# If two hooks support weights on a string across a span, how much weight can it support?

Assume there are two hooks, each rated as able to hold 5 pounds. They are both mounted to a wall, one at the north end of the wall and the other at the south end of the wall. The wall is 10 m long. If a 10 m string, weighting 1 pounds is tied to each hook and goes along the wall horizontally, and weights placed at regular intervals along the string, what amount of weights can this setup hold?

J---------------------------J
|   |   |   |   |   |   |
o   o   o   o   o   o   o


Does the fact that there are two 5-lb hooks mean it can hold 10 lb or just 5 lbs still? Does the fact that the weights aren't directly under the hooks create a lever-like situation that changes the hook's ability to support as many weights as if the weights were just hung directly on the hooks?

Interesting question! I'm going to consider a more simple example first because I'll probably make a mistake otherwise.

Consider a light string 10 meters long, strung up between 2 hooks that are 10 meters apart. I'm going to fasten a mass 'm' to the middle. The gravitational force this object exerts is F = mg, so the question becomes: How heavy can I make this mass before the hooks can't hold anything?

This is going to sound crazy, but in this situation any positive mass would cause the hooks to snap off! Let's think about why this is. If we draw a free-body diagram, we're going to have a T shape, where the force of the mass is pointing down to the ground, and the opposing tension forces are pointing perpendicular to the hooks. So basically the hooks have to exert a sideways force, to counteract the downward force, which can't be done! You'd have to have infinite tension in the strings.

Another way think about this geometrically is to understand that the tension force of the string is going to be proportional to $$\tan\theta$$. We see we run into a problem when $$\theta = \frac{\pi}{2}$$!

So basically this means that if you want to hang anything from a horizontal string, you always have to have some slack.

Edit : A diagram was requested:

• If I add a little slack then, can it hold some weight? 5 lb? 10 lb? – Village Aug 15 '19 at 3:09
• So the amount of weight the string can hold is dependent on how far apart the hooks are. We see that when the hooks are 10m apart they can't hold anything. When we bring them closer together, they can hold more and more because there's less sideways tension. (Remember, sideways tension is proportional to $\tan{\theta}$. The max amount the string can hold is when the hooks are right next to each other, meaning they'll hold just a little under 10 pounds. – Visipi Aug 15 '19 at 3:15
• A diagram and some math would go a long way here. The answer doesn't really answer the central question either. – Gert Aug 15 '19 at 3:31
• I'll post a diagram in a bit. If you want to talk about how a uniform distribution of weights along a string would look and how much weight it can support, this is a problem which depends on the number of weights you want to consider. In the continuous limit where the rope has a given linear density $\sigma$ You'll actually end up with a string that looks like a hyperbolic cosine when it hangs. It's called a Catenary. Here's the wiki for it. en.wikipedia.org/wiki/Catenary – Visipi Aug 15 '19 at 4:06
• certainly a string with no slack can hold some amount of weight, otherwise guitar strings would snap when touched – Adrian Howard Aug 15 '19 at 5:24