The sun’s mass is about 2.7 x 107 times greater than the moon’s mass. The sun is about 400 times farther from Earth than the moon. How does the gravitational force exerted on Earth by the sun compare with the gravitational force exerted on Earth by the moon?

- anonymous

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- BAdhi

for easiness take newtons gravitational equation as,
\[F \propto \frac{M}{r^2}\] (since \(m\), \(G\) are common for both scenarios)
for moon=>
\[F_m\propto \frac{M_m}{r_m^2}\] for sun, \[F_s \propto \frac{M_s}{r_s^2} \]
by dividing both, get the equation

- anonymous

im a little lost

- BAdhi

well this is a bit of a easy manipulation method, but if its confusing i'll describe,
The gravitational field equation is,\[F=\frac{GmM}{r^2}\]
If the gravity force between the moon and earth is \(F_m\) and mass of moon is \(M_m\) and the distance between moon and earth is \(r_m\) and the mass of the earth is \(m\) the above equation becomes,\[F_m = \frac{GmM_m}{r_m^2}= (Gm) \frac{M_m}{r_m^2}\]
For force between the sun and earth is \(F_s\) and mass of the sun is \(M_s\) and distance is \(r_s\) , equation becomes,\[F_s = \frac{GmM_s}{r_s^2} = (Gm) \frac{M_s}{r_s^2}\]
Is it clear so far?

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## More answers

- anonymous

yes

- BAdhi

now what you need is to compare \(F_s\) with \(F_m\) or get a value for \(\displaystyle \frac{F_s}{F_m}\)
For that what should you do?

- anonymous

divide F8/Fm

- BAdhi

there you go.... any problems after that?

- anonymous

only curious what do you plug into it so you can divide

- BAdhi

first divide and see what terms get eliminated by division... Then try to plugin the values given as data...
Remember what is given is as ratios too... For example, they say sun is \(2.7\times 10^7 \) times larger than moon in mass. Simply what that mean is,
\(M_s = 2.7\times 10^7 M_m\)

- anonymous

8/m?

- BAdhi

whats that?

- anonymous

F8/Fm F cancels rights

- BAdhi

you mean \(\displaystyle \frac{F_s}{F_m} = \frac{s}{m}\) ??

- anonymous

yah...is that wrong

- BAdhi

oh dear...
\(F_s\) and \(F_m\) are symbols as whole Just like \(A\) or \(B\) or any other symbols.. you jst cannot cut out parts of the symbol... simply what you are saying is like,
\[\frac{V}{W} = \frac{1}{V}\] :P (by cutting out a apart of W with V since W is like two Vs)

- anonymous

my bad so what do i do than

- BAdhi

If you are not comfortable with the notation, Jst replace \(F_m,F_s, M_m, M_s, r_m, r_s\) with normal english letters such as \(a,b,c,d,e,f\) etc. and try out..
And please try to show what you've tried out

- anonymous

no its not the symbols i just dont understand what im supposed to do to divide F8 by Fm

- BAdhi

I have given you two equations for F_s and F_m. What you have to do is divide like,
\(F_s = abc\), \(F_m = pqr\)
\[\frac{F_s}{F_m} = \frac{abc}{pqr}\]
Havent you done equation division before?

- anonymous

idk

- BAdhi

ok here is a problem solve it for me..
find \(\frac{F}{T}\)
\(F=ba^2\)
\(T=bc\)

- anonymous

2b*c*a^2

- BAdhi

I need the steps how you got there... not jst the answer..

- anonymous

idk how to explain it i just factored it out

- anonymous

i know if you divide a equation by an equation you actually combine and multiply the variables

- anonymous

there are 2 b(s) hence 2b theres 1 c hence times c and 1 a^2 hence times a^2

- anonymous

correct?

- anonymous

so f8/fm becomes 2f8m

- anonymous

\[2F _{8}m\]

- BAdhi

well thats not how it works...
I think you should work out more with your equations and symbols. before getting into this kinda applications.

- anonymous

its the last question i have than im done :(

- BAdhi

\[\frac{F}{T} = \frac{ba^2}{bc}=\frac{a^2}{c}\] when the left hand side is divided same goes to the right hand side.. then if there are common symbols they jst cross out jst like what happen to the b here

- BAdhi

Sorry I cannot jst give out the answer to this...

- anonymous

im not asking you to just help me understand what i need to do

- anonymous

ohhh ok i get it

- BAdhi

So would you mind showing us how to get the answer..

- anonymous

yes but i just need to know what am i pluging in as the right hand side against F8/Fm

- anonymous

nvm i know

- anonymous

nvm i know what i need to plug in

- anonymous

\[\frac{ F _{8} }{ Fm }=\frac{ \frac{ (Gm)M _{m} }{ r _{m}^2} }{ \frac{ (Gm)M _{8} }{ r _{8}^2 } }=\frac{ \frac{ M _{m} }{ r _{m}^2 } }{ \frac{ M _{8} }{ r _{8}^2 } }\]

- anonymous

?

- BAdhi

good

- BAdhi

\[\frac{\left(\frac ab\right)}{\left( \frac c d\right)} = \frac a b \times\frac d c\] use this now

- anonymous

\[\frac{ M _{m} }{ r _{m}^2 }*\frac{ M _{8} }{ r _{8}^2 }=2M _{m8}r _{m8}^4\]

- anonymous

?

- anonymous

?

- BAdhi

note that when the division is changed to multiplication, c/d is changed to d/c...so what you've written there is wrong
also... as i have mentioned before, \(M_s\times M_m\) cannot be further simplified to \(2M_{ms}\) (u can, maybe with some other algebra :P). It should jst be \(M_sM_m\) and treat \(M_s\) as a whole which cannot be separated and same goes to \(M_m\)

- anonymous

ohh ok so how would I write it @Badhi

- BAdhi

\[\frac{M_m}{r_m^2}\frac{r_s^2}{M_s}\] and then adjust it like this,
\[\left(\frac{M_m}{M_s}\right)\left(\frac{r_s}{r_m}\right)^2\]

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