How to Keep a Sign from Collapsing

[postfan]

Homework Statement

. The sign shown below consists of two uniform legs attached by a frictionless hinge. The coefficient of friction
between the ground and the legs is µ. Which of the following gives the maximum value of θ such that the sign will
not collapse?

(A) sin θ = 2µ
(B) sin θ/2 = µ/2
(C) tan θ/2 = µ
(D) tan θ = 2µ
(E) tan θ/2 = 2µ

Diagram here : http://www.aapt.org/physicsteam/2014/upload/exam1-2013-1-6-unlocked.pdf
Page 2, Diagram 6

Homework Equations

The Attempt at a Solution

I drew a FBD and got a force of gravity for each leg with force mg down (relative to the surface on the ground) , normal force of mgcos((180-theta)/2) up (relative to the surface of the leg), and friction of mu*N towards the interior of the sign. The question is where do I go from here?

 

[phinds]

Since the two forces you have computed don't oppose each other, you aren't going to get far that way. How about you find the force that the friction force IS opposing?

 

[postfan]

Friction opposes the horizontal component of the normal force which is 1/2*mg*sin(180-theta), right?

 

[phinds]

Friction opposes the horizontal component of the normal force which is 1/2*mg*sin(180-theta), right?

Well, what do you think?

 

[postfan]

Well I am pretty sure that that's it, but I want to make sure.

 

[phinds]

Well I am pretty sure that that's it, but I want to make sure.

I didn't look at your figures but you have the right idea. The friction is a horizontal force so obviously the only force that can directly oppose it is a horizontal force.

 

[postfan]

So is the next same just to equate the horizontal component of the normal force with the force of friction?

 

[phinds]

So is the next same just to equate the horizontal component of the normal force with the force of friction?

You seem very hesitant to embrace the obvious. I suggest that you spend some time studying force vectors in various kinds of problems, not just ones where the forces that matter happen to be horizontal.

 

[postfan]

I'm hesitant because I already equated it with each other and got mu=cos(90-theta/2) which is not one of the answer choices.

 

[phinds]

I'm hesitant because I already equated it with each other and got mu=cos(90-theta/2) which is not one of the answer choices.

That could be due to a calculation error, but can't be due to the forces not cancelling each other when the end is right at the point of moving.

My point is that once you've done enough of these problems you'll have an instinctive feel for what forces have to cancel each other and an answer that seems wrong will just lead you to recheck your calculations, not your logic.

 

[BvU]

I always learned that normal forces are just that: normal forces. Normal (perpendicular) to the horizontal plane in this case. So I don't think there is such a thing as a horizontal component of the normal force in this exercise.

 

[postfan]

Ok so in that case m*g*sin(theta) = mu*m*g*cos(90-theta/2) ?
If so anyway to simplify cos(90-theta/2)?

 

[phinds]

I always learned that normal forces are just that: normal forces. Normal (perpendicular) to the horizontal plane in this case. So I don't think there is such a thing as a horizontal component of the normal force in this exercise.

That's correct. But there IS a horizontal component to the ladder's force on the floor. There are two resultant forces from the ladder's force on the floor, the normal force and the horizontal force.

 

[postfan]

OK I'm confused.

 

[phinds]

OK I'm confused.

To such an utterly vague statement, I can only say, well that's unfortunate. I'm not. Do you have a question? Do you understand force vectors at all? If you don't then that's where to start.

 

[postfan]

You two are telling me conflicting things.

 

[phinds]

You two are telling me conflicting things.

OK, I certainly don't mean to be. What two things are you talking about? Do you want me to guess or would you like to tell me?

 

[postfan]

You are telling me to use the horizontal component of the normal force but BvU is saying to just use the normal force without decomposing it.

 

[phinds]

You are telling me to use the horizontal component of the normal force but BvU is saying to just use the normal force without decomposing it.

No, I most emphatically am NOT telling you to use a component of the normal force. Please re-read what I said in posts #6 and #13

 

[postfan]

I was using the horizontal component of the normal force.

Friction opposes the horizontal component of the normal force which is 1/2*mg*sin(180-theta), right?

And you said ...

I didn't look at your figures but you have the right idea.

 

[phinds]

I was using the horizontal component of the normal force.

Ah ... that's where the confusion set it. Sorry. Since there IS no horizontal component to the normal force, I just read this as meaning that you were using the horizontal component for the force that the ladder exerts on the floor, which is the right thing to do. Since, as I said, I did not look at your calculations, I did not see that that apparently was not what you were doing.

I AGAIN urge you to read what I said in post #13.

I'm done with this conversation. Since you clearly do not understand force vectors, I can only say once again that what you need to do is learn about force vectors in general before trying to solve a problem involving force vectors.

 

[BvU]

One place where there is a horizontal force in action is at the floor level. There is one other place.
Look under the relevant equations to see what you have available as conditions for equilibrium (i.e. non-collapse). It may well be that the resultant there is zero, but it helps in the math for each of the legs.

