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?

#### 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.

(F_R)_x=5-F\text{ sin30}^0

(F_R)_x=5-0.5F

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

Notice that F\text{ 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.

(F_R)_y=F\text{ cos30}^0-4

(F_R)_y=0.8660F-4

(Again, we simplified the equation as cosĀ 30^0 = 0.8660)

Now, we will use the Pythagorean theorem to find the magnitude of the resultant force, F_R.

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_R\frac{\text{d}F_R}{\text{d}F}=2F-11.93

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

(Set \dfrac{\text{d}F_R}{\text{d}F}=0)

0=2F-11.93Solving 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:

(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.