Finding the tensionin the cords

[Dell]

what i have done is used my equations for equilibrium on the structure, the problem being that i only have 2 that i can effectively use here and i have 4 unknowns

$$\Sigma$$Fy= Ay + By + Cy + Dy - P = 0
$$\Sigma$$Mc= D*L - B*L - 2*A*L = 0


what else can i do here, i tried "cutting" the beam at different places but this exposes another moment and another force so that doesn't help either.

 

[Q_Goest]

Hi Dell.
2 hints in the wording of the question: The bar is "rigid" and there are "four identical wires".

Consider the bar to be infinately rigid and that the wires, being identical, also stretch in proportion. In other words, the wires produce a force as if they were springs. If these were springs, would that help?

Also, consider the way these wires are in tension. Let's look at A, B, C and D in series and rewrite the tension in each of these in tension instead of in the way they provided.

Ta = P/10
Tb = P/5 = 2P/10
Tc = 3P/10
Td = 2P/5 = 4P/10

Notice anything?

 

[Dell]

1, 2, 3, 4, that i saw before, but that's only because they gave me the answers

what does rigid and identical wires help me? since the bar is rigid it has no deformation and if i take the wires as springs F=kx, and i can say that Ka=Kb=Kc=Kd correct??

but from here what do i do?

 

[Redbelly98]

To rewrite what Q_Goest said slightly differently,

Ta = 0.1
Tb = 0.2
Tc = 0.3
Td = 0.4​

Now do you see a pattern?

Since the bar is straight and rigid, the amount by which the wires stretch must be of the form

Δy = mx + b​

...i can say that Ka=Kb=Kc=Kd correct??

Correct.

 

[Dell]

okay that i didnt think of
y=mx+b

very logical, like you said the bar is rigid,

$$\Delta$$A = m*0 + b
$$\Delta$$B = m*L + b
$$\Delta$$C = m*2L + b
$$\Delta$$D = m*3L + b

now what do i so with this, how do i connect this to the tension,
can i say F=k*y using the y's above

then i have 6 equations and 7 unknowns A,B,C,D, K,m,b

 

[D H]

You don't need to know the specific values of the spring constant k, the displacement b, or that scale factor m.

Suppose you know the tension in strings A and B. Given just the facts that the bar is rigid and the strings are identical, you should be able to determine the tensions in strings C and D. Can you do that?

Addendum
Do this, you will have only two unknowns (e.g., T_A and T_B). Use these in the force and torque balance equations and voila! you will get the answer.

 

[Dell]

A + B + C + D - P = 0
D*L - B*L - 2*A*L = 0

if A and B are given i can solve it easily, but they arent,
what am i not seeing here?

 

[D H]

A + B + C + D - P = 0
D*L - B*L - 2*A*L = 0

if A and B are given i can solve it easily, but they arent,
what am i not seeing here?

You did not write C and D in terms of A and B. That the rod is rigid and the strings are identical means you can do just that.

 

[Dell]

A + B + C + D - P = 0
D*L - B*L - 2*A*L = 0

D = B+2A
C = P-3A-2B

now what

 

[Dell]

F=k$$\Delta$$Y

and since the function is linear

Ta=Ta
Tb=2Ta
Tc=3Tc
Td=4Ta

Ta(1+2+3+4)=p

Ta=P/10

and so on,

is this a way to solve it,

do i not need to define $$\Delta$$Y like i did before ??

 

[Dell]

obviously i do, i need that to prove that the force is linear

 

[D H]

F=k$$\Delta$$Y

and since the function is linear

Ta=Ta
Tb=2Ta
Tc=3Tc
Td=4Ta

This happens to give the correct answer here. Hang the weight anywhere other than below points C and this will be wrong. There is however, a generic way to express the tensions in strings C and D in terms of the tensions in strings A and B.

 

[Dell]

i got it thanks

since $$\Delta$$Y is linear, F=KY is also linear, therefore since the forces are equally placed,

Ta=Ta
Tb=2Ta
Tc=3Ta
Td=4Ta
$$\Sigma$$=10Ta=P

Ta=P/10
Tb=2P/10
Tc=3P/10
Td=4P/10

 

[D H]

No, you don't have it! You got lucky here. You do not know ahead of time that Tb=2Ta. For example, if the weight was suspended from the middle of the rod, the tensions in all four strings would be equal.

 

[Dell]

i know, i solved the next question where the weight was at 2/3L but i just recalculated the equations and managed to get it right,

i found 2 equations with P in them, then found the ratio between mL and b, from there i can find the relations of Ta->Tb->Tc... then Ta+Tb+...=P

from there its easy