What is the Tension in Each Cord of a Suspended Object with Three Variables?

[EvanQ]

Homework Statement

Find the tension in each cord in the figure if the weight of the suspended object is w.

Homework Equations

F=ma

The Attempt at a Solution

I've been able to handle these questions when 2 variables are involved, but having 3 has me stumped at the moment.

so far i have only attempted system A. in this i focused on the horizontal aspects first, getting equations for Ta in terms of Tb and vice versa, by using the 0 horizontal components of both w and Tc.

when subbing these into the vertical formula (TaSin30+TbSin45+TcSin90-w=0) i could only manage to get a new formula of Tc in terms of w and wither Ta or Tb, eg. Tc=w-1.366Ta

and here i seemingly hit a dead end.
please help, I'm extremely confused.

 

[learningphysics]

I think you're making a mistake with your freebody diagrams. A good idea is to draw a circle (just a closed loop... not exactly a circle) around what exactly you're taking a free-body diagram of... and see what chords cut the circle... only the chords that cut the circle exert a force on whatever is inside the circle. you still have to consider the weight of whatever is inside the circle... you can assume the chords are massless for this problem.

For example, for part a) if I draw a circle around the suspended object cutting the C chord... so taking the freebody diagram of whatever is in the circle... I only need to worry about the chords that cut the circle ie C... So using this freebody diagram I have only two forces... w and Tc... so it should be clear what Tc equals...

 

[EvanQ]

ahh ok to Tc = w. and then i can attack the other 2 as if Tc is not there...
if i do that i got values of 1.115w for Tb, and 0.5w for Ta, the program then told me that my answer was off by a multiplicative factor. then i thought i'd discovered something seeing as they could both me multiplied by 0 and seeing as c is seemingly taking the full w, that they were both really 0w.

so i tried this and failed, and yes, now down to 1 attempt at submission for each.

can you see where i could have gone wrong with a multiplicative factor?

 

[learningphysics]

ahh ok to Tc = w. and then i can attack the other 2 as if Tc is not there...
if i do that i got values of 1.115w for Tb, and 0.5w for Ta, the program then told me that my answer was off by a multiplicative factor. then i thought i'd discovered something seeing as they could both me multiplied by 0 and seeing as c is seemingly taking the full w, that they were both really 0w.

so i tried this and failed, and yes, now down to 1 attempt at submission for each.

can you see where i could have gone wrong with a multiplicative factor?

Can you write out the equations? It'll be easier for me to check that way...

 

[rootX]

I guess you should pratice a lot!
only then, you would be able to solve these questions in less than 2 minutes.

 

[EvanQ]

ok so I'm trying to follow an example from my textbook, but it's only dealing with 2 strings so that is probably where i am getting stuck.

Ta+Tb+Tc+w=ma
as net force is 0, ma = 0.

then into x and y components:
x:
Tax+Tbx+Tcx+wx=0
-Tacos30+Tbcos45+Tccos90+0=0

y:
Tay+Tby+Tcy+wy=0
Tasin30+Tbsin45+Tcsin90-w=0

 

[EvanQ]

I guess you should pratice a lot!
only then, you would be able to solve these questions in less than 2 minutes.

ha i know.
nothing wrong with starting now and seeking a little help to get on top of it though right?

 

[rootX]

yep,

Ta+Tb+Tc+w=ma
as net force is 0, ma = 0.

then into x and y components:
x:
Tax+Tbx+Tcx+wx=0
-Tacos30+Tbcos45+Tccos90+0=0

y:
Tay+Tby+Tcy+wy=0
Tasin30+Tbsin45+Tcsin90-w=0

assuming that you are analyzing that system at point between A, B, and C threads..(where three threads meet together)
from where did w come in there?

 

[learningphysics]

Your equations aren't right. careful with your freebody diagrams... don't just assume that you need Ta, Tb, Tc and w all in the same equation.

You already know Tc = w.

