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Optimization variables problem

  1. Dec 2, 2006 #1
    This problem has to do with physics but it is from my calculus book, and for my calc class so I put it here:
    1. The problem statement, all variables and given/known data
    "A component is designed to slide a block of steel with weight W across a table and into a chute. The motion of the block is resisted by a frictional force proportional to its apparent weight. (Let k be the constant of proportionality.) Find the minimum force F needed to slide the block and find the corresponding value of theta. (Hint: FcosTheta is the force in the direction of the motion, and FsinTheta is the amount of force tending to lift the block. So the apparent weight is W-Fsintheta.)"


    2. Relevant equations
    Apparent weight = W-FsinTheta



    3. The attempt at a solution

    I set the equation FcosTheta = k(w-Fsintheta)
    I really don't know where to go from here. In every optimization problem we have found 2 equations a primary and a secondary and used the secondary to relate to get rid of variables. Can someone please point me in the right direction I'm pretty lost.
     
    Last edited: Dec 2, 2006
  2. jcsd
  3. Dec 2, 2006 #2

    AlephZero

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    Think back to basic calculus, sketching graphs of function, etc. If you have an equation y = f(x), how do you find the value(s) of x where the maximum or minimum value(s) of y occur?

    Here you have an equation with two variables F and Theta, and you want the minimum value of F.
     
  4. Dec 2, 2006 #3
    the answer is F = kW/(sqrt(k^2 +1)) if that helps anyone (its in the back of the book) I'm really not sure how to get there
     
    Last edited: Dec 2, 2006
  5. Dec 2, 2006 #4

    AlephZero

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    Your equation looks OK.

    Do you know the answer to the question "how to find the min or max of a function y= f(x)" using calculus? If you drew a graph of a function y = f(x), what is the slope of the graph when y = a minimum or maximum? How do you find the slope of a graph?

    When you do that, you will have another equation in F and Theta and you can solve the 2 equations.
     
  6. Dec 2, 2006 #5
    yeah so I just take the derivative of my equation and set it equal to zero right? The only thing I'm not sure is what to set the equation equal to before I find the derivative should I make it F = (rest of equation).

    I think my problem is in actually finding the derivative I get this for the derivative
    F' = (-kw(-sinTheta+kcosTheta))/(cosTheta+ksinTheta)^2
     
    Last edited: Dec 2, 2006
  7. Dec 3, 2006 #6

    AlephZero

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  8. Dec 3, 2006 #7
    I see! That makes a lot of sense I thought about deriving it that way at first, I just didn't realize to plug 0 in for dF/dt. Thanks alot! To get the Force I'm thinking I can plug in tan theta for k into the original equation, thanks for all the help.
     
  9. Dec 3, 2006 #8

    AlephZero

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    You got it.
     
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