 

[BvU]

I'm sorry to see that Phinds is worn out a little. My impression is that's partially caused by the medium we are working with: it's easy to trigger misunderstandings with such indirect communication. Can't be helped at the moment. I am completely convinced of Phinds good will and expertise. I am also convinced of postfans sincerity, curiosity and courage in tackling a challenging set of AAPT contest questions, so I'll go along a bit more.

Going back to square 1 (post #1?) with this exercise (usually going under the tags ladder - wall, e.g. here -- check out post #4 for some relevant equations !), I think a lot of hassle would have been avoided with the picture (good for 1k words, they say) you drew ! (Can you still show it ?)

Things start well in post #1with $2mg$ down acting at the center of mass of the whole contraption -- awkward place, but probably cancels anyway. Easier to deal with if we use symmetry and look at one leg only: $mg$ halfway the ladder. (You can see I'm too lazy to carry the 2 around, so I use m for the mass of one leg only).
[edit:} Can't do that: too confusing.

Things start well in post #1 with $mg\$ down acting at the center of mass of the whole contraption -- awkward place, but probably cancels anyway.

Easier to deal with if we use symmetry and look at one leg only: ${1\over 2}mg\$ halfway the ladder. Straight down.​

Then a second force "normal force $mg\; \cos\left ( (180^\circ - \theta)/2 \right )$ up (relative to the surface of the leg) " -- here you can see how useful a picture would have been ! As well as a bit of basic trig: $\cos (\pi/2 - \alpha) = \sin\alpha$. Apparently we are projecting $mg$ (That is the actual normal force) on a line perpendicular to the leg. For the right leg would point a bit up and to the right. Project that onto the floor and presto: there is a horizontal component where there was no horizontal component before. No good.

Would be wiser here to not lump the two legs together. One normal force is ${1\over 2}mg$ pointing straight up at the point the right leg is on the floor. You can guess the second normal force is ${1\over 2}mg$ pointing straight up at the point the left leg is on the floor. Together they nicely add up to $mg$ pointing straight up and this is compensating gravity: the stand won't sink into the floor, nor will it float up. Both normal forces are equal, which is good too: the thing won't play dog and lift one leg :) or start rotating in a vertical plane any other way.
​

Third force in post #1: $\mu N$ inwards. Let me not be corny and go on about lumping together. There is $\mu {1\over 2}mg$ pointing to the left at the point the right leg is on the floor, etc. Left and right add up to 0. Everybody happy -- if that were all. And it isn't, as the bloke who looked at each leg separately has noticed already: the legs lean against each other and a fourth force comes in the picture. Two actually: left on right and vice versa.

Now we can have a sidebar about the direction of these forces -- acting at the hinge, which is frictionless, thank goodness.
Is it purely horizontal, or is it in the direction the tip would want to go (i.e. perpendicular to the leg), so that the legs don't just lean against one another, but also ON one another. I'm in favor of the latter (thinking of one heavy leg and one light leg). I hope someone else reads this far and is knowledgeable enough to settle this.
The good news is it doesn't matter, because of the symmetry: both normal forces at the bottom will remain at ${1\over 2}mg$ . And at the top we only need the horizontal component -- to compensate $\mu {1\over 2}mg$ at the bottom.
​

As they say: one picture... The above isn't far short of a thousand words (and cost me a lot of time).


Back to work: four forces per leg, dragged together in such a way that the force balance is in proper order. One more condition for equilibrium to fulfil. My bet is the ${1\over 2}mg\$ divides out :) and the L too.

 

[postfan]

Thanks for your confidence BvU! I've read and reread your post several times (plus drawn a few FBDs!) and this is the part I still don't understand.

Then a second force "normal force mgcos((180∘−θ)/2)mg\; \cos\left ( (180^\circ - \theta)/2 \right ) up (relative to the surface of the leg) " -- here you can see how useful a picture would have been ! As well as a bit of basic trig: cos(π/2−α)=sinα\cos (\pi/2 - \alpha) = \sin\alpha. Apparently we are projecting mgmg (That is the actual normal force) on a line perpendicular to the leg. For the right leg would point a bit up and to the right. Project that onto the floor and presto: there is a horizontal component where there was no horizontal component before. No good.

How is mg "the actual normal force" and not mg*sin(theta/2)?

 

[BvU]

Would the normal force be 0 when theta is zero ?

 

[postfan]

No , in that case the normal force would be mg.

 

[BvU]

Normal force is always mg, straight up. Because weight is always mg, straight down. It doesn't sinkl into the ground, it doesn't float away into thin air.
And from symmetry you can see each leg presses on the ground with a force mg/2

 

[postfan]

That makes sense. So are there 2 "normal forces" once acting perpendicular to the leg half way up and one acting straight up from the ground?

 

[BvU]

For each leg there is one normal force. Where the leg touches the ground. Magnitude mg/2. Straight up.

There is nothing that could possibly exercise a normal force halfway up.
Gravity, however, effectively acts at the center of mass. magnitude mg/2 (m/2 is the mass of one leg). Halfway up the ladder, and pointing straight down.

These two give $\Sigma \vec F = 0$ but not $\Sigma \vec \tau = 0$. That we have to achieve from friction at the bottom and the hinge at the top. By choosing a decent center of rotation, we can minimize the amount of calculation work.