Determining Maximum Lamp Mass for Coplanar Force System

[Robb]

Homework Statement

Determine the maximum mass of the lamp that the cord system can support so that no single cord develops a tension exceeding 350 N

Homework Equations

The Attempt at a Solution

F(x)=0; F(ED)cos30 - F(CD)= 0
F(y)=0; F(ED)sin30 - W

F(ED)= 2W
F(CD)= 1.73205W

F(X)=0; 1.73205W - F(AC)(3/5) - F(CD)cos45 = 0
F(y)=0; F(AC)(4/5) - F(BC)sin45 = 0

F(AC)= F(BC).88388

This is where I start getting confused. With all the substitutions.

 

[Tallus Bryne]

F(X)=0; 1.73205W - F(AC)(3/5) - F(CD)cos45 = 0

This is incorrect. Check your FBD and make sure you're writing out your equation with the correct subscripts.

 

[Robb]

From the lower fbd:
F(x)= 0; 1.73205W - F(AC)(3/5) - F(BC)cos45

 

[Tallus Bryne]

From the lower fbd:
F(x)= 0; 1.73205W - F(AC)(3/5) - F(BC)cos45

Alright, so substitute in for F(AC) in this equation. Use this to write F(BC) in terms of W. Afterwards you can then write F(AC) in terms of W. You will then have expressions for all the forces of tension in the ropes in terms of W. Comparing all of these you should be able to determine which rope develops the maximum tension in this system.

 

[Tallus Bryne]

Also, as a quick alternative to going through all the math: Examine each FBD carefully. Assuming static equilibrium of the ring at point D, it can be said that F(DE) must be larger in magnitude than F(CD) since only one component of F(DE) balances the force F(CD). A slightly more careful consideration of how this strategy applies to the FBD of point C should lead you to find which rope is under the maximum tension. As you can see, most of the trouble in this problem is finding that particular rope.

 

[Robb]

So, F(AC) = .88388F(BC)
1.73205W - F(BC).53033 - F(BC).7071 = 0
-1.23743F(BC) + 1.73205W
F(BC) = 1.39971W

F(AC) = 1.23718W

W = 150

The answer is 17.8? Not sure how to get there.

 

[Robb]

Also, as a quick alternative to going through all the math: Examine each FBD carefully. Assuming static equilibrium of the ring at point D, it can be said that F(DE) must be larger in magnitude than F(CD) since only one component of F(DE) balances the force F(CD). A slightly more careful consideration of how this strategy applies to the FBD of point C should lead you to find which rope is under the maximum tension. As you can see, most of the trouble in this problem is finding that particular rope.

I like the insight. Now that you mention it it looks obvious.

 

[Robb]

So, F(AC) = .88388F(BC)
1.73205W - F(BC).53033 - F(BC).7071 = 0
-1.23743F(BC) + 1.73205W
F(BC) = 1.39971W

F(AC) = 1.23718W

W = 150

The answer is 17.8? Not sure how to get there.

See edit to W.

 

[Robb]

Ok. I got it. Sorry. Thanks for your help!

 

[Tallus Bryne]

W = 150

Ok. I got it. Sorry. Thanks for your help!

No problem. Just one last thing.
Not sure if your edit on your post for the value of W worked. Just to clarify it should be 175 N. Although I will assume that you must have gotten that result if you calculated 17.8 kg as the mass.