# Homework Help: Vectors minimum component

1. Mar 10, 2016

### newbphysic

1. The problem statement, all variables and given/known data
The truck is to be towed using two ropes. If the
resultant force is to be 950 N, directed along the positive x
axis, determine the magnitudes of forces FA and FB acting
on each rope and the angle q of FB so that the magnitude of
FB is a minimum. FA acts at 20° from the x axis as shown.

2. Relevant equations

3. The attempt at a solution
I have no idea where to start because there is only 2 known values so i cant use sine law or cosine law. What's the first step ?

2. Mar 11, 2016

### ehild

Write the equations for the x and y components of the forces FA and FB. Solve for FB in terms of the angle q. At what q is FB minimum?

3. Mar 11, 2016

### newbphysic

FAx = FAcos(20)
FAy = FAsin(20)

FBx = FBcos(q)
FBy = FBsin(q)

can you help me with q ?
i know q must make
FBy=-FAy
FBx + FAx = 950
But i dont know how to find it.

4. Mar 11, 2016

### ehild

Plug in the expressions with the angles for the force components.

5. Mar 11, 2016

### newbphysic

FBx + FAx = 950
FBcos(q) + FAcos(20) = 950
FB cos(q) + 0.94FA = 950

FBy=-FAy
FB sin(q) = - 0.34 FA

6. Mar 11, 2016

### ehild

Now eliminate FA.

7. Mar 11, 2016

### newbphysic

Fb cos(q) + 0.94 Fa = 950
Fb sin(q) + 0.34 Fa = 0
--------------------------------------------- -

$$\frac{Fbcos(q)}{0.94} - \frac{Fbsin(q)}{0.34}= \frac{950}{0.94}$$

8. Mar 11, 2016

### ehild

Isolate FB.

9. Mar 11, 2016

### newbphysic

$$Fb ( \frac{cos(q)}{0.94} - \frac{sin(q)}{0.34})= \frac{950}{0.94}$$

$$Fb = \frac{950}{0.94} / (\frac{cos(q)}{0.94} - \frac{sin(q)}{0.34})$$

10. Mar 11, 2016

### ehild

Fb is function of q and you need to find the minimum of that function. Have you studied Calculus?

11. Mar 11, 2016

### newbphysic

only calculus 1

12. Mar 11, 2016

### ehild

What is the derivative of a function at a minimum or maximum?

13. Mar 11, 2016

### newbphysic

zero

14. Mar 11, 2016

### ehild

Take the derivative of Fb with respect to q.

15. Mar 11, 2016

### newbphysic

derivative of cos and sin will result another sin and cos
what should i do with it?

16. Mar 11, 2016

### ehild

That expression would be equal to zero. You will be able to solve that equation for q.

17. Mar 11, 2016

### newbphysic

hmm,it looks like mission impossible . Do you know another method ehild ?

18. Mar 11, 2016

### ehild

It is a fraction equal to zero. What do you know about the numerator, if the fraction is zero?

19. Mar 11, 2016

### newbphysic

numerator must be zero which means q must be 0 for sine and 90 for cosine

20. Mar 11, 2016

### ehild

q is the same angle both for sine and cosine. What equation do you have for q?

21. Mar 11, 2016

### newbphysic

my guess q must be 135(-225) or 315(-45) because sin and cos have different sign there

22. Mar 11, 2016

### ehild

Do not ques, solve. What is the equation?

23. Mar 11, 2016

### newbphysic

950 sin(q) + 2626.47 cos(q) = 0

24. Mar 11, 2016

### ehild

OK. divide the equation by cos(q). Replace sin(q)/cos(q) by tan(q). Solve for tan(q).

25. Mar 11, 2016

### newbphysic

All right, thanks a lot ehild.
One more question though, there is other method that says in order for Fb to be minimum it must perpendicular to Fa ? Can you explain why it works ?