# Answered: Determine The Magnitude Of The Force F⃗ ., A Force Is Given As F=X+5Y

Determine the magnitude of force F so that the resultant F_R of the three forces is as small as possible. What is the minimum magnitude of F_R?

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

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Solution:

Let’s first draw the vector components as follows:

The dashed orange arrows represent the x and y components of the force, F.

We can write down these x and y components.

The next step is to add all the x components together, and to add all the y components together. To do this, we will first establish which sides we consider to be positive. We will choose forces acting up and forces acting to the right as positive.

+
ightarrowsum(F_R)_x=sum(F_x)(F_R)_x=5-F ext{ sin30}^0(F_R)_x=5-0.5F

(we simplified the equation as sin 30^0 = 0.5)

Notice that F ext{ sin30}^0 is negative. This is because the x component is acting to the left and we said forces acting to the right is positive.

+uparrowsum(F_R)_y=sum(F_y)(F_R)_y=F ext{ cos30}^0-4(F_R)_y=0.8660F-4

(Again, we simplified the equation as cos 30^0 = 0.8660)
F_R=sqrt{(F_R)_x^2+(F_R)_y^2}F_R=sqrt{(5-0.50F)^2+(0.8660F-4)^2}(Expand the brackets inside the square root and simplify)

F_R=sqrt{F^2-11.93F+41}(square both sides to get rid of the square root)F_R^2=F^2-11.93F+41

Because we need to find the minimum magnitude of F_R we must take the derivative.

(Take the derivative of both sides)

2F_Rfrac{ ext{d}F_R}{ ext{d}F}=2F-11.93

We can now use this to find the minimum resultant force. If we equate frac{ ext{d}F_R}{ ext{d}F} to 0, we will find the minimums.

2F_Rdfrac{ ext{d}F_R}{ ext{d}F}=2F-11.93

(Set dfrac{ ext{d}F_R}{ ext{d}F}=0)

0=2F-11.93

Solving for F gives us:

F=5.964 kN

Now, we can substitute this value back into our square root equation (look above). Our equation was the following:

F_R=sqrt{F^2-11.93F+41}

(Now that we know that F=5.964 kN, we can substitute it in)

F_R=sqrt{5.964^2-11.93(5.964)+41}

Solving for F_R gives us: F_R=2.330,kN

And so, we have our answers. If F=5.96 kN, it will produce the minimum resultant force. The minimum resultant force, F_R=2.330,kN.

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 2, question 2-52.