@absicht_MAUERzuBAUEN said in #43:
I am especially thankful for your easy mathematical explanation, but I have to ask, what is the <Unit> used for energy?
E=m (gramm?) * c(length/time) ^2 * y (length/time/(lengh/time))
So the unit to measure and describe energy equals
{weight * (lengh/time)^2 * lengh/time?}
My pleasure! Physics rambling incoming:
The formula is E = γmc^2
Let's do so called "dimensional analysis" to find the units:
[E] = [γ][m][c]^2
Here square brackets denote the dimension (or unit) of the parameter. The Lorentz factor γ (denoted by the greek lower case letter gamma) is said to be "dimensionless". Fancy word, but it means nothing more that it has no unit. It's simply some number.
If v is 99.5% of the speed of light for instance, the Lorentz factor is the number 10.
Returning to the units we can write:
[γ] = 1
Why? Because all units cancel out in the formula for γ:
γ = 1/(√(1 - (v/c)^2))
Let's focus in on the v/c bit (as the rest of the formula for γ already are numbers without units). That's a speed (that of the object) divided by the speed of light. Both have units of length divided by time, but that isn't really important. It's enough to observe that both have the same units, therefore the units cancel out when you divide one by the other:
[v/c] = [v]/[c] = (L/T)/(L/T) = 1
You can treat units in the same way as you treat numbers. When you have the fraction 15/25, you know that you cancel the common factor 5 in the numerator and the denominator: 15/25 = (53)/(55) = (5/5)(3/5) = 1(3/5) = 3/5
In the same way you can cancel units of length (denoted by L) in the numerator with units of length in the denominator.
Let's look at mass next: It has, well, units of mass (like the kilogram for example):
[m] = M
The speed of light has units of length divided by time (meters per second for example):
[c] = L/T
Therefore energy has units of mass times units of length squared divided by units of time squared:
[E] = M*(L^2)/(T^2)
I've done this in the most general way possible, that way you can choose which practical units you want to measure in. You could choose the metric system, using the kilogram (kg) as a unit of mass M, the metre (m) as a unit for length L and the second (s) as a unit for time T.
In the metric system energy has units of kg*(m^2)/(s^2), spoken as "kilogram square metre per square second". This is a bit of a mouthful and therefore physicists have defined the derived unit of energy called the Joule (J), which is exactly that:
1 J = 1 kg*(m^2)/(s^2)
When measuring electric energy, you often don't use the Joule itself, but another derived unit, the kilowatt-hour (kWh). The watt is the unit of power (energy per unit time): 1 W = 1 J/s (Joule per second)
The k in kWh means the prefix kilo- (a factor 1000). And an hour has 60 minutes with 60 seconds each. So:
1 kWh = (1000 J/s)(6060 s) = 3,600,000 J
This unit (the kWh) is more convenient because the power drawn by electrical home appliances (washing machine, fridge, computer, etc.) is usually given in Watts or kilowatt. So you can easily calculate how many hours you can run them if you only want to pay for x kilowatt-hours.
Other unit systems exist (like the cgs system, which uses grams instead of kilograms for mass and centimetres instead of metres for length) and are helpful in certain contexts (like electromagnetism).
In particle physics Einstein's E_0 = m*c^2 is used to measure rest masses in terms of the associated energy (divided by the speed of light squared). So the rest mass of a proton would be given as
m_p ≈ 938.3 MeV/c^2 (megaelectronvolts per speed of light squared),
where 1 MeV is 1,000,000 times the kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt (1 V) in vacuum. You see, units can get really weird, really fast. The MeV/c^2 is a useful unit for particle masses though :-)
In the metric system you would give the mass of a single proton as m_p ≈ 1.6726*10^(-27) kg (kilograms), which looks a lot uglier (to physicists) than 938.3 MeV/c^2 does, because the kilogram is a unit invented to measure tangible things commonly found around humans like other humans, a sack of potatoes or some piece of furniture. The kilogram is not designed to measure the incredibly tiny mass of individual protons, which is why you need to add the unimaginable 10^(-27) factor, i.e. 0.000000000000000000000000001, when you try to measure the proton mass in kilograms.
The MeV/c^2 however it up to the job. I was designed with elementary particles (namely electrons) in mind.
There's also the system of customary units in the US, which still uses units defined by some English king in the middle ages on the basis of the length of a barley corn (and other things intuitively accessible to the people of the time). They use feet (ft) for length, pound (lb) for mass, etc.
