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.

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

Beginner2022-06-23Added 20 answers

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

polivijuye

Beginner2022-06-24Added 16 answers

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.

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

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

$\frac{4{\pi}^{2}{r}^{5}}{{T}^{2}}=GM$

$\frac{4{\pi}^{2}{r}^{5}G}{{T}^{2}}=M$

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.