Optimization variables problem

[physstudent1]

This problem has to do with physics but it is from my calculus book, and for my calc class so I put it here:

Homework Statement

"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.)"

Homework Equations

Apparent weight = W-FsinTheta

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.

 

[AlephZero]

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.

 

[physstudent1]

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

 

[AlephZero]

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.

 

[physstudent1]

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

 

[AlephZero]

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).

Yes, take the derivative and set it equal to zero.

You can do it your way and write F = (rest of equation) if you want, but it's might be easier just to differentiate it as it is.

I'm going to write t for Theta 'cos I'm lazy so your equation was

F cos t = k(w - F sin t)
F cos t = kw - kF sin t
Differentiate with respect to t (remember k and w are constants)
Using the rule for differentiating a product of two functions f and g: d(f.g)/dt = f dg/dt + g df/dt

dF/dt cos t - F sin t = - k dF/ft sin t - kF cos t

At the minimun dF/dt = 0 so

- F sin t = - kF cos t
tan t = k

Actually your way gives the same answer: you have

-kw(-sinTheta+kcosTheta)/(cosTheta+ksinTheta)^2 = 0

Multiply both sides by (cos Theta + k sin Theta)^2

-kw(-sin Theta + k cos Theta) = 0
k and w are not = 0 so -sin Theta + k cos Theta = 0
tan theta = k

 

[physstudent1]

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 a lot! To get the Force I'm thinking I can plug in tan theta for k into the original equation, thanks for all the help.

 

[AlephZero]

To get the Force I'm thinking I can plug in tan theta for k into the original equation, thanks for all the help.

You got it.