Author Topic: TSSM Mistake...?! (Read 1156 times) Tweet Share

Mao · « **on:** March 26, 2008, 09:53:45 am »

From the TSSM Summer Head Start program, a list of formulas were given in the notes for gravity and satellite motion.
Here is a list of formulas that are wrong:

Please confirm my workings, and people with these notes please make changes accordingly (found on page 19)

Quote from: TSSM

$T^2=\frac{4 \pi R^3}{GM}$

An equation similar to this can be derived from:

$a=\frac{v^2}{R}=\frac{4 \pi^2 R}{T^2}=g=\frac{GM}{R^2}$

$\implies \frac{4 \pi^2 R}{T^2}=\frac{GM}{R^2}$

$\implies \frac{4 \pi^2}{GM}=\frac{T^2}{R^3}$

$\therefore T^2=\frac{4 \pi^2 R^3}{GM}$

Quote from: TSSM

$a^3=\frac{16 \pi^2 GM}{T^4}$

An equation similar to this can be derived from:

$a=\frac{v^2}{R}$

$\implies a^3=\frac{v^6}{R^3}$

Since $v=\frac{2 \pi R}{T}$

$\implies a^3=\frac{2^6 \pi^6 R^6}{T^6} \cdot \frac{1}{R^3}$

$\implies a^3=\frac{64 \pi^6 R^3}{T^6}$

Also Since $\frac{4 \pi^2}{GM}=\frac{T^2}{R^3}$ (from previous)
$\implies R^3=\frac{GMT^2}{4 \pi^2}$

$\implies a^3=\frac{64 \pi^6}{T^6} \cdot \frac{GMT^2}{4 \pi^2}$

$\therefore a^3=\frac{16 \pi^4 GM}{T^4}$

Quote from: TSSM

$T^2=\frac{4 \pi G^3 M^3}{v^6}$

An equation similar to this can be derived from:

$v=\frac{2 \pi R}{T}$

$\implies v^3=\frac{8 \pi^3 R^3}{T^3}$

Since $R^3=\frac{GMT^2}{4 \pi^2}$ (from previous)

$\implies v^3=\frac{8 \pi^3}{T^3} \cdot \frac{GMT^2}{4 \pi^2}$

$\implies v^3=\frac{2 \pi GM}{T}$

$\therefore T=\frac{2 \pi GM}{v^3}$

$\implies T^2=\frac{4 \pi^2 G^2 M^2}{v^6}$ <-------- what was the point of that!

Mmmmm some tea?

I don’t think I made any mistakes…. Please confirm

cara.mel · « **Reply #1 on:** March 26, 2008, 09:59:57 am »

Yeah, it should be pi^2
Although, the 2nd and 3rd one I have never seen before and I'd have no idea why you'd want to do that

1st one is useful

Collin Li · « **Reply #2 on:** March 26, 2008, 11:43:57 am »

The first one is Kepler's law! You should know how to derive this one

cara.mel · « **Reply #3 on:** March 26, 2008, 12:06:09 pm »

Quote from: coblin on March 26, 2008, 11:43:57 am

The first one is Kepler's law! You should know how to derive this one

Kepler's 3rd law

I learnt the first two this year =)

iamdan08 · « **Reply #4 on:** March 26, 2008, 04:39:24 pm »

Good find Mao!

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Author Topic: TSSM Mistake...?! (Read 1156 times) Tweet Share

Mao

TSSM Mistake...?!

cara.mel

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Collin Li

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cara.mel

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iamdan08

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