Determining Range of Values of P for Taut Cables

[yaro99]

Homework Statement

Two cables are tied together at C and loaded as shown. Determine the range of values of P for which both cables remain taut.

Homework Equations

$\Sigma F_x=0$
$\Sigma F_y=0$

The Attempt at a Solution

$\Sigma F_x=\frac{ 4 }{ 5 }*P-\frac{ 600 }{ 650 }*T_{AC}=0$
$\Sigma F_y=\frac{ 250 }{ 200 }*T_{AC}+T_{BC}+\frac{ 3 }{ 5 }*P-480=0$

I am ending up with 2 equations and 3 unknowns. How can I eliminate the variables?

 

[rock.freak667]

You'll need to take moments about a point then. Try about point A.

 

[yaro99]

My knowledge on this is a bit rusty. Taking the moment about point A, I know for BC I would have: 600*T_(BC)
I am not sure about the others, but here is my guess: $ƩM_A=0.6*T_{BC}+0.6*(600/650)*T_{AC}-0.6*480+0.6*(2/5)*P=0$

Is this correct?

 

[haruspex]

You need to use the critical condition that the cables remain taut. If P is too small for that, which cable will go slack? What if P is too great?

 

[yaro99]

Setting P=0, it seems that cable AC will go slack. I am not sure about if P is too great. It looks like it would be BC. I am still confused.

 

[haruspex]

Setting P=0, it seems that cable AC will go slack. I am not sure about if P is too great. It looks like it would be BC. I am still confused.

You may be confused, but you're getting there . So what equation do you get for BC going slack?

 

[yaro99]

You may be confused, but you're getting there . So what equation do you get for BC going slack?

This is where I'm confused. I took an arbitrarily large number and set it equal to P. Plugging this into both equations, T_(BC) becomes a large negative value. Not sure if I'm doing this correctly.

 

[Chestermiller]

Homework Statement

Two cables are tied together at C and loaded as shown. Determine the range of values of P for which both cables remain taut.

Homework Equations

$\Sigma F_x=0$
$\Sigma F_y=0$

The Attempt at a Solution

$\Sigma F_x=\frac{ 4 }{ 5 }*P-\frac{ 600 }{ 650 }*T_{AC}=0$
$\Sigma F_y=\frac{ 250 }{ 200 }*T_{AC}+T_{BC}+\frac{ 3 }{ 5 }*P-480=0$

I am ending up with 2 equations and 3 unknowns. How can I eliminate the variables?

Your coefficient of T_AC in the y force balance is incorrect. It should be 250/650.

Solve this pair of equations for T_AC and T_BC as a function of P. Make a graph or a table of T_AC and T_BC versus P. Each of the cables will go slack if the tension in the cable is less than zero. Find out the range of P that makes this happen for each of the cables. For example, you can immediately see from the x- force balance that cable AC will go slack if P is less than zero.

 

[haruspex]

This is where I'm confused. I took an arbitrarily large number and set it equal to P. Plugging this into both equations, T_(BC) becomes a large negative value. Not sure if I'm doing this correctly.

You don't need to try plugging in an arbitrary value for P. As P increases from 0, what will T_BC be at the point where BC goes slack?
(Also, note the correction Chestermiller mentions to your Fy equation.)

 

[yaro99]

Your coefficient of T_AC in the y force balance is incorrect. It should be 250/650.

Solve this pair of equations for T_AC and T_BC as a function of P. Make a graph or a table of T_AC and T_BC versus P. Each of the cables will go slack if the tension in the cable is less than zero. Find out the range of P that makes this happen for each of the cables. For example, you can immediately see from the x- force balance that cable AC will go slack if P is less than zero.

Thank you! I did this and got the correct answer. Here is what I did:

I rearranged the equations like this:
$T_{AC}=P*\frac{4}{5}*\frac{650}{600}$
$T_{BC}=480-\frac{250}{650}*T_{AC}-\frac{3}{5}*P$


Then I created these tables:

Since T_BC is positive at P=514 and negative at P=515, 514 must be the maximum value of P.

Is this the only way to do this problem? Is there any easier method that takes less time?

 

[haruspex]

Is there any easier method that takes less time?

Yes - answer my question in post #9.

 

[yaro99]

You don't need to try plugging in an arbitrary value for P. As P increases from 0, what will T_BC be at the point where BC goes slack?
(Also, note the correction Chestermiller mentions to your Fy equation.)

T_BC would be 0 where BC goes slack. But then what do I set T_AC equal to?

 

[haruspex]

T_BC would be 0 where BC goes slack. But then what do I set T_AC equal to?

With T_BC = 0 you now have two equations and two unknowns. Solve them.

 

[yaro99]

With T_BC = 0 you now have two equations and two unknowns. Solve them.

Ah, right, I wasn't thinking

Indeed this yields the same answer. Thanks!