anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
phi
  • phi
do you have a formula for G ?
midhun.madhu1987
  • 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
  • midhun.madhu1987
You can find G by substituting other values..

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midhun.madhu1987
  • midhun.madhu1987
|dw:1437838350097:dw|
anonymous
  • anonymous
the formula I'm using for G is G=Fg d2/m*M
anonymous
  • anonymous
@phi
phi
  • phi
now put in numbers in place of the symbols
phi
  • 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
  • anonymous
sorry typo, I meant 6.432
phi
  • 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
  • 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
  • anonymous
1*10^22
phi
  • 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
  • anonymous
okay
phi
  • 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
  • phi
Let's just do the "numbers" (the leading coefficients) what is (6.432*4)/(6*6.4) ?
anonymous
  • anonymous
0.67
phi
  • phi
ok. now let's do the top exponent part 10^14 * 10^22 = ?
anonymous
  • anonymous
10^36
phi
  • phi
now do the bottom exponent part. what do we get ?
phi
  • phi
10^24 * 10^23 = ?
anonymous
  • anonymous
10^47
phi
  • 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
  • anonymous
divide the top by the bottom?
phi
  • phi
yes, but to do that you do 36-47 to get the new exponent
phi
  • phi
for example \[ \frac{100}{10} = 10 \] using exponents \[ \frac{10^2}{10^1} = 10^{2-1} = 10^1 = 10 \]
anonymous
  • anonymous
0.765
phi
  • phi
? do 10^36 / 10^47
phi
  • phi
the answer is 10 ^ new exponent the new exponent is 36 - 47
anonymous
  • anonymous
-11
phi
  • phi
yes, \(10^{-11}\)
phi
  • 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
  • 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
  • anonymous
1^-12
phi
  • phi
yes 10^-12 so the answer is 6.7 x 10^-12
anonymous
  • anonymous
now I need to rewrite the formula to solve for one of the mass values
phi
  • 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
  • anonymous
how would I do that?
phi
  • 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
  • anonymous
okay got it
phi
  • 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
  • phi
any idea what m/m is ?
anonymous
  • anonymous
not really
anonymous
  • anonymous
m?
phi
  • phi
if it were numbers: 2/2 or 3/3 or 7/7 ?
anonymous
  • anonymous
1
phi
  • 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
  • phi
now multiply both sides by \(\frac{1}{G} \)
anonymous
  • anonymous
mG/g=Fg*d^2/m*1/G
phi
  • 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
  • anonymous
it's 1
phi
  • phi
and what does 1* m simplify to ?
anonymous
  • anonymous
1m
phi
  • phi
or just m
phi
  • 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
  • anonymous
okay
phi
  • phi
in other words you get \[ m=\frac{F_g \cdot d^2}{G\cdot M} \]
phi
  • phi
the bottom could also be written M*G. you can change the order when multiplying
anonymous
  • anonymous
okay

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