Hanging a chair with chain

R

razortag

New member
Joined
Jan 26, 2009
Messages
4
Sorry to bother you guys with a relatively simple physics question. It's just that I posted it on a physics site and it got no traffic.

I have a hanging chair that I'm having some trouble installing in my new apartment.

I don't think my ceiling is strong enough, so I've decided to stretch a 13-foot chain across my room from two hooks on opposite walls, and hanging the chair (and 230 pounds of me) from the middle of it. Fully installed, the chain's lowest point in the middle (where it connects to the chair) is one foot below either wall-hook. How much force must each hook hold while I sit in it?

The geometry involved is an upside-down 1-foot high isosceles triangle, with the shorter two sides being 6.5 feet.

I'm just not good at the physics part of it. But am I right in guessing each hook must hold MUCH MORE than 230 pounds?

Thanks for any input!
- Dave
 
Sorry to bother you guys with a relatively simple physics question. It's just that I posted it on a physics site and it got no traffic.

I have a hanging chair that I'm having some trouble installing in my new apartment.

I don't think my ceiling is strong enough, so I've decided to stretch a 13-foot chain across my room from two hooks on opposite walls, and hanging the chair (and 230 pounds of me) from the middle of it. Fully installed, the chain's lowest point in the middle (where it connects to the chair) is one foot below either wall-hook. How much force must each hook hold while I sit in it?

The geometry involved is an upside-down 1-foot high isosceles triangle, with the shorter two sides being 6.5 feet.

I'm just not good at the physics part of it. But am I right in guessing each hook must hold MUCH MORE than 230 pounds?

The chain makes an angle of arctan(1/6.5) = 8.746º with the ceiling

The supported weight is the weight of the chair plus 230 or (W + 230).

Each leg of the chain supports 1/2 the weight.

The load in the chain is therefore P = [.5(W + 260)/sin8.75º) = 3.286(W + 260)

If the chair weighs 40 lb., the tension load in each leg of the chain would be 300(3.286) = 986 lb. You would be wise to put a conservitive design safety factor of 1.5 on this due to the uncertainty of the wall stud and hook load limits making the ultimate chain, stud, and hook design loads ~2000 lb. each.

Of course, the lower the chain to chair connection to the floor, the lower the loads. For example, if the chain legs could make an angle of 30º to the ceiling, the design loads would be reduced to 450 lb. At 10º, P = 1295 lb. At 15º, P = 870 lb. At 20º, P = 658 lb. At 25º, P = 532 lb.
 
galactus said:
I hope I am interpreting correctly. I have posted a diagram.

I am going to assume a weight of 40 lbs for the chair as Will did. You can adjust accordingly.

The angle is derived from \(\displaystyle sin^{-1}(\frac{1}{6.5})=8.85^{\circ}\)

Code: 
Force                   x component                      y component

T1                       -T1*cos(8.85)                      T1*sin(8.85)

T2                        T2*cos(8.85)                       T2*sin(8.85)

T3                         0                                       -270 lbs.

The first condition for equilibrium gives us \(\displaystyle (1): \;\ \sum F_{x}=T_{2}cos(8.85)-T_{1}cos(8.85)=0\)

\(\displaystyle (2): \;\ \sum F_{y}=T_{1}sin(8.85)+T_{2}sin(8.85)-270=0\)

From (1), T1 and T2 must be equal in magnitude and from (2) the sum of the vertical components must balance the weight.

Solve (1) for T2 in terms of T1 and sub into (2):

The, we can see that \(\displaystyle T_{1}=T_{2}\), which we can easily see since the chain is the same on both ends.

\(\displaystyle 2T_{1}sin(8.85)-270=0\)

\(\displaystyle T_{1}=T_{2}=\frac{135}{sin(8.85)}\approx 877.5 \;\ \text{pounds}\)

Which is close to what Will said.

Make sure you have it re-enforced.
Click to expand...

AHHAAAAA! The chain is 13 feet long and the walls move . I assumed that the distance between the hooks was 13 feet and the chain length varied. Thanks for catching that galactus.
 
galactus said:
It really makes little difference. Your tan is 8.75 compared to sine which is 8.85. I think it only makes about a 10 pound difference. No big deal.
Click to expand...
Well, not so bad Razor: you only need to lose 10 pounds :wink:
 
TchrWill,

Thank you so much for such a well thought-out answer!

I have more questions though...
You didn't intend to change (W + 230) to (W + 260), did you?
Also, how could [P at 10º = 1295 lb], when [P at 8.7º = 986 lb] and [P at 15º = 870 lbs]?
Is this vector math or trig?

Again, thanks for your insight!

Sitting comfortably in mid-air, Dave
 
And now I see galactus's answer as well. Thanks to you as well! I imagined myself drawing a dorky lil diagram as well, but I'm glad you did it. It's much better. =)
 
TchrWill,

Thank you so much for such a well thought-out answer!

I have more questions though...
You didn't intend to change (W + 230) to (W + 260), did you?
Also, how could [P at 10º = 1295 lb], when [P at 8.7º = 986 lb] and [P at 15º = 870 lbs]?
Is this vector math or trig?

Again, thanks for your insight!

Sitting comfortably in mid-air, Dave

Yes I goofed Dave. I knew I should have waited a tad longer after my surgery to get back to work.

I mistakenly assumed that the supporting walls were 13 feet apart as opposed to moving the walls closer together, which seemed impractical. And yes, I added 30 pounds to your weight. Sorry. The corrections to my wrong approach are as follows.


The chain makes an angle of arctan(1/6.5) = 8.746º with the ceiling.

The supported weight is the weight of the chair plus 230 or (W + 230).

Each leg of the chain supports 1/2 the weight.

The load in the chain is therefore P = [.5(W + 230)/sin8.75º) = 3.286(W + 230)

If the chair weighs 40 lb., the applied tension load in each leg of the chain would be 270(3.286) = 887 lb. You would be wise to put a conservitive design safety factor of 1.5 on this due to the uncertainty of the wall stud and hook load limits making the ultimate chain, stud, and hook design loads ~1331 lb. each.

Of course, the lower the chain to chair connection to the floor, the lower the loads. For example, if the chain legs could make an angle of 30º to the ceiling, the design loads would be reduced to 405 lb. At 10º, P = 1166 lb. At 15º, P = 782 lb. At 20º, P = 592 lb. At 25º, P = 479 lb. Each case includes the 1.5 design safety factor and a variable length chain.

Galactus has given you the correct answers for a fixed length chain od 13 feet.
 
You must log in or register to reply here.
Top