Now draw a circle around what you want to take a freebody diagram of... there are 2 options I see... take the the freebody diagram of the junction of the three chords... draw a little circle around the junction... so you have 3 forces which must sum to 0... Ta, Tb and Tc (notice that I haven't included w because w acts on the suspended object which is not inside the circle). You can use this to get Ta and Tb...

the other option... draw a circle that encloses the C chord and the suspended object... again you have three forces... Ta, Tb and w (notice that Tc isn't here because it is inside the circle and not an external force on whatever is inside the circle)...

Both these options give the same results since Tc = w...

Let me know if this makes sense... then write out the equations again

 

[EvanQ]

ok so say i take the first of the 2 options:

x component:
Ta+Tb+Tc=0
Ta+Tb=0 (as there is no x direction of Tc)

-Tacos30+Tbcos45=0
Tacos30=Tbcos45
Ta=Tbcos45/cos30

y component:
Tasin30+Tbsin45-Tcsin90=0 (Tc is in the negative direction?)
Tb(cos45/cos30)(sin30)+Tbsin45=w (subbing Tc for w, sin90=1)
0.408Tb+0.707Tb=w
1.115Tb=w
Tb=w/1.115

?

 

[learningphysics]

ok so say i take the first of the 2 options:

x component:
Ta+Tb+Tc=0
Ta+Tb=0 (as there is no x direction of Tc)

-Tacos30+Tbcos45=0
Tacos30=Tbcos45
Ta=Tbcos45/cos30

y component:
Tasin30+Tbsin45-Tcsin90=0 (Tc is in the negative direction?)
Tb(cos45/cos30)(sin30)+Tbsin45=w (subbing Tc for w, sin90=1)
0.408Tb+0.707Tb=w
1.115Tb=w
Tb=w/1.115

?

Yup. That looks right. So I'd right Tb as 0.896575w and Ta as 0.73205w

 

[EvanQ]

sweet thanks.
got them both right.

now need to try for part b :p

do i apply the same rules or does it change up since one cord is coming from the side now?

 

[learningphysics]

sweet thanks.
got them both right.

now need to try for part b :p

do i apply the same rules or does it change up since one cord is coming from the side now?

No it's the same. Just do it the same way you did the previous problem. The main difference is that Ta will have its y-component pointing downwards...

 

[EvanQ]

horizontal component:
-Tacos60+Tbcos45=0
Tacos60=Tbcos45
Ta=Tb(cos45/cos60)
=Tb(2^.5)

vertical component:
-Tasin60+Tbsin45-Tcsin90=0
-Tb(2^.5)(sin60)+Tbsin45=w
0.707Tb-((2^.5)(sin60))Tb=w
Tb=-w/0.5176

which i put in and was wrong. so i misc decided to make the a component positive and put in w/1.932 and was wrong again. then realized i'd put them in the A string place not B string and put them both in for B string, and got them both wrong again.

so down to 1 attempt on both, can you see where i went wrong?

 

[learningphysics]

horizontal component:
-Tacos60+Tbcos45=0
Tacos60=Tbcos45
Ta=Tb(cos45/cos60)
=Tb(2^.5)

vertical component:
-Tasin60+Tbsin45-Tcsin90=0
-Tb(2^.5)(sin60)+Tbsin45=w
0.707Tb-((2^.5)(sin60))Tb=w
Tb=-w/0.5176

which i put in and was wrong. so i misc decided to make the a component positive and put in w/1.932 and was wrong again. then realized i'd put them in the A string place not B string and put them both in for B string, and got them both wrong again.

so down to 1 attempt on both, can you see where i went wrong?

Your angle for A is wrong... I'm sorry, I should have said something about that earlier... The horizontal component for A is Tasin60... and the vertical component is -Tacos60.

post your answers before submitting. I'll check to see that I get the same thing.

 

[EvanQ]

just subbed those in, and for Tb i got w/1.115? same as in example a.

and then for Ta = w/0.724

 

[learningphysics]

just subbed those in, and for Tb i got w/1.115? same as in example a.

and then for Ta = w/0.724

I'm getting Ta = 2.732w and Tb = 3.346w. Can you show exactly how you got your numbers?