I’m a maths tutor, so this isn’t asking for help to complete homework. One of my students is also studying physics, and asked me about the problem shown on page 24 on the following link:
He, myself and just about everyone else think that the force is going to be exerted at 90 degrees to the horizontal rod. However, the mark scheme claims it follows the line of action of support rod Y, so acts at 35 degrees. I can’t think of any earthly reason why this should be the case - can anyone weigh in and explain how this might be true?
Something missing from the explicit statement of the problem but implied by the paucity of information in it is that member Y has to be assumed to be pin-connected (that is having a rotational degree of freedom at each end making it incapable of accepting externally applied moments) to both member X and the column. Because of this, it can only experience axial loads, and reactions are also aligned with the axis of Y. You know this because if member Y were permitted ‘rigid’ connections with no degrees of freedom, the load condition would be statically indeterminate and you would require geometric and mechanical properties of the members which are not provided.
That’s interesting. So is this what you’re saying:
The lack of information provided means that the only sensible model is one where the support rod experiences axial loads- yes?
I’m curious now as to your opinion on something. Given that every idea you used to explain the answer is not covered by the syllabus for this exam, is the question fair?
Thanks for the reply. It does make sense - but I’m not enough of a practical engineering type to ever spot that. Neither was my student, or their teacher!
Well, that depends. If this question were being posed in an introductory physics class those assumptions of rigidity and the rotational degree of freedom should explicitly stated. In an engineering ‘statics’ (basic static mechanics) course, the default assumption is that when you are looking at a truss frame load scenario (of which this is a trivial example) you are dealing with rigid members that are (generally) pin connected because analysis dealing with flexible members or negative degrees of mechanical freedom generally requires applying principles of deformable members and mechanics of materials, which is a more advanced class. In a more advanced engineering physics or mechanical/civil/aerospace engineering course, part of the pedagogical method is to encourage students to understand the limitations of the knowledge about the problem and apply appropriate assumptions such as (in this case) assuming a pin-connected frame even though that isn’t stated explicitly or shown in figure. That is a reasonable assumption for this case because assuming that members X and Y are slender (small in sectional dimensions compared to the length) then that assumption will give a reasonable approximation even if the members are connected with no rotational degree of freedom, i.e. welded, bolted, or otherwise ‘rigidly’ connected to the column and each other. However, that may not be intuitive to someone who hasn’t done a series of these kinds of problems.
By the way, the third line of the problem is not correct, hence the note that it only gets the first two marks. The third line calculates the vertical component of the reaction from member Y onto X but there is also a substantial component in the horizontal direction, so the 662.145 N⋅m term should be divided by 0.57 (sin(35°) or cos(55°)) to get the combination of the horizontal and vertical components. The solution should also show and carry units all the way through (or state what units are being used for each quality), and should be limited to two significant figures since the stated mass and lengths are each two figures.
Thanks for the comprehensive reply - I appreciate it.
" If this question were being posed in an introductory physics class those assumptions of rigidity and the rotational degree of freedom should explicitly stated."
These concepts are not covered in the syllabus for this high school level physics course. If you’re at all curious - and I don’t blame you if you’re not - here’s the specification:
It’s very, very basic. As far as any of them are aware, reaction forces are “normal” - end of discussion. The kind of depth and detail that you’re discussing and explaining is far removed from what they are learning at this level.
Yeah, agreed for a high school level course this assumption is non-intuitive. In engineering mechanics courses for statics and mechanics of materials we have labs that go along with the coursework to reinforce this intuition with physical examples that the students have to work through. For a problem like this, as an example, you might have a setup with an inline load cell on X, and then insert members of different length (and thus angle) on Y and different masses, have them develop the formula to predict the tensile load on X, and make a plot of the relationship in Matlab or Python/Matplotlib. But that is probably beyond the scope of a high school physics lab unless it is an honors or university credit class.
This is indeed just a standard course for 16-18 year olds - not an advanced class. This question was on the final, national exam for the qualification though, which is why I find it so interesting. Not just some random textbook question getting a bit carried away, or engaging in stretch and challenge enrichment - the actual final. The examiner’s report contains no discussion on the issue, apart from noting that “many students assumed the reaction force to be normal to the horizontal” - which of course they would, at this level.
