Determine the length of l (equilibrium)

[ND3G]

A ring of negligible size is subjected to vertical force P. Determine the required length l of the cord AC such that tension acting in AC is TAC. Also, what is the force acting in cord AB?

P = 300N, d = 0.2m (distance between A & B), angle ABC = 45 degrees, TAC = 250N

The graph basically looks like a "Y" with C & B at the tops, A at the centre and P the downward force.

Fx = 0; Fab(cos45) - 250cosx = 0
Fy = 0; Fab(sin45) + 250sinx - 300 = 0

Fab = (250cosx/cos45)

(250cosx/cos45)sin45 + 250sinx - 300 = 0

This is about as far as I get before I start mucking it up. Can some one help me through the steps to solve for "x" so I can eventually find l?

 

[weatherhead]

okay, if you're sure that's the equation you want to solve, then you should realize it's a pretty nasty beast. You're unlikely to get an exact answer, but you can tidy it up so it looks a lot nicer.
First, try dividing through by 50.
then, think what sin 45/cos 45 is (hint - trig identities)
in the end you've still got an equation that looks like this:
a cosx +b sin x +c = 0
this is an absolute pig to solve, your only chance is to do it numerically, or "translate" one of the functions into the other - do you know how to do this?? i.e cos(a)=sin(a+x)

 

[m2003]

I would approach this problem by first defining the variables and their relationships. In this case, we have the force P acting on the ring at point C, which creates a tension TAC in the cord AC. The cord AB is also under tension, but we do not know the magnitude of this tension. The distance between points A and B is given as d, and the angle ABC is 45 degrees.

To find the required length l of the cord AC, we can use the equilibrium condition, which states that the sum of all forces acting on an object must be equal to zero. In this case, we can write the following equations:

ΣFx = 0: TACsin45 - Tabcos45 = 0
ΣFy = 0: TACcos45 + Tabsin45 - P = 0

We can solve these equations simultaneously to find the value of TAC, which is given as 250N. Substituting this value into the first equation, we can solve for Tab, which comes out to be approximately 176.78N.

Now, to find the length l of the cord AC, we can use the trigonometric relationship:

sinx = opposite/hypotenuse

In this case, the opposite side is d/2 (since the ring is at the midpoint of the cord), and the hypotenuse is l. Therefore, we can write:

sin45 = (d/2)/l

Solving for l, we get l = d/√2. Substituting the value of d (0.2m), we get l = 0.141m or approximately 14.1cm.

Finally, to find the force acting in the cord AB, we can use the relation:

cosx = adjacent/hypotenuse

In this case, the adjacent side is d/2 and the hypotenuse is l. Therefore, we can write:

cos45 = (d/2)/l

Solving for l, we get l = d/√2. Substituting the value of d (0.2m), we get l = 0.141m or approximately 14.1cm.

Therefore, the force in the cord AB is equal to Tabsin45 = 176.78N.