Asked and Answered 3.4
Hello, and here we are again. I thought I was done with this series on hanging pictures, but it seems physics never dies — it just gets more complicated.
Some commenters on my previous articles (Why Frames Tilt Forward, The “Hang It with Two Hooks” Calculator, and The Physics of Hanging Pictures) asked how they could hang items on wall studs, if the studs are off-center from the desired hanging spot. This seemed to be a rather specialized topic, and beyond the scope of my series, so I deferred until now. But a recent commenter rekindled my interest and finally inspired me to take a look.
Before I proceed, however, I must mention that I’m not the first to address this problem. The number-one result (as of now) I found in searches for “hang item on off-center studs” is this article on instructables.com by an author named MolecularD. The author describes the principles involved and offers a set of equations (minus the math) that are meant to show the reader where to place the wall hooks. Unfortunately, some readers commented that they did not get the desired result when they followed the author’s instructions.
The solution provided in the inscrutables.com article is such a complicated equation that there is no way for me to verify it without essentially solving the problem myself. Which is what I will do now, taking a somewhat simpler, more intuitive approach.
The four consecutive views in Figure 1 demonstrate the concept:
View (A) depicts a frame hanging on a wall, centered at our desired position (dotted line), using a wire on a single hook. Because of the symmetry of the system, there is no tendency for the frame to rotate one way or the other. Ignore for now the fact that the wire extends above the top of the frame.
View (B) shows the studs in the wall behind the frame (we use a stud-finder to spot them). The two studs are different distances from the center of the frame. We drive a nail into the center of each stud, just touching the underside of the wire. This does not cause the frame to rotate.
In View (C), we attach a piece of wire (blue) to the original wire, from the point where the first nail touches the wire to where the second nail touches the wire, without any slack. The load is now shared between the central hook and the nails in the studs. But this still does not cause the frame to rotate.
In View (D), we snip away the original wire where it touched the nails, leaving our new wire in place. The nails in the studs now assume all the load, with the higher nail bearing more than the lower. Still the frame does not rotate, so we have found the solution.
Obviously, I don’t expect readers to repeat these steps to hang their pictures — this was just a demonstration of concept. Instead I will offer a calculator, with instructions for taking measurements, placing the hooks and cutting the wire, to help the reader achieve the final result.
That is, if you really insist on using studs. Personally, I think it would be easier in most cases to forget about the studs and use the Hang-It-With-Two-Hooks calculator that I presented in my earlier article. You would fasten the hooks to the wall with toggle bolts, which can hold a significant amount of weight when paired with the appropriate hooks. (This video shows how to install them.) But in the end, it’s your call.
Oh, you’re still here! This must mean that you really, really want to use two studs to hang your item. Okay then, onto the intricate details. Please consult Figure 2 (below) to get a sense of the important lengths and measures:
Start by measuring the height H and the weight of the item you want to hang. Then mark the spot 0n the wall corresponding to the top-center of the item. All other measurements will refer to this point.
Next, use your stud-finder to measure XA, the distance from top-center to the center of the closest stud, and XB, the distance from top-center to the center of the next-closest stud.
Now inspect your hanging hardware. You want to (ideally) hide all your hardware behind the item you are hanging, which means the higher hook (A) should not show. Therefore, you should choose a value for ZA, the distance from the top of the frame to the bottom of Hook A, that is slightly greater than the length of the hook.
While you are it, measure the length (D) of the D-rings attached to the item. If you plan to attach the wire directly to the item, then this length is zero.
Your next measurement is WD, the distance between the D-ring attachment points. If you have not yet attached the D-rings to your item, then mark the spots where you think they should be attached, and measure the distance between those marks.
Note that I have not asked you to specify Y, the distance from the top of the frame to the D-ring attachment point, or ZB, the distance from the top of the frame to the bottom of Hook B, or S, the length of wire to cut. These values will be returned by the calculator.
There is one last thing you may have noticed on the diagram: to make the item hang true, you need to install a guide hook below Hook A to equalize the slack in the wire — and the forward tilt — on the left and right sides. More on this later.
The Math and The Calculator
I provide geometric and algebraic solutions in this attachment. The result we are most interested in is:
ZB = ZA+ (XB – XA) tan θ
where θ is the wire angle, tan θ = (Y – ZA)/(WC – XA) and WC = ½ WD.
The formula for ZB assumes that Y, the D-ring attachment point, is a given. But I don’t ask you to specify Y directly, as this involves a judgment call. Ideally, the ratio Y/H would be about 1/5 (the “one-fifth rule”) to minimize forward tilt of the item. But this might call for too small a wire angle and create too much tension in the wire. On the other hand, if the wire angle is too large, and Y/H is greater than 1/3, then the forward tilt could be excessive.
So what I did in the calculator is ask you to specify the wire angle, with 30° as the default. (The minimum entry is 20° and the maximum is the angle corresponding to Y/H = 1/3.) The wire angle is used to back-calculate Y as described in the attachment.
If the default angle seems to provide a reasonable value for Y/H, then go ahead with it, assuming the wire tension is not too high.
If the calculator flags one of your entries as out-of-bounds, don’t ignore it. The calculator will not report any results if Y/H is greater than 1/3, and it will warn you if the estimated wire tension exceeds 25 lbs. (You are responsible for selecting the appropriate hardware.)
Results are reported to the nearest one-eighth-inch. The calculator provides guidance on positioning the guide hook and attaching the wire to the D-rings. The suggested length of wire S includes 6 extra inches (3 inches per side) for tying the loose ends to the D-rings.
It may be a challenge to hang your item on three hooks. I suggest you find a helper, if only for you to have someone to complain to while attempting it. (Still, watch your language.) You might start by feeding the slack of the wire through the guide hook and onto Hook A. Then slide the item toward Hook B and feed the wire over Hook B.
You ask, do I have to use the guide hook? If your item weighs much of anything, then yes. The farther that Hook B is from the center, the more the item will tilt forward at Hook A, since there is more slack in the wire on that side. And the more front-heavy the item, the more uneven the forward tilt will be. The guide hook helps keep the wire close to the wall on the Hook A side.
I end with my usual disclaimer. My calculator makes it easier for a person to hang an item on two off-center studs using hooks and wire. But whether this method is suitable in your situation is a judgment only you can make. You assume full responsibility for your project. I offer this calculator as a convenience, but I accept no liability for damage of any kind, even if the suggestions offered in this post are followed exactly.
If you’re not confident how things are going to work out, you can always do a mock-up in your garage before you mark up your walls.
With that out of the way, good luck. I would be interested to hear about any successes, failures or problems. As always, your suggestions and feedback are welcome.