c. When the center of Earth is 2 × 1011 meters from the center of Mars, the force of gravity between the two planets is about 64.32 × 1014 newtons. The mass of Earth is about 6 × 1024 kilograms, and the mass of Mars is about 6.4 × 1023 kilograms. Using these values, estimate the gravitational constant.

- anonymous

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- phi

do you have a formula for G ?

- midhun.madhu1987

The universal law of gravitation states that:
\[F = GMm \div d ^{2}\]
where F = Gravitational Force
G= Universal Gravitational constant
M = Mass of Earth
m = Mass of mars
d = distance between the 2 objects

- midhun.madhu1987

You can find G by substituting other values..

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## More answers

- midhun.madhu1987

|dw:1437838350097:dw|

- anonymous

the formula I'm using for G is G=Fg d2/m*M

- anonymous

@phi

- phi

now put in numbers in place of the symbols

- phi

2 × 1011 meters from the center of Mars, the force of gravity between the two planets is about 64.32 × 1014 newtons. The mass of Earth is about 6 × 1024 kilograms, and the mass of Mars is about 6.4 × 1023 kilograms.
\[ G = \frac{F_g \cdot d^2}{m\cdot M}\\
G= \frac{64.32\cdot 10^{14} \ N \cdot (2\cdot 10^{11})^2 \ m^2}{6\cdot10^{24} \cdot 6.4\cdot 10^{23} \ kg^2}
\]
are you sure about the force being 64.32 x 10^14 (normally the leading number would be 6.432)?

- anonymous

sorry typo, I meant 6.432

- phi

so the expression is
\[ G= \frac{6.432\cdot 10^{14} \ N \cdot (2\cdot 10^{11})^2 \ m^2}{6\cdot10^{24} \cdot 6.4\cdot 10^{23} \ kg^2}\]

- phi

first, square 2*10^11
which is 2*10^11 * 2 * 10^11
or, reordering
2*2 * 10^11*10^11
I assume you know 2*2 is 4
what is 10^11 * 10^11 ?

- anonymous

1*10^22

- phi

10*10 is 100
or using exponents \( 10^1 \cdot 10^1 = 10^2 \)
or
100*100= 10000
using exponents \(10^2 \cdot 10^2 = 10^4 \)
do you see a pattern ?
if you have the same base (and we do, it is 10)
then when you multiply them, you add their exponents

- anonymous

okay

- phi

\[ G= \frac{6.432\cdot 10^{14} \ N \cdot 4\cdot 10^{22} \ m^2}{6\cdot10^{24} \cdot 6.4\cdot 10^{23} \ kg^2} \]

- phi

Let's just do the "numbers" (the leading coefficients)
what is (6.432*4)/(6*6.4) ?

- anonymous

0.67

- phi

ok. now let's do the top exponent part
10^14 * 10^22 = ?

- anonymous

10^36

- phi

now do the bottom exponent part. what do we get ?

- phi

10^24 * 10^23 = ?

- anonymous

10^47

- phi

so we now have
\[ 0.67 \cdot \frac{10^{36}}{10^{47}} \]
when you divide numbers with the same base, do top exponent - bottom exponent
can you do that ?

- anonymous

divide the top by the bottom?

- phi

yes, but to do that you do
36-47 to get the new exponent

- phi

for example
\[ \frac{100}{10} = 10 \]
using exponents
\[ \frac{10^2}{10^1} = 10^{2-1} = 10^1 = 10 \]

- anonymous

0.765

- phi

?
do 10^36 / 10^47

- phi

the answer is 10 ^ new exponent
the new exponent is 36 - 47

- anonymous

-11

- phi

yes, \(10^{-11}\)

- phi

so we now have
\[ 0.67 \cdot \frac{10^{36}}{10^{47}} \\ 0.67 \cdot 10^{-11} \]
now we put the answer in standard form. we want 6.7 instead of 0.67
to do that we multiply by 10. and then divide by 10, like this
\[ 0.67 \cdot 10 \cdot \frac{10^{-11}}{10^1} \]

- phi

0.67*10 is 6.7
we have
\[ 6.7 \cdot \frac{10^{-11}}{10^1} \]
can you do the 10 part? remember : top exponent - bottom exponent

- anonymous

1^-12

- phi

yes 10^-12
so the answer is
6.7 x 10^-12

- anonymous

now I need to rewrite the formula to solve for one of the mass values

- phi

if you start with
\[ G = \frac{F_g \cdot d^2}{m\cdot M} \]
multiply both sides by m
(which means write m * on both sides)
can you do that?

- anonymous

how would I do that?

- phi

multiplying by a letter (like "m") is easy... you write m on both sides
example, say you have x=y
and you multiply both sides by m. you write
mx = my

- anonymous

okay got it

- phi

like this
\[ mG = m\frac{F_g \cdot d^2}{m\cdot M}\]
you can write the right-hand side as
\[ mG =\frac{m}{m} \frac{F_g \cdot d^2}{ M}\]

- phi

any idea what m/m is ?

- anonymous

not really

- anonymous

m?

- phi

if it were numbers: 2/2 or 3/3 or 7/7 ?

- anonymous

1

- phi

yes. anything divided by itself is 1
(which is why I did what we did)
so we have
\[ mG =\frac{m}{m} \frac{F_g \cdot d^2}{ M} \\
mG =1\cdot\frac{F_g \cdot d^2}{ M} \\
mG =\frac{F_g \cdot d^2}{ M}\]
1 times anything is the anything, which is how we get the last line

- phi

now multiply both sides by \(\frac{1}{G} \)

- anonymous

mG/g=Fg*d^2/m*1/G

- phi

it is clearer if you use G (not g) on the left side
on the left side you have
\[ \frac{mG}{G} \]
what happens to the G/G ?

- anonymous

it's 1

- phi

and what does 1* m simplify to ?

- anonymous

1m

- phi

or just m

- phi

so you have
\[ m= \frac{F_g \cdot d^2}{ M}\cdot \frac{1}{G} \]
on the right side, to multiply fractions, you multiply top*top and bottom * bottom

- anonymous

okay

- phi

in other words you get
\[ m=\frac{F_g \cdot d^2}{G\cdot M} \]

- phi

the bottom could also be written M*G. you can change the order when multiplying

- anonymous

okay

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