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January 26th, 2019, 03:15 PM   #11
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Originally Posted by paulm View Post
Why not the surface area of a circle or some multiple of pi?
As JeffM1 says, you are thinking too specifically.

Area of a square of side $r$: $r^2$
Area of a circle of radius $r$: $\pi r^2$
Area of a right triangle with short legs of $r$: $\frac12 r^2$
Surface area of a sphere of radius $r$: $4\pi r^2$
Surface area of a cylinder of height and radius $r$: $4\pi r^2$
Surface area of a cone of base radius $r$ and side $r$: $2\pi r^2$
Surface area of a cube of side $r$: $6r^2$

All of these have $r^2$ in common because when we measure area we always multiply two distances together. Area is proportional to the square of linear measures. Proportional means that there may be a constant term in there multiplying the linear measure, but that term doesn't change with the linear measure.
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Last edited by skipjack; January 26th, 2019 at 06:52 PM.
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February 2nd, 2019, 03:11 AM   #12
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Originally Posted by v8archie View Post
Area is proportional to the square of linear measures. Proportional means that there may be a constant term in there multiplying the linear measure, but that term doesn't change with the linear measure.
Great insight, thanks.
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February 7th, 2019, 08:43 AM   #13
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As already mentioned, the force of gravitational and electrical attraction varies as 1/r^2.

If a particle disintegrates, it radiates energy E in proportion to the mass (double the mass, double the energy). So

E = kM, where k is a constant.

The speed c of the energy radiated (light) is constant relative to the disintegrating particle, and E has to have units of energy. So

E=mc^2
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February 7th, 2019, 12:19 PM   #14
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Originally Posted by zylo View Post
...If a particle disintegrates, it radiates energy E in proportion to the mass (double the mass, double the energy)...
Sorry, I can't let these two go by. They are easy misinterpretations to make and are fairly common, so they need some attention.

What zylo is trying to say is that the total energy of a particle is given by $\displaystyle E = \sqrt{p^2c^2 + m^2 c^4}$. When we are in the reference frame where the particle is stationary, p = 0, so we get the famous $\displaystyle E = mc^2$. And the rest follows.

"Disintegrates" is also a word I'd like to draw attention to. Disintegration to my mind means "nothing left over". The better word is "decays", which implies there are some particles left over. As there is no such thing as "pure energy", decay is more the word we want to look for here.

-Dan
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Last edited by skipjack; February 7th, 2019 at 04:18 PM.
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