This system of measurement still has its uses (interpreting cooking recipes for example), but it's not designed for science and engineering which is why in the US, military, spaceflight, engineering, scientific research, etc. has largely switched over to the metric system (at least under the hood).
Sorry, I went on a long tangent there. Bottom line is: The most sensible unit for (kinetic) energy of common objects (like a baseball flying through the air or a plate falling to the floor) is the Joule. Other than that, there are many energy units which are more convenient in certain contexts.
What is E_0? Does it mean the energy equals 0? The E_0*y must be 0,right?
No, it's not equal to 0 (for objects with nonzero mass). Sorry for the confusion.
It's my way to write E with a subscript 0. This is often done to denote the base state (lowest energy state) of a system. In this case, it means the rest energy of the object. Rest energy is the energy that is intrinsic to the object, it's there even if the object isn't moving at all.
I did this in order to differentiate it from the total energy of the object E (which includes its kinetic energy, i.e. energy of motion).
1 kg of motionless matter (be it gold or carbon or water or milk or whatever) always has the same amount of rest energy E_0. You can calculate this intrinsic energy via Einstein's famous E_0 = mc^2.
If we plug in m = 1 kg and c^2 ≈ 910^16 (m/s)^2, we obtain a rest energy of
E_0 ≈ 9*10^16 J = 90000 TJ (terajoule)
You might be familiar with the prefix tera- from your computer (1 TB = 1000 GB of memory).
This is an enormous amount of energy. It's equivalent to 21.5 million tons of TNT explosives. In a single kilogram of matter (a litre of milk for example)!
And indeed this is the principle that allows nuclear weapons to be so destructive. A small amount of matter (only a few hundreds of kilograms) is fully converted into energy during the explosion of a nuclear weapon. Luckily you need some pretty extreme conditions (fissionable material being bombarded by high energy neutrons for example) to get this nuclear reaction going, so you don't need to worry about your litre of milk spontaneously blowing you and your entire city to smithereens. It's perfectly stable and won't blow up.
The "dragon" is there (yes, even in your can of milk), but it is sleeping deeply. I sincerely hope we humans will never again waken the nuclear dragon.
@absicht_MAUERzuBAUEN said in #43:
> I am especially thankful for your easy mathematical explanation, but I have to ask, what is the <Unit> used for energy?
> E=m (gramm?) * c(length/time) ^2 * y (length/time/(lengh/time))
>
> So the unit to measure and describe energy equals
> {weight * (lengh/time)^2 * lengh/time?}
My pleasure! Physics rambling incoming:
The formula is E = γ*m*c^2
Let's do so called "dimensional analysis" to find the units:
[E] = [γ]*[m]*[c]^2
Here square brackets denote the dimension (or unit) of the parameter. The Lorentz factor γ (denoted by the greek lower case letter gamma) is said to be "dimensionless". Fancy word, but it means nothing more that it has no unit. It's simply some number.
If v is 99.5% of the speed of light for instance, the Lorentz factor is the number 10.
Returning to the units we can write:
[γ] = 1
Why? Because all units cancel out in the formula for γ:
γ = 1/(√(1 - (v/c)^2))
Let's focus in on the v/c bit (as the rest of the formula for γ already are numbers without units). That's a speed (that of the object) divided by the speed of light. Both have units of length divided by time, but that isn't really important. It's enough to observe that both have the same units, therefore the units cancel out when you divide one by the other:
[v/c] = [v]/[c] = (L/T)/(L/T) = 1
You can treat units in the same way as you treat numbers. When you have the fraction 15/25, you know that you cancel the common factor 5 in the numerator and the denominator: 15/25 = (5*3)/(5*5) = (5/5)*(3/5) = 1*(3/5) = 3/5
In the same way you can cancel units of length (denoted by L) in the numerator with units of length in the denominator.
Let's look at mass next: It has, well, units of mass (like the kilogram for example):
[m] = M
The speed of light has units of length divided by time (meters per second for example):
[c] = L/T
Therefore energy has units of mass times units of length squared divided by units of time squared:
[E] = M*(L^2)/(T^2)
I've done this in the most general way possible, that way you can choose which practical units you want to measure in. You could choose the metric system, using the kilogram (kg) as a unit of mass M, the metre (m) as a unit for length L and the second (s) as a unit for time T.
