not enough pressure in the core to do this. Because the outward pressure of energy is no longer maintained, the core collapses and sends a shockwave outwards, and the star’s outer layers are cast off in a planetary nebula, with the resulting core becoming a white dwarf. The core is made almost of pure carbon (like coal), and glows white because it still possesses a lot of left-over heat. It now also has much less mass because it has shed its outer layers, and any planets it has would move to much farther orbits or be completely ejected from the system, if they had not been engulfed by the star in the expanded red giant phase. The white dwarf is destined to drift in space for millennia as it slowly cools. Most have an approximate size of the Earth (8,000 km (7,500 miles) diameter), and has a density such that a matchbox’s worth would weigh about as much as an elephant. It has a maximum weight of 1.4 solar masses. As it cools, it will grow dimmer, and will eventually become a black dwarf – a frozen lump of carbon floating though space.

But what if the star was not a normal star like our sun? I’m not going to details this time, cause it may make some of you bored. If the star is less than about 9 (but more than 1.4) solar masses, the core will collapse into a neutron star – a star made entirely of neutrons. If the star possesses more mass, it will continue to collapse into a black hole – a point of theoretically infinite density that possesses such a strong gravitational pull that not even light can escape its pull. Another momentous fact about a black hole is that it has got the highest entropy in the universe we know. Why? As I said previously, while the entropy decreases by the gravitational pull among the particles, it also increases by the energy it gives out. But what about the black holes? They allow nothing to escape their gravitational pull, thus, they keep the energy inside them, making their entropy go higher and higher:

S_{\text{BH}} = \frac{kA}{4\ell_{\mathrm{P}}^2}

When these stars explode and the outer layers get released freely in space, sometimes, these outer layers form planets like our own earth. Sometimes they’re tremendously supermassive themselves that the core become supermassive black holes and the outer layers form  galaxies orbiting around the center which are the supermassive black holes themselves.

How are we so sure about the big bang? Because of a couple of reasons we have in hand. The most important one is that all of the galaxies are getting far from each other. After the big bang we can expect such a behavior in our universe. It’s like blowing a balloon; when you do that, each point on the balloon’s surface recedes from any other point on the other part of balloon’s surface. Actually, the following image can help you visualize the expansion of the universe after the big bang, vividly:

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Image source: http://en.wikipedia.org

On basis of our understanding of the past and present universe, we can think about its future. The prime question is whether gravitational attraction between galaxies will one day slow the expansion and ultimately force the universe into contraction, or whether it will continue to expand and cool forever. The current rate of expansion and the average density of the universe determine whether the gravitational force is strong enough to halt expansion.

\Omega \equiv \frac{\rho}{\rho_c} = \frac{8 \pi G\rho}{3 H^2}.

Where ρ represents the actual density and the ρc represents the critical density c=(1.1) * 10^(-26) kg per cubic meter). With Omega less than 1, the universe is called “open” (forever expanding). If Omega is greater than 1 the universe is called “closed”, which means that it will contract and eventually collapse in a Big Crunch. In the unlikely event that Omega = 1, the expansion of the universe will slowly slow down until it becomes virtually invisible, but it won’t collapse.

Some scientists think it not impossible that the universe is going back and forth between eras of expansion and contraction, where every Big Bang is followed by a Big Crunch. Stephen Hawking pointed out the possibility that such a universe must not necessarily start and end in singularities, think of questionable points in spacetime where physical theories, such as General Relativity, break down while energy and density levels approximate infinity. Although everything points towards Big Bang, the future reversal and contraction of the universe is rather uncertain. Big Crunch is at most a hypothesis, because only about 1% of the matter needed for Omega=1 can be observed.

In spite of this, galaxies and star clusters behave as if they would contain more matter than we can see. It is almost as if these objects were engulfed by invisible matter. This “dark matter” is one of the open questions in cosmology. Dark matter is thought to make up 23% of the universe.

Now that we are aware of definite extinction of the human race in the far future (If we will be able to travel transgalactical to avoid sooner extinction), what should we do? I think we had better think of transuniversal travels (travels to the parallel universes)!

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Problem #9

In how many different ways can you divide 26 cards through 52 cards in two equal halves, between two people? (To see the solution, click on the Continue Reading)

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