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Archive for May, 2012

Orbits of Different Mass Bodies

The scenario from last post (Orbits of Same Mass Bodies) was a very simple one. Lets kick it up a notch, and change one of the masses.

Two orbiting bodies of different mass.

 

The first thing to notice is that the center of mass is no longer in the middle of the system. This point is called the barycenter, and can be calculated with the following formula:

M is the total mass of the system, and m and r are the mass and position of the individual bodies. This gives the position in any number of dimension, from any origin.

For example, if you have a 1 kg mass orbiting 1 m away from a 2 kg mass, this formula tells us that the heaviest mass is 0.33 m from the barycenter, while the lightest is 0.67 m.

 

Since the two bodies have different masses, this changes the centripetal force a bit:

 

Unintuitively, the acceleration of a body in orbit depends on the mass of the other body, not its own. The Earth accelerates towards the moon at 3.33E-5 m/s², while the moon accelerates towards the Earth at 2.70E-3 m/s². About a hundred times faster.

 

For the velocity, the formula could be simplified to this:

 

Once again, it is worth noting that the velocity of a body doesn’t care about its own mass. The Earth’s velocity around its orbit is around 113 m/s, while the moon’s is about 1019 m/s.

 

With these two formulas, we can calculate how two bodies orbit each other, if the orbit is stable.

 

There is no solution for a three-body system, though I may present the problem in a future post…

Categories: Science, Thinking

Orbits of Equal Mass Bodies

When you have two objects, they influence each other through gravity. Take, for example, two marbles. If we assume a weight of 5 grams each, and a distance of 10 centimeters between the centers, we can calculate how much force they exert on each other.

where G is the gravitational constant, 6.67E-11 N m² / kg².

So the attraction between the two marbles is 1.7E-17 N (0.000000000000000017 N). That’s tiny compared to the force of friction that keeps the marbles on the floor. If the marbles are on a wood floor, the friction would be around 0.1 N, meaning the marbles would need almost 6 quadrillion times more force than their mutual attraction. To give some point of reference, the attraction between the marble and the entire Earth is 0.5 N. Between you and the entire Earth? 740 N. How about the earth and moon? 2E26 N.

 

To better understand orbits, let’s imagine the simplest system. There are two bodies of equal mass in a stable orbit around each other, in a universe that contains absolutely nothing else.

Bodies of equal mass in orbit.

What forces are in this system? The easiest one to guess would be gravity, towards the other body, equal to the formula at the beginning of the post. Since the two objects are of equal mass, we can simplify the equation to

We can also calculate how fast they move towards each other, using Netwon’s second law of motion,

Solving for the acceleration, we find that

There is not only gravity, because the two objects would just go straight for each other and collide. To be a stable orbit, there should be a speed, perpendicular to the acceleration, and can find it from the centripetal force formula:

In our case, the gravitational attraction is the centripetal force, and, solving for the velocity, we find that

Forces acting on one of the bodies.

 

If there are two marbles in a stable orbit around each other in an otherwise empty universe, their mutual attraction is still 1.7E-17 N, and, assuming they would be one meter away from each other, they would accelerate towards the other marble at 3.3E-16 m/s². Calculating their velocity in the orbit results in a speed of 5.8E-7 m/s. We know that a circle with a diameter of one meter has a circumference of 2Π meters (6.3 m), so our marbles would do one complete revolution in roughly 4 months.

Right now, if we look at the formulas to calculate the acceleration and the velocity, we know that the distance and mass are the only things that can have an effect on the system.

In the next post, we will calculate what happens if the two bodies are not the same mass.

Categories: Science, Thinking

Astrology and Work

Did you ever read the horoscope section in a newspaper? Did you notice how broad and fake they all sound?

The people who make these “predictions” use phrases that reach as many people as possible. Some people are paid a lot of money to make them, and their goal is to make everyone who read it feel like the message is personal, that it reaches them inside.

For example, The Globe And Mail’s gemini horoscope for today (2012-05-08) is: You’ve found it difficult to trust certain people in recent weeks, which has had a negative effect on relationships. As from today your confidence in them will be restored. Hopefully they feel the same about you.

I think everyone feels like that often in any given year. Recent weeks can easily refer to two months ago, and there is no detail about these “certain people“. Most could relate to that, myself included. It continues by saying that your confidence in them will be restored, which is a power you have in yourself, and no prediction at all.

 

There is a community of people who study astrology, or the way the sun, moon, planets, and stars influence our actions.

My father knows a lot about that kind of astrology (which I call the real astrology, in comparison to the fakes in the newspapers and on the Internet). He has many books on the subject, and bases important decisions on these extra-terrestrial influences.

For example, before making a 300km trip, he could check when would be the best day to leave. The biggest influence in this case would be the weather, but does his knowledge of astronomy prevent him from catching a huge traffic along the way, just by leaving two hours later? Or maybe it’s just in the head – that his mind, by just thinking / knowing that everything is better, makes him think that the trip went for the better?

Being a scientist-minded person, I try to find logical explanations on what he comes up with. So far, and most of the time, my reasoning is just that “Physics doesn’t work that way“. But exactly what do we know about the world around us, aside from classical physics, Einstein’s relativity, quantum physics, and the myriad of developments and discoveries made in our history?

