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Gravitational Force and True Weight

  1. Mar 3, 2009 #1
    1. The problem statement, all variables and given/known data[/b]
    At a distance H abouve the suface of a planet, the true weight of a remote probe is one percent less than its true wieght on the surface. The radius of the planet is R. Find the ration H/R.

    2. Relevant equations[/b]

    W=mg=G*(M(of planet)/r(squared))*M(of probe)

    where G is the universal gravitational constant= 6.673x10-11 N*m2/kg2
    r is the distance between the middle of the planet and the probe



    3. The attempt at a solution[/b]

    I tried to equate two equations, one for the weight on the probe on the surface, and one for the mass of the probe on H above the surface of the planet. I then tried to solve for H. That however didn't work at all as the provided answer is H/R= 0.005

    Any help!!!???
     
  2. jcsd
  3. Mar 3, 2009 #2
    Where have you made use of the 1% apparent loss in weight? You need to re-think the problem, remembering that the mass is the same in both places.
     
  4. Mar 3, 2009 #3
    wouldn't the 1% loss in true weight mean that W=mg at a distance H from the planet would be 1% less than W=mg at the planets surface due to a decrease in g at a distance H from the surface? I set up my equations as follows
    G*(M(of planet)/(r + H) (squared))*M(of probe) = 0.99(G*(M(of planet)/r(squared))*M(of probe))

    is this not the right idea??
     
  5. Mar 3, 2009 #4

    lanedance

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    sounds reasonable to me, why don't you like your answer?
     
  6. Mar 4, 2009 #5

    turin

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    Do you know about the Taylor expansion and when you can ignore higher order terms? Of course, in principle, you can solve this exactly as well. You should realize that a lot of stuff cancels out.
     
  7. Mar 4, 2009 #6
    well when I try and solve it from the two equations pretty much everything cancels out and I'm left with H=1.01. Is this right? because the answer in the back of the book it H/R=0.005
     
  8. Mar 4, 2009 #7

    lanedance

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    Sorry I didn;t understand from your post that your were calculating the wrong answer. Its usually always quicker & easier if you show your working.

    can you show how you get 0.01?

    cancelling terms you should get
    1/(r+h)^2 = 0.99/r^2

    if i work from this I get the required (remember you're finding the ratio)
    h/r = 0.005
     
  9. Mar 4, 2009 #8
    So this is what I did exactally:

    G*(M/(H+r))^2*m=0.99(G*(M/r^2)*m)

    then I carried the 0.99 into the equation to get rid of the brackets (can you cancel terms before carrying the 0.99 through?):

    G*(M/(H+r)^2)*m=(0.99G)(0.99M/r^2)(0.99m)

    then I cancelled terms that I saw on both sides of the equation:

    1/((H+r)^2) = 0.99*(0.99/r^2)*0.99

    then I got all my variables to one side of the equation:

    (r^2)/((H+r)^2) = 0.97

    then I cancelled my square:

    r/(H+r) = 0.97

    then I isolated for the variables:

    r=0.97(H+r) = 0.97H + 0.97r

    so r=0.97H + 0.97r

    then:

    r-0.97r = 0.97H

    0.03r = 0.97H

    0.03 = 0.97 H/r

    H/r= 0.03/0.97 = 0.03

    this is wrong!! I think maybe my math skills are lacking
     
  10. Mar 4, 2009 #9

    lanedance

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    fewe errors strating off
    your 1st line: starting out your square is in the wrong place
    your 2nd line: you have multiplied the right hand side by 0.99^2, without multplying the left hand side, this is not consistent

    i think you're making it a little harder than it needs be, this is how I would approach it

    starting eqation
    GMm/((H+r)^2)=0.99*GMm/(r^2)

    cancel GMm as its on both sides
    1/((H+r)^2)=0.99/(r^2)

    invert each side
    (H+r)^2 = (r^2)/0.99

    so try from here, i would start by taking the sqrt of everything.

    The main rule in algebraic manipulation is: as long as you do the same to the Left Hand Side as you do to the Right Hand Side its ok... you preserve the equal sign
     
  11. Mar 5, 2009 #10

    turin

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    Yeah, just be careful not to divide by zero and take the appropriate branch of sqrt.
     
  12. Mar 5, 2009 #11
    ok so at this point it would go like this:

    square root of both sides results in:

    H+r = r/0.994

    rearrange equation to give:

    0.994H + 0.994r = r

    further rearrange to give:

    0.994H = 0.0050r

    divide both sides by r:

    0.994H/r = 0.0050

    isolate for H/r:

    H/r= 0.0050/0.994 = 0.005

    YES THAT IS THE RIGHT ANSWER!!!

    thanks so much for your help! good to know I had the right idea and it's just my crappy math skills that suck!
     
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