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.

Question

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.

phi · Answer

do you have a formula for G ?

midhun.madhu1987 · Answer

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 · Answer

You can find G by substituting other values..

midhun.madhu1987 · Answer

|dw:1437838350097:dw|

anonymous · Answer

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

anonymous · Answer

@phi

phi · Answer

now put in numbers in place of the symbols

phi · Answer

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 · Answer

sorry typo, I meant 6.432

phi · Answer

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 · Answer

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 · Answer

1*10^22

phi · Answer

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 · Answer

okay

phi · Answer

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

phi · Answer

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

anonymous · Answer

0.67

phi · Answer

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

anonymous · Answer

10^36

phi · Answer

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

phi · Answer

10^24 * 10^23 = ?

anonymous · Answer

10^47

phi · Answer

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 · Answer

divide the top by the bottom?

phi · Answer

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

phi · Answer

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

anonymous · Answer

0.765

phi · Answer

?
do 10^36  /  10^47

phi · Answer

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

anonymous · Answer

-11

phi · Answer

yes,  $10^{-11}$

phi · Answer

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 · Answer

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 · Answer

1^-12

phi · Answer

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

anonymous · Answer

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

phi · Answer

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 · Answer

how would I do that?

phi · Answer

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 · Answer

okay got it

phi · Answer

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 · Answer

any idea what m/m is ?

anonymous · Answer

not really

anonymous · Answer

m?

phi · Answer

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

anonymous · Answer

1

phi · Answer

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 · Answer

now multiply both sides by $\frac{1}{G} $

anonymous · Answer

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

phi · Answer

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 · Answer

it's 1

phi · Answer

and what does 1* m simplify to ?

anonymous · Answer

1m

phi · Answer

or just  m

phi · Answer

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 · Answer

okay

phi · Answer

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

phi · Answer

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

anonymous · Answer

okay