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Energy

Tuesday, 22 February 2011


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Energy Equations
· Energy E is a scalar. The energy of an object is given by

E = p c +m c
2 22

24 0

where p is the momentum, m0 the rest mass and c the speed of light. There is also (potential) energy in fields such as gravitational or electric fields:

In a given collection of objects (a system), energy is conserved unless an external force f is applied to this system, in which case the change in energy of the system is:
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Gm1m2 1 q1q E= + r 4 0 r

2

dE = v .F dt
The dot is the vector dot product


That's actually all you need to know
· But now we'll talk about what it actually means...

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Energy
· The basic principle is very familiar things don't do stuff without energy. · Energy can take many forms, and change from one to another, but it is always conserved and you can't get it for nothing. · This can sometimes let you solve seemingly impossible problems with the greatest of ease 5
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Impossible without details
· You don't get something for nothing · Perpetual motion machines can't exist. · If you see something doing stuff, there must be an energy source hidden somewhere. · If you don't have enough energy, no matter what, you can't do something.
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For example
· "All this talk of space travel is utter bilge", Professor Woolley, ANU, 1956 · Launch of Sputnik 1, 1957 · Argument - the energy liberated by a kilogram of TNT is less than the energy needed to lift one kilogram into space. · So the most explosive materials known cannot lift even themselves into space.
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What's wrong with this?

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Two things
· You can get much more energy per unit weight from things like petrol compared to explosives - the explosives have less energy, but liberate it faster. · You can burn tones of fuel to get 1kg of payload into space - most of the fuel is burned low down.
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· http://www.youtube.com/watch? v=wvWHnK2FiCk

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Energy
· Throw a ball into the audience · Let's see what you know. · Write down on a scrap of paper what is going on with energy while a ball is thrown across the classroom.

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Less kinetic energy and more gravitational potential energy Back to Kinetic energy, mostly in the ball but a bit in air currents Kinetic energy Heat energy in the ball and my hand. Maybe some sound energy?

Chemical energy in my muscles
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Is this plausible?

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On energy grounds alone...

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Not plausible

He goes higher at the peak than at the start. Where did the energy for this come from?
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Maths behind this?
· The chemical energy is really hard to measure (at least without dissecting you followed by a cell-bycell chemical analysis) · But you can work out how much energy you used using the law of conservation of energy. · The kinetic energy in the ball must have come from your muscles. · And your arm probably warmed up a bit from the exercise. · Add this heat energy to the kinetic energy and that's how much chemical energy you must have used.
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Kinetic energy
· The true energy of a moving object is 2 22 24 given by:

E = p c + m0 c E = m0 c

If you set the momentum P to zero (i.e. the ball isn't moving), you get: 2 24 Take the square root of this, and you get an equation you may recognize:

E = m 0c

2

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Relativity
· Relativity (covered in PHYS1201) tells us that matter and energy are the same thing. · The true energy equation takes this into account - even a stationary ball has lots of energy (you multiply the mass in kg by the velocity of light squared...) · But in most everyday situations you don't need to worry about this - you can ignore this rest-mass energy and just look at the change in energy due to the motion. · If the speed of an object is much less than the speed of light, you can approximate this using another familiar equation: 12 KE = mv 2
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Kinetic and potential energy
· So if you know how fast the ball was moving when it left my hand, you can work out the kinetic energy. · But as it moves higher into the air, it will slow down. · Energy has been lost by the ball, and gained by the gravitational field of the Earth. This is called Potential Energy.

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Potential Energy
· The true equation for gravitational potential energy (well, almost - it does need some corrections for General Relativity which I won't go into here) is:

-GM m E= r

where M in this case would be the mass of the Earth, and m the mass of the ball (or vice versa - it makes no difference). r is distance between the centre of the ball and the centre of the Earth.

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Approximation
· You can use that full equation - r might start off at 6400 km and go to 6400.005 km. · But over this small range in r (the distance to the centre of the Earth), you can use a simpler approximate form of the potential energy equation:

PE = mgh
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Where h is the height, m the mass of the ball, and g = 9.8 m s-2.


Straight up?
· If I threw it straight up, all the kinetic energy would turn into potential energy for a moment when it's at the top of its arc. · So you could work out how high it would go, using conservation of energy. · The Kinetic energy when it leaves my hand must equal the potential energy at the top of its motion.
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12 mv = mgh 2
Cancel m

12 v = gh 2
Divide both sides by g, and write it down backwards

1v h= 2g
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Other forms of energy
· Energy in a spring:
·

1 E = kD 2

2

where k is the spring constant and D the distance by which the spring is extended or compressed.

· Rotational Energy:

1 E = I 2

2

· where I is the moment of inertia and is the angular velocity 25
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General Procedure
· Pick states (like beginning and end) · Write down all the various forms of energy at each state · Set them equal to each other · Solve for whatever it is you want to know.

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