The lateral force at one point has to equal the friction, and at the other equal or exceed it, in order for the member AB to be in impending motion. (You can intuit this by imaging a frictionless contact at B and realizing that it still can’t move until friction is exceeded at A.) Note that if the movement of A is impending toward C, then B must be impending away, and vice versa.
I looked through the formulation developed in the solutions manual and it uses a geometric approach that I suspected existed but seemed overly complex. It does have the merit of providing an elegant solution to find the range of θ for static stability analytically, but it is completely dependent upon ACB being a right triangle, and so it isn’t the approach I would use for a general solution for a rod suspended between to curves.
To illustrate what I’m saying, let’s look at a simpler problem: Both walls are at 45º, the rod is horizontal, and the problem is just to find the normal and frictional forces at the ends. Both frictional forces being zero is a valid solution to this problem, because static friction is allowed to be less than mu*N. Both frictional forces being maximal is also a valid solution, as is any symmetric solution with a ratio between f and N anywhere between 0 and mu.
If even this simpler problem does not have a unique solution, then how can the more complicated one?
Yes, there isn’t enough information to solve for all the forces in that case, but that’s not what the problem is asking for. Imagine sliding the rod a bit to the left and letting it go; friction will hold it in place. If you keep doing that, eventually you’ll reach a point where the static friction isn’t enough, and it will slide back down and to the right. The problem is asking for where that limiting point is, at which the rod will just barely stay in place. And the answer will be the angle θ.
So we know that at least one end of the rod will be at the limit of static friction, where friction/normal = 0.25. you can then replace those with a single resultant force at a known angle.
My question still is, does that apply at both ends of the rod at the same time (as the online solution uses), or should it be treated as a case of finding which end is more critical than the other?
I’m beginning to think that the online solution is valid. The rod isn’t moving, but if you slide it just a smidgen more, it will slip back down. Motion is impending at both ends, and friction at both ends is helping to hold it in place. I suspect that the rod will stay in place at a higher angle with friction at both ends than figuring either end in isolation.
I know that the simplified problem I posted isn’t the same problem. But it illustrates the same issue: The rod can be laid in gently, or it can be jammed in, and if all we know is the positions of all of the objects, then we can’t tell which case we’re in. In the original problem, a jammed-in rod will stay stable at angles more extreme than the gently-laid rod. The solution you found online, where both ends are assumed to have maximal friction, is the solution to the maximally-jammed case, while the assumption you made, that friction is zero in one of the ends, will give you the solution for the gentle case.
In the real world, maybe, since real-world friction is complicated. But in textbook-problem world, the whole point of a coefficient of friction is that it’s only dependent on the materials in question. In the jammed-in case, both the normal forces and the frictional forces would be increased.
The bar is constrained to slide along both surfaces. If one end slides so does the other. If one of the ends is at its limit and the other end is not, it won’t slide. So the limiting case is both ends at their limit.
The two relations between normal force and friction coefficent, equilibrium of force in two directions, and moment equilibrium allow the limiting value of theta to be found. Force equilibrium in normal direction at point A, gives an expression for the normal force at A in terms of the weight and shear force at B. Force equilibrium in normal direction at point B gives expression for friction force at A in terms of weight and normal force at B. Using the limiting values of shear forces at A and B gives normal force at B in terms of only the weight, alpha and friction coeficent. Moment equilibrium (I took it about point A), results in an expression in terms of just friction coefficent, alpha and theta, which can be solved for theta (length and weight of the bar cancel out of the expression). That gives theta is alpha - 2atan(mu) as one solution and alpha + 2atan(mu), depending on which direction it slides.
This is one of those times when I wish I was taking a class, and could ask the professor. As much as I like this place and appreciate all the answers, it’s a little tough to ask these questions over text. It would be nice to have a whiteboard.
It’s going to take me a while to do the math, but I will come back to this answer.
There is no mistake. As has been repeatedly noted, this is a textbook for a statics course which is intended to teach the principles of mechanics on rigid, non-moving bodies and structures which do not deform or store energy as an internal stress/strain condition. The member AB cannot be “jammed in” in a way that puts it into compression or has a variable internal strain energy condition which will result in a statically indeterminate condition with two points of contact.
As previously discussed, the values for the static coefficient of friction at contacts A and B can be considered as free parameters which vary independently from 0.00 ≥ μ ≥ 0.25 as required order to solve for the condition with all forces in balance. In the particular case of α = θ = 45°, that is a singular case in which the value of μ is indeterminate because there is no frictional force required in that symmetrical condition, but the problem isn’t asking for a solution at that point; it is asking the student to find the limit conditions on angle θ for the defined angle α at which static friction will just hold the member from impending motion. This is a solvable problem by at least two different methods which do not require any consideration for deformation of the member or any details about the contact other than the stated relationship between the normal force and static friction.