In the metric system energy has units of kg*(m^2)/(s^2), spoken as "kilogram square metre per square second". This is a bit of a mouthful and therefore physicists have defined the derived unit of energy called the Joule (J), which is exactly that:
1 J = 1 kg*(m^2)/(s^2)
When measuring electric energy, you often don't use the Joule itself, but another derived unit, the kilowatt-hour (kWh). The watt is the unit of power (energy per unit time): 1 W = 1 J/s (Joule per second)
The k in kWh means the prefix kilo- (a factor 1000). And an hour has 60 minutes with 60 seconds each. So:
1 kWh = (1000 J/s)*(60*60 s) = 3,600,000 J
This unit (the kWh) is more convenient because the power drawn by electrical home appliances (washing machine, fridge, computer, etc.) is usually given in Watts or kilowatt. So you can easily calculate how many hours you can run them if you only want to pay for x kilowatt-hours.
Other unit systems exist (like the cgs system, which uses grams instead of kilograms for mass and centimetres instead of metres for length) and are helpful in certain contexts (like electromagnetism).
In particle physics Einstein's E_0 = m*c^2 is used to measure rest masses in terms of the associated energy (divided by the speed of light squared). So the rest mass of a proton would be given as
m_p ≈ 938.3 MeV/c^2 (megaelectronvolts per speed of light squared),
where 1 MeV is 1,000,000 times the kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt (1 V) in vacuum. You see, units can get really weird, really fast. The MeV/c^2 is a useful unit for particle masses though :-)
In the metric system you would give the mass of a single proton as m_p ≈ 1.6726*10^(-27) kg (kilograms), which looks a lot uglier (to physicists) than 938.3 MeV/c^2 does, because the kilogram is a unit invented to measure tangible things commonly found around humans like other humans, a sack of potatoes or some piece of furniture. The kilogram is not designed to measure the incredibly tiny mass of individual protons, which is why you need to add the unimaginable 10^(-27) factor, i.e. 0.000000000000000000000000001, when you try to measure the proton mass in kilograms.
The MeV/c^2 however it up to the job. I was designed with elementary particles (namely electrons) in mind.
There's also the system of customary units in the US, which still uses units defined by some English king in the middle ages on the basis of the length of a barley corn (and other things intuitively accessible to the people of the time). They use feet (ft) for length, pound (lb) for mass, etc.
This system of measurement still has its uses (interpreting cooking recipes for example), but it's not designed for science and engineering which is why in the US, military, spaceflight, engineering, scientific research, etc. has largely switched over to the metric system (at least under the hood).
Sorry, I went on a long tangent there. Bottom line is: The most sensible unit for (kinetic) energy of common objects (like a baseball flying through the air or a plate falling to the floor) is the Joule. Other than that, there are many energy units which are more convenient in certain contexts.
> What is E_0? Does it mean the energy equals 0? The E_0*y must be 0,right?
No, it's not equal to 0 (for objects with nonzero mass). Sorry for the confusion.
It's my way to write E with a subscript 0. This is often done to denote the base state (lowest energy state) of a system. In this case, it means the rest energy of the object. Rest energy is the energy that is intrinsic to the object, it's there even if the object isn't moving at all.
I did this in order to differentiate it from the total energy of the object E (which includes its kinetic energy, i.e. energy of motion).
1 kg of motionless matter (be it gold or carbon or water or milk or whatever) always has the same amount of rest energy E_0. You can calculate this intrinsic energy via Einstein's famous E_0 = m*c^2.
If we plug in m = 1 kg and c^2 ≈ 9*10^16 (m/s)^2, we obtain a rest energy of
E_0 ≈ 9*10^16 J = 90000 TJ (terajoule)
You might be familiar with the prefix tera- from your computer (1 TB = 1000 GB of memory).
This is an enormous amount of energy. It's equivalent to 21.5 million tons of TNT explosives. In a single kilogram of matter (a litre of milk for example)!
And indeed this is the principle that allows nuclear weapons to be so destructive. A small amount of matter (only a few hundreds of kilograms) is fully converted into energy during the explosion of a nuclear weapon. Luckily you need some pretty extreme conditions (fissionable material being bombarded by high energy neutrons for example) to get this nuclear reaction going, so you don't need to worry about your litre of milk spontaneously blowing you and your entire city to smithereens. It's perfectly stable and won't blow up.
The "dragon" is there (yes, even in your can of milk), but it is sleeping deeply. I sincerely hope we humans will never again waken the nuclear dragon.