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Thursday, 29 December 2011

The Irregularity of Time [2/2]

A satellite orbits the earth. Due to it's speed, special relativity says time should be going slower than if it was on earth for it, but in fact, it goes quicker. Why?

This is the third of three posts on time. We have already discussed The Special Theory of Relativity in the first post, and the concept of space-time in the second one. I would recommend reading them first, for this most may not make much sense if you have not. In true "TheCompBlog" style (a great tech blogger/friend-of-mine), this post is being written whilst on a airplane flight, and just how topical it is! We learnt in the aforementioned posts, that the faster an object move, the slower time runs for it. So up here, in this plane, time for me is moving quicker than for someone standing below me right? Well, there is more to it than that. You see, another of Einstein's discoveries was that not only is time affected by speed, it is also affected by gravity. How does this come about? well really, it is just applying Special Relativity:

The best way to explain Gravitational Time Dilation, this process of time being slowed by gravity, is through imagining a perfect vacuum in the middle of space. You are in a rocket in this vacuum. No external forces, including gravity, are acting on the rocket. At the front end of rocket is a lamp, emitting regular pulses of light every second. At the other end is a target.
The pulses arrive at the target at the same rate as when they left the lamp, namely 1 per second. The time at the front of the rocket would be moving at exactly the same rate as the time at the back. However, what would happen if the rocket started accelerating? By the time the pulse reached the other end of rocket, the rocket would have aquired a speed. We know that velocity equals acceleration times time taken, or v=at, and the time it takes for the light pulse to travel the distance is distance divided by the speed of light, or t=xc. Therefore the speed gained whilst the pulse is traveling is v=a xc. Well firstly we have to look at something called the Doppler shift. We have all experienced the effect when an ambulance passes you. The siren sound seems to drastically change as it is approaching, passing and moving away. This is because the frequency changes. This change can be measured with this formula:

f'=f(1±  v  )
c
The bottom of that fraction is similar to that from our other equations from Special Relativity, though this is only because, just like in time in special relativity, the frequency is affected by velocity. For very large velocities, time dilation has to be taken into account, but in this situation, the speeds are not large enough for it to matter. The equation can be rearranged to be
(f'-f) ≈ -fvc
and from our previous equation for v, we get
(f'-f) ≈ -fahc2
The pulses of light reach the target at a different frequency as when they started:

Now imagine that the rocket is not in space but instead on a earth. The lamp is on the floor of the rocket now, and the target is on the top:
Gravity is, in essence, just an acceleration. If a feather and hammer are dropped in a near vacuum, they will fall at exactly the same rate. On earth, that rate is 9.81 m/s2. So the rocket, in this situation, is in fact feeling an acceleration downwards, just as we feel pushed towards the ground. The person standing at the top would conclude that the pulses are being released at longer intervals a person standing next to the lamp would measure the frequency to be, this is called the gravitational redshift. Now imagine that the pulses are some sort of clock, each one emitted a second after the last. Seeing that the frequency of this would be less for the person at the target, he would, correctly, conclude that the time closer in the gravitational field was moving slower.

Many experiments have been made to prove this theory. In 1960, Robert Pound and Glen Rebka verified it by firing gamma radiation up and down a 22.5m tower. The shift would only have been about -2.5 x 10-15 but it was verified with a margin for error of only 1%. When the first GPS satellites were launched they did not take into account time dilation from gravity or from speed and were therefore very far off when calculating someones position after only a day or two.

I really hope you like this post and this series of posts. I may revisit it one more time, just to run over the main points, sort of as a conclusion, but I hope that this has interested you in the mean time. Please email me at theaftermatter@gmail.com, comment or get in touch on Twitter if you have any suggestions,  complaints or just want to have a chat about physics/maths/engineering, I love hearing from you.

Check out my last two posts: 
What is Energy? - The word energy is used so loosely these days, so what does it actually mean?
"We don't serve neutrinos in here!"...A neutrino walks into a bar - What are neutrinos and what is all the fuss about?

Main source for this and the other articles based around relativity is the book about relativity here. It goes into more detail on all these subjects and the other books they do are brilliant too, I recommend them a lot.

Wednesday, 7 December 2011

What is Energy?

Renewable Energy, Wasting Energy, 'He's Got Too Much Energy' - None Of These Are True. So What Is Energy?

We have all heard of energy. A word used in all sorts of situation. Energy in food, people and objects are all very common things to hear, but what exactly is energy? Well the easiest way to describe energy is through the statement: Energy is the ability to do work. Ok...That doesn't bring us any closer to knowing what energy actually is, so let me first start of by explaining Work:

Before we all get confused, physical Work is not work in the sense that we use it everyday. The incredibly clever person in your class, who does extra homework does not have more energy than you. Work, shown with the letter W, can be expressed easily with this simple formula:
W = Fx
In words, this means that the work done is the Force applied to an object multiplied by distance.  For example, if I am pushing an object with a force of 10 Newtons, and I push it 5 meters, the work done would be 50 Joules, 10 x 5, because F (the force) is 10 and x (the distance) is 5. 
As you can tell Work is very simple. The only thing you need to remember when thinking about Work is that the distance is not just any distance, it is the distance in the direction of the force. This may not seem very significant, but let me give you this example: If I have a steel ball suspended by a string. I start spinning it around, in a circle of circumference 5 meters, with the same force as before, 10 newtons, and it does one rotation. Surely the work done would be the same as in the first example?