What if astrology was really a field of science that was so faint to observe that we would readily dismiss everything as “coincidence”? Quantum physics partially works that way – as far as we know, radioactive decay is random, and an electron’s position is defined as a probability wave.

Even though I find myself thinking that physics is but a set of formulas, I know that “knowing the future” would require knowledge of all the atoms in the universe, and that’s when omitting quantum probabilities.

 

Sometimes, he makes predictions that are too accurate to be simply waved off as a coincidence, or postdictions of things that were only known to myself.

In February, he made the remark that there was something in my job which would affect me, that was going to happen around late-April – July. Back then, I had recently learned that the company where I work was moving 13km farther, and hadn’t told him yet. A company moving is pretty rare, so this sounded like a good prediction to test, since we were supposed to move in March.

As on cue, the move was split into three groups. Being in the IT department, we were in the second group, set to move in early April. Close enough!

Then the move was reported to mid-April. Then to the last week of April. That started to sound freeky – somehow, my father knew!

True, the prediction was in the range, but that was admittedly a pretty huge range. Plus, as I kept reminding myself, I am someone who is not easily bothered by such things. My working place was now 12km from home instead of 3, but still in the opposite side of the traffic, so why would anyone “see” anything, since I didn’t care that much?

What if the move brought unwanted changes with it? Changes that would take me about two months to accept? That started to seem like a better alternative.

On cue again, the company did not have enough electricity input to power the huge central AC/heat, and it would take until late May to have the extra power. Being on the farthest extension to the building, I spent the first two days programming with gloves (not a fun thing to do, and hard on the backspace key).

The weather warmed up quickly though, and life was back to normal pretty quickly. In fact, it was so quick that I just assumed my father “saw” influences for my company move and some unimportant hardships. I totally forgot about it.

 

Since then, I continued working on my personal development. About a week ago, I started reading more about alternate ways to work, about how working for myself, and receiving passive income instead of receiving a salary and having a boss, could work and actually be a more logical way to live. I am more and more in love with the idea, with each passing day, and I am almost constantly thinking about it now.

My father and I spoke last week-end about his February prediction (that’s why I remembered it, and have been planning this post since yesterday). He sees a spike in early June, and an “important decision that I have to make“, one that would make “my parents and sister happy“. I have no idea how “quitting my job” would make my family happy, but something is supposed to happen soon, for better or for worse. Maybe it’s a decision not to quit that makes them happy.

 

In hindsight, this led me to a lot of thinking. Did my father really see something important in June, or did my brain make it happen? I am never one to underestimate the power of the brain, but maybe my father planted a seed in my thoughts, and my subconscious mind worked its magic from there.

Or maybe the extra-terrestrial bodies in the universe really exert an influence on how we live. Maybe they push us along a path, and we have the decisions to follow it along the way.

Maybe this is what we call “having a star”.

 

Having a choice to believe in the power of the mind or the power of the stars, I would tend to choose the power of the mind. But that couldn’t explain some other predictions and postdictions, the tales of which would have to wait for a future post.

Until then, I’ll continue to grow the sapling of logical life that I have in my mind.

Categories: Thinking

The Toastmasters Club

Monday evening, I went to my first session of Toastmasters International. I had heard about it from Steve Pavlina’s blog, and noticed a few months ago (in the local newspaper) that there was a club near where I live.

It took me a long time to decide to give it a try. For someone like me, speaking (voluntarily!) in front of a small audience is masochist, akin to paying to run a marathon. I knew how it was going to look, even with the first step towards the podium.

The first thing that happened after the meeting started was that we all stood up, one by one, and presented ourselves,what we did for work, what company we work for, and whether we believe in superstition or not (it was the word of the day).

My turn went nicely, although I knew I was speaking too quickly. After I sat back down, I noticed an elevated heart rate and blood pressure.

Then we went into a round of improvisation. The idea is to be able to speak without having studied a subject beforehand. Everyone went, one by one, letting their imagination run, based on a random subject related to superstition. Since I was a guest for the evening, they asked me if I was interested to do the same (every member has to go), so I decided to give it a try and went in front. I usually have a wild imagination, so this is going to go well, I thought.

The improvisation subjects the others received were not bad. The first one was “When you look in a mirror on top of a well, you’ll see your past loves; if you look at a mirror under your bed, what will you see?” The lady had to imagine the rest and speak about it for two minutes. The other subjects were similar.

The “president” searched through his notes for an “easy one”, and I had a sinking feeling while he spoke. It wasn’t in the same format, and it took me… well, not by surprise, but let’s say I wasn’t prepared. It sounded like “When the bride leaves the altar (or doesn’t go, I don’t really remember), cancelling the marriage, what will the other family members do?” So I had to completely make up a superstition, not make up an effect for one. I stayed there for about five seconds, not speaking. A lot flashed through my head, some about what to respond, others about how an idiot I was looking like. Nothing I could have said would last two minutes. I declined and went back to my seat, feeling my face totally flushed (for the next hour or so).

Then there were three members making a speech they had prepared for the evening, then rounds of evaluation about these speeches and the improvisations. After two hours, the meeting ended.

I left the place, feeling like a masochist inside, because I knew I was going back the next week.

 

I know I will hate it when I will have to speak, I mean really HATE it, but having witnessed my “performance” at the improvisation, I know this will only help me with my social anxiety.

Categories: Thinking