Rod Y is stationary, so the sum of moments at any point along the rod must be zero. If we use as a reference the end of the rod attached to the column, then there is only one force acting on the rod - therefore the component of the force normal to the rod must be zero for the moment to be zero. Hence the force applied to the rod must be along the length of the rod.
If you miss this shortcut you can also solve for zero net moment along rod B from (a) the calculated force normal to rod A (which is 125 degrees from normal on rod B) and (b) an unknown force acting along rod A (which is 215 degrees from normal on rod B), than calculating the resultant. More math, same result.
This seems more like a problem from Introduction to Statics in college rather than high school physics, but maybe kids these days are smrter than we were back then…
Hi Marvin, thanks for weighing in. Your explanation makes sense to me on a more intuitive level - I hadn’t thought of considering moments on the support, but I really should have.
One clarification though I’d like to ask you about. This makes pretty obvious sense in this case because the support rod is considered to be massless. So, as you say, the sum of moments around any point must be zero, which explains the line of action of the force being along it’s length.
But…what if the rod is NOT considered massless? It seems to me (the simple-minded high school math teacher) that this changes the situation, or at least gives us a longer walk to the same conclusion. I’m probably missing something, but I’m curious as to the answer.
Members X and Y being massless or not don’t really have any bearing on the fact that member X is assumed to be a rigid member in pure compression; if they had masses would just add more loads which would make for a more complicated problem but it doesn’t really change the general state of the problem.
If you put every technique and standard assumption in the syllabus, you wouldn’t have a syllabus any more; you’d have a textbook. I would expect a syllabus to include a list of topics, which in this case would be something like “static torque problems”.
If a problem like that is showing up in homework, then (assuming any basic level of competence from the teacher) problems similar to it would also have shown up in class, and the teacher would have shown how to do them (including all of the implicit assumptions).
If the rods have mass, you would add the mass as “weights” hanging down from the center of mass of each rod (the midpoint if the rods are uniform). Then solve the two moment equations with these additional moments added.
This assumes the rods are perfectly rigid. If not, this goes way past high school
A valid point. But nothing like that is mentioned in the syllabus, or has come up on previous final exams. It seems very much outside the scope of the syllabus, which is why it’s such an odd question.
Also, as a reminder. Not my subject, just my student. So my competence is very much not being put on trial here
Seems to me that they/you are overthinking it. The angle is irrelevant in this.
If you were lifiting the assembly at the point X meets Y, what force would you have to exert to prevent it from pivoting at the column end? it’s a simple lever calculation. The force is the force needed to maintain the system as static. No more, no less.
The angle is relevant to examining the forces’ breakdown but does not change the total.
True, interesting rivets or pins or whatever could change the answer, but absent any such information, we have to assume that the X-to-column point has no support strength, at least unless Y compresses or bends. If it did, it would be stated in the question to affect the answer. With that we’d be getting well into Engineering design.
Huh? Y has compression, X is subject to bending moment
You are making the same mistake as the OP. The angle does indeed make a difference. Your answer is correct if rod B is vertical so all the force it applies to rod A is also vertical. Once rod B is at another angle there are both vertical and horizontal components to the force - rod B both applies the force required to hold rod A horizontal (your lever calculation) and applies a force which pushes rod A to the left (and which is countered by as equal rightward force from the post, preventing rod A from moving).
Interesting. Now I’m trying to imagine the forces involved if B were vertical, and if B were also purely horizontal but contacting A only at the same point. What are the forces on A if there were no B?
Also, the force of B on the column must balance the force of A on the column for a static situation.
If we dig into this, it becomes a very complex problem I would think… hence this thread. Well above high school level?
No. The document the OP linked to shows the correct solution using moments. The OP’s misunderstanding was due to an erroneous simplification they made (assuming rod B only provides a vertical force to rod A). We can use the same moment technique to calculate the forces exerted by the pole to rods A and B.
