Gravitational Force and True Weight

[rascal323]

1. Homework Statement [/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. Homework 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!?

 

[Dr.D]

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.

 

[rascal323]

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??

 

[lanedance]

sounds reasonable to me, why don't you like your answer?

 

[turin]

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.

 

[rascal323]

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

 

[lanedance]

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

 

[rascal323]

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 canceled 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 canceled 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

 

[lanedance]

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

 

[turin]

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

Yeah, just be careful not to divide by zero and take the appropriate branch of sqrt.

 

[rascal323]

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!