That’s fine if you’re teaching a course just for fun, consisting of meaningless brainteasers. It’s completely useless, though, if the point is to learn anything at all about the real world. Which, last I checked, is what engineering is all about, and hence the way that the vast majority of engineering courses are structured.
I assume the textbook already says in the beginning of the chapter/section to assume or approximate that the body is rigid, in which case you cannot expect them to reiterate each time that the rod is not bent or jammed in there.
This problem, as well as the one you critiqued previously, are not “meaningless brainteasers”. They are impelling the student to assess how to solve problems by examining the underlying conditions and find simplifying assumptions and methods rather than to use rote methodology in a brute force approach, which is a critical thinking skill being rapidly lost by successive generations of engineers who have become far to conditioned to just feed all analyses into integrated simulation tools without thinking about whether the question can be bounded with a more simple calculation and some rational assumptions. Working through these kinds of problems such as finding a limit condition are crucial to developing that intuition for how to approach “real world” problems in the most efficient manner, and provide fundamentals to build upon to apply more complex phenomena such as deformation, contact phenomena, et cetera, in the same way that having a first semester chemistry student do basic product/reaction equations for even though the reality is that many chemical reactions are highly dynamic and will not come to a stoichiometric equilibrium quickly if at all.
As it happens, not only have I taken an wide array of courses in the mechanical engineering curriculum but have also been a teaching assistant for for both Statics and Mechanics of Materials courses and the lab for the latter; we used Hibbeler but Beer & Johnson is a popular choice. I also worked on development of tutorial software for statics, dynamics, and mechanics of materials to illustrate critical principles to students, as well as subsequently having worked as a working mechanical engineer approaching three decades in heavy machinery and aerospace industries, so I think I have a little bit of authoritative knowledge about how “engineering courses are structured” and whether the problem sets drawn from this text are solvable and illustrative of useful principles.
I’m not talking about bending the rod. You can do a lot by assuming that objects don’t have any strain at all, and a lot more by assuming that strains are infinitesimal. But the ambiguity in this problem doesn’t come from strain; it comes from stress. And you can’t do any statics at all by assuming that stresses are zero or infinitesimal.
I was feeling kind of impressed with myself for taking on the challenge of this textbook just for the joy of learning; no credits, no diploma, no job advancement. It’s a little humbling to be reminded of how much I still don’t know about the subject.
I think I said before that I’m using an old edition of the textbook. There are some computer exercises and tools; they’re on a 5.25" floppy disk inside the back cover.
I was thinking it’s analogous to Physics classes; solving for objects moving under constant forces, with no friction or air resistance, etc. It’s not a perfect representation of the real world, but it’s still useful. I think I’m using a roughly first-year textbook, and the complications would be taught later.
This problem has nothing to do with stress, and no information is given to calculate stress in the member or at the contacts. It is purely a loads analysis.
Although the principles that are being taught in that text are ‘basic’, they are being applied in ways that people often have a poor native intuition without the experience of working through actual problems. Working through the text on your own is no mean feat, and even though you are not going it with the goal of occupational advancement or formal credit, it will equip you to be able to understand how and why civil structural failures occur, or why some structure can’t be made arbitrarily light and cheap.
That is correct; the next course in the sequence is Mechanics of Materials, where you learn about elasticity, the relationship between stress and strain (and the difference between ‘engineering strain’ and true strain), how to calculate deflection from shear and bending moments, Poisson’s ratio and volumetric stress, distortion from torque and combined loading, Mohr’s circle and applying stress/strain criteria, and how to determine or at least estimate stresses in statically indeterminate structures. Some schools break that into two courses depending on whether they are on a semester or trimester/quarter system, and how far into the weeds they get into some of the more advanced areas of buckling, fatigue and fracture, et cetera.
After that, it depends upon the engineering discipline but you generally take a couple of material science-type courses, and then more applied engineering classes like civil structural analysis, geotechnical engineering, machine design, mechanism synthesis, aerostructural analysis, composite structural design, piping and pressure vessel design, continuum mechanics, et cetera where those fundamentals are applied to problems closer to being representative of real-world engineering challenges where you often need that intuition about how to appropriately frame or simplify the load and boundary conditions to make the problems readily solvable.
Solve what? The problem doesn’t ask for the student to determine contact stresses (which definitionally aren’t within the scope of a point contact between two rigid bodies), and it isn’t necessary to get the answer to the problem as stated, to wit determine the angles for which the rod is stable with static friction. You seem to be trying to “solve” a problem that isn’t anywhere in the text, and hijacking the thread in order to do so rather than to just answer the question of the o.p.