 
Firstly, when looking at the work done, you have to analyse the forces. As we can see, the only two forces acting on the ball is that of me, which is F1 and that of the string, F2. The string is pulling the ball towards me, whilst, to keep the ball spinning, my force is at a right angle to F2. The resultant force is in a different direction than either F1 or F2, therefore, there is no work done.

So now that we know what work is, we can move on to energy. If I asked you to list all types of energy you knew, what would you say? If you said Nuclear, chemical, electric, thermal etc. you would be right, but they can be put more simply into two groups, potential energy, Ep, and kinetic energy, Ek. An object has kinetic energy when it is doing work, this can be in the case of electrical energy, in an exothermic reaction or simply when something is moving. An object has potential energy when it has the capacity to do work due to it's position or configuration. This can be in the case of a compressed spring, something being held any distance above the ground or, in fact, any matter, as famously described in Einstein's E=mc2, anything with mass also has energy.

Both of these types of energy have formulas to work out their values. These will change for certain situations, for example, with electrical energy, but in simple mechanics the formula for kinetic energy is Ek=12mv2 and the one for gravitational (potential energy possessed by an object due to it being pulled by gravity) potential energy, in it's simplest kind, is Ep=mgh. Both of these equations can be obtained using the equation for work, as previously mentioned, and Newton's second law: F=ma.
The potential energy equation is very simple. If you substitute ma, from F=ma, into the definition for work , Fx , for the F, you get: Ep=max. When using this to describe gravitational potential energy, you can substitute h, the height the object is lifted too, in for x, which just means distance. As we know, gravity is just an acceleration. The earth's gravitationally constant is 9.81m/s2 so we can put g in for a. That is how we get to the equation Ep=mgh.
The kinetic energy equation is also very simple. One of the main equations of motion is that final velocity squared equals initial velocity squared plus 2 times the acceleration times the time taken, or v2=u2 + 2ax. I do not feel it is relevant to explain how we got this equation, although, if the idea is popular, I may do a post on the basics of mechanics. Mainly Newton's laws of motion and the equations of motion, but anyway, kinetic energy: In a system with kinetic energy, the object must have started at a stop. Therefore the initial velocity is 0 and can be removed from the equation. Now we have v2=2ax. The equation F=ma can also be expressed as a = Fm and therefore we can change our equation to v2=2Fxm. Look what we have found! Above the m is Fx, or work done, and energy is the just the ability to do or be doing work, so, by rearranging the equation we get: Ek=12mv2.
So now we understand energy, right? Well first there are a few things to go over. Energy does not get destroyed or created. When I drop a pen, no energy is being lost, or created. Instead the potential energy of the pen is being converted into kinetic energy. In fact, if we forget about friction, all of the potential energy is lost over the fall. Let me give you an example: I lift a pen, mass 1kg (It is a heavy pen) 3 meters and then drop it. What is the velocity when it hits the ground?
 
Since we are ignoring friction, we know that:
   Loss in Ep= Gain in Ek
    mgh =12mv2
1Kg x 10m/s2 (We are rounding this up, gravity is actually more like 9.81 m/s2) x 10m= mgh
         12mv2 = 12 x 1 x v2  
0.5v2 (Kinetic Energy) = 30J (Potential Energy)
   v2= 60J
     v ≈ 7.45 m/s

The amount of energy in a isolated system, even in the air or other apparently inactive objects, must be the same, no matter what form they are in. This is the law of conservation of energy!

Now we understand energy. It isn't a physical particle or even a force, but instead a capability to do something. For the next section, let me ask you a question. If you take a large piece of coal and burn it, and then take a spatula of gunpowder and light it. Which reaction would you say is more violent? Probably the gunpowder, but the coal released more heat, and therefore, we will assume, it had more energy. Why does this happen? Well it is all down to something called Power:
Power is the rate at which work is done. It is given by the formula: P=Wt . P is power, W is work done and t is time taken to do the work. Therefore, even though the coal does more work, the gunpowder does it's work in a shorter time and is therefore more powerful.

So those are the basics of energy. Of course, there is further you can go, looking into the different forms of potential energy, for example. In fact, string theory, something I looked at in my last post, says that ever bit of matter, when broken down as far as you can, past atoms, protons, electrons and quarks are just 1 dimensional strings of energy. So next time you hear someone talk about an energy crisis, renewable energy or wasting energy, you can correct them, our energy is not running out, instead, we are just running out of easy sources of it.

Anyway, I hope this cleared up any misconceptions you had about energy, and helped you understand one of the most important physical aspects of the universe. Thank you for reading.
Check out my last two posts: 
"We don't serve neutrinos in here!"...A neutrino walks into a bar - What are neutrinos and what is all the fuss about?
The Irregularity of Time <1.5/2> (You might want to check out the first post on this topic before this) - Why time isn't as constant as you think.