Oakey1w

2022-06-22

Find equation for mass in gravity
A satellite is moving in circular motion round a planet.
From the physics we know that
$\mathrm{\Sigma }{F}_{r}=m{a}_{r}=\frac{GMm}{{r}^{2}}$
So I wanted to find the equation for $M$ knowing also that
$v=\omega r=\frac{2\pi r}{T}$
and
${a}_{r}=\frac{{v}^{2}}{r}$
Thus,
$m{a}_{r}=\frac{GMm}{{r}^{2}}$
${a}_{r}=\frac{GM}{{r}^{2}}$
$\frac{{v}^{2}}{r}=\frac{GM}{{r}^{2}}$
$\frac{{\left(\frac{2\pi r}{T}\right)}^{2}}{r}=\frac{GM}{{r}^{2}}$
$\frac{\frac{4{\pi }^{2}{r}^{2}}{{T}^{2}}}{r}=\frac{GM}{{r}^{2}}$
$\frac{4{\pi }^{2}{r}^{3}}{{T}^{2}}=\frac{GM}{{r}^{2}}$
$\frac{4{\pi }^{2}{r}^{5}}{{T}^{2}}=GM$
$\frac{4{\pi }^{2}{r}^{5}G}{{T}^{2}}=M$
However, this is wrong! It should be:
$M=\frac{4{\pi }^{2}{r}^{3}}{G{T}^{2}}$
What was my mistake in Mathematics? Please don't migrate it to physics because my misunderstanding is on math.
Note: I would be very happy if you show my mistake, instead of showing me another way to get to the equation.

tennispopj8

$\frac{4{\pi }^{2}{r}^{2}}{{T}^{2}}/r=\frac{4\pi r}{{T}^{2}}$ and G should go down not up in numerator.

polivijuye

The mistake lies in these steps:
$\frac{\frac{4{\pi }^{2}{r}^{2}}{{T}^{2}}}{r}=\frac{GM}{{r}^{2}}$
$\frac{4{\pi }^{2}{r}^{3}}{{T}^{2}}=\frac{GM}{{r}^{2}}$
$\frac{4{\pi }^{2}{r}^{5}}{{T}^{2}}=GM$
$\frac{4{\pi }^{2}{r}^{5}G}{{T}^{2}}=M$
Actually, it should have been:
$\frac{\frac{4{\pi }^{2}{r}^{2}}{{T}^{2}}}{r}=\frac{GM}{{r}^{2}}$
$\frac{4{\pi }^{2}r}{{T}^{2}}=\frac{GM}{{r}^{2}}$
$\frac{4{\pi }^{2}{r}^{3}}{{T}^{2}}=GM$
$\frac{4{\pi }^{2}{r}^{3}}{G{T}^{2}}=M$
1.In the second step, the numerator will have $r$ and not ${r}^{3}$
2.In the last step, $G$ will be in the denominator.

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