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anonymous

  • one year ago

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.

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  1. phi
    • one year ago
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    do you have a formula for G ?

  2. midhun.madhu1987
    • one year ago
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    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

  3. midhun.madhu1987
    • one year ago
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    You can find G by substituting other values..

  4. midhun.madhu1987
    • one year ago
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    |dw:1437838350097:dw|

  5. anonymous
    • one year ago
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    the formula I'm using for G is G=Fg d2/m*M

  6. anonymous
    • one year ago
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    @phi

  7. phi
    • one year ago
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    now put in numbers in place of the symbols

  8. phi
    • one year ago
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    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)?

  9. anonymous
    • one year ago
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    sorry typo, I meant 6.432

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

  11. phi
    • one year ago
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    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 ?

  12. anonymous
    • one year ago
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    1*10^22

  13. phi
    • one year ago
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    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

  14. anonymous
    • one year ago
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    okay

  15. phi
    • one year ago
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    \[ G= \frac{6.432\cdot 10^{14} \ N \cdot 4\cdot 10^{22} \ m^2}{6\cdot10^{24} \cdot 6.4\cdot 10^{23} \ kg^2} \]

  16. phi
    • one year ago
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    Let's just do the "numbers" (the leading coefficients) what is (6.432*4)/(6*6.4) ?

  17. anonymous
    • one year ago
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    0.67

  18. phi
    • one year ago
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    ok. now let's do the top exponent part 10^14 * 10^22 = ?

  19. anonymous
    • one year ago
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    10^36

  20. phi
    • one year ago
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    now do the bottom exponent part. what do we get ?

  21. phi
    • one year ago
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    10^24 * 10^23 = ?

  22. anonymous
    • one year ago
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    10^47

  23. phi
    • one year ago
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    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 ?

  24. anonymous
    • one year ago
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    divide the top by the bottom?

  25. phi
    • one year ago
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    yes, but to do that you do 36-47 to get the new exponent

  26. phi
    • one year ago
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    for example \[ \frac{100}{10} = 10 \] using exponents \[ \frac{10^2}{10^1} = 10^{2-1} = 10^1 = 10 \]

  27. anonymous
    • one year ago
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    0.765

  28. phi
    • one year ago
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    ? do 10^36 / 10^47

  29. phi
    • one year ago
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    the answer is 10 ^ new exponent the new exponent is 36 - 47

  30. anonymous
    • one year ago
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    -11

  31. phi
    • one year ago
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    yes, \(10^{-11}\)

  32. phi
    • one year ago
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    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 \[ 0.67 \cdot 10 \cdot \frac{10^{-11}}{10^1} \]

  33. phi
    • one year ago
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    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

  34. anonymous
    • one year ago
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    1^-12

  35. phi
    • one year ago
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    yes 10^-12 so the answer is 6.7 x 10^-12

  36. anonymous
    • one year ago
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    now I need to rewrite the formula to solve for one of the mass values

  37. phi
    • one year ago
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    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?

  38. anonymous
    • one year ago
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    how would I do that?

  39. phi
    • one year ago
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    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

  40. anonymous
    • one year ago
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    okay got it

  41. phi
    • one year ago
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    like this \[ mG = m\frac{F_g \cdot d^2}{m\cdot M}\] you can write the right-hand side as \[ mG =\frac{m}{m} \frac{F_g \cdot d^2}{ M}\]

  42. phi
    • one year ago
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    any idea what m/m is ?

  43. anonymous
    • one year ago
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    not really

  44. anonymous
    • one year ago
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    m?

  45. phi
    • one year ago
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    if it were numbers: 2/2 or 3/3 or 7/7 ?

  46. anonymous
    • one year ago
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    1

  47. phi
    • one year ago
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    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

  48. phi
    • one year ago
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    now multiply both sides by \(\frac{1}{G} \)

  49. anonymous
    • one year ago
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    mG/g=Fg*d^2/m*1/G

  50. phi
    • one year ago
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    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 ?

  51. anonymous
    • one year ago
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    it's 1

  52. phi
    • one year ago
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    and what does 1* m simplify to ?

  53. anonymous
    • one year ago
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    1m

  54. phi
    • one year ago
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    or just m

  55. phi
    • one year ago
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    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

  56. anonymous
    • one year ago
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    okay

  57. phi
    • one year ago
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    in other words you get \[ m=\frac{F_g \cdot d^2}{G\cdot M} \]

  58. phi
    • one year ago
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    the bottom could also be written M*G. you can change the order when multiplying

  59. anonymous
    • one year ago
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    okay

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