How Do You Calculate Force in Equilibrium Using Calculus?

[BigFlorida]

Homework Statement

• The figure shows a heavy weight suspended by a cable and pulled to one side by a force F. We want to know how much force F is required to hold the weight in equilibrium at a given distance x to one side (say to place a cornerstone correctly). From elementary physics, T cos θ = W , and T sin θ = F .
  1. (a) Find F/W as a series of powers of θ.
  2. (b) Usually in a problem like this, what we know is not θ, but x
     
     and l in the diagram. Find F/W as a series of powers of x/l.
     
  
  

  Homework Equations

  [/b]
  
  Tcos(theta) = W
  Tsin(theta) = F
  sin(theta) = (x/l)
  cos(theta) = sqrt(l^2-x^2) = sqrt(1-(x/l)^2)[/B]
  
  

  The Attempt at a Solution

  
  
  The first part was a relatively simple solve, it was just a power series of the tangent function.
  
  The second part is what I am stuck on.
  
  I have (F/W) = (x/l)(1/sqrt(1-(x/l)^2)), but I have no clue how to turn the right side of this equation into a power series. Should I just create 2 different functions (one where x is a function of l and another where l is a function of x) and write 2 power series, or is there a way that I am not seeing? I have also tried just letting theta = arcsin(x/l) and substituting that into the power series I found in part 1, but that solution was incorrect. I am at a complete loss. Any help would be much appreciated.
  
  Thanks in advance!

 

[BigFlorida]

Never mind, I solved it. It was a simple binomial expansion that I was overlooking.

 

[anvesh210591]

I am still struck at the second part. can you please explain how you have solved it.
Thanks in advance