Statics Solved Problems Pdf

[Catrin Muzquiz]

Auniform rod of mass m and length l is hanging vertically from the pivot O. A horizontal force F acts at the lower end of the rod. If F always remains horizontal then maximum angular displacement of the rod is______.

First I thought the problem is of statics but later thought that the rod is actually moving so I finally thought it should be dynamics problem. Now I don't know how to approach to this problem. Am I lacking knowledge or am I not intelligent enough.My only question is how tricky or hard problems are solved by begginers like me because textbooks gives theory and leave us with problems where we either confuse or didn't know the process of solving. Sometimes it feels like demotivating because I didn't see any progress in solving problems after spending a lot of time.Should I again read the theory or just think what is happening in the problem by imagination. Or should I experiment the problem with real objects and predict the solution methodology (Which I don't know).I know it is a general question but I really want to know what the textbooks really expecting from students. Please tell how to tackle problems of hard level and process required for achieving that level.

First, if you think a question is ambiguous, meaning you think the words have more than one legitimate interpretation in terms of what is physically going on and what is being asked, then a correct scientific response is to state clearly, in your answer, that you think there is this ambiguity, and then make clear in your own work what physical problem you will address. If you are pressed for time, then you would probably pick whichever is the simple reading of the problem, or whichever seems to match best to the course content you have followed. But if you have plenty of time then you can answer all versions of the problem! Just make a list and answer them one by one. In the present example a static interpretation is legitimate and simpler, so you should probably answer that way in the first instance. But then you could add that the wording could be construed as allowing a dynamic case, and then you could tackle that. (If you are in an exam with time limits then probably you should not spend too much time on that if you think the static answer has a good chance of being what the question really meant).

(Regarding ambiguity, I would like to recount a nice story which I read in a book by Martin Gardner. It concerns a question which asked "How would you use a barometer to measure the height of a tall building?" A clever student gave a whole list of answers, ranging from "dangle it on a rope till it touches the ground then measure the length of the rope and add the height of the barometer" to "take it to the caretaker and offer to give him the barometer if he will tell you the height of the building". Of course the answer the examiner expected was to discuss the air pressure variation with height. A good student would have known this and so would have written that type of answer. It is valuable to note that almost any statement can be interpreted in more than one way, and one aspect of scientific understanding is to have good instincts on how to interpret statements.)

Here is an example of an extremely simple case of a statics problem: you have a bowl, you release a marble at an arbitrary point. The marble rolls down, overshoots, rolls back, etc, etc. After a little while friction has removed all motion. When the marble has come to rest, where has it come to rest?

The marble-in-a-bowl case is very simple, but this approach of finding where the potential is lowest is the way to mathematically derive things like the shape of a soap film between two circular pieces of wire

So you do need to take into account, mathematically, in what way change of position of the rod affects the force balance. Still, to find the answer it does not matter along which trajectory the rod arrives at the final position.

This is a dynamics problem. You have to equate the work done by the force $F$ with the energy gained by the rod when it has reached its maximum angular displacement from the vertical. Note that at this point of maximum displacement the angular speed of the rod is zero, so its kinetic energy is also zero.

I would base this decision on context. If the problem is from a section of the course (or book) dealing with statics, treat it as a statics problem. Find the angle where the torque produce by the horizontal force balances the torque produced by gravity. As a dynamics problem you want an angle where the work done by (F) equals the increase in the potential energy of the rod and the K.E. = 0.

In the absence of context, I would be inclined to treat it as a statics problem as I would interpret "maximum angular displacement" to mean the maximum possible angular displacement where static equilibrium is possible for a purely horizontal applied force. See the figures below. But I can see it interpreted as being a dynamics problem as in the other answers.

Now to your actual problem: not understanding problems is sadly an important part of physics and of science in general. Physics can be quite complicated and most things you can't just reason on your own: you have to see a similar problem done or you have to know the theory. So don't be afraid to ask help, either on this site or from your friends or your teachers. How are you supposed to know that you can view this force $\vec F$ as a force similar to gravity so if you add those two together you get a new gravity $\vec F'_g=\vec F_g+\vec F$ and if you rotate your setup until the new force points down the problem is similar to a pendulum only experiencing gravity starting at some angle $\theta$ and this angle $\theta$ is just the angle between $\vec F'_g$ and $\vec F_g$? So again it is normal to struggle a bit and even the smartest people have experienced times where they felt dumb or inadequate. But if you're struggling it does mean that you have to put in work to get to a level where you're struggling less. It might take some time though before you notice the effects of your work.

When working submerged surface problems in statics, remember that all submerged surfaces have a fluid acting upon them, causing pressure. You must compute two pressures: the hydrostatic pressure resultant and the fluid self weight.

Shear and moment diagrams are a statics tool that engineers create to determine the internal shear force and moments at all locations within an object. Start by locating the critical points and then sketching the shear diagram.

In many statics problems, you must be able to quickly and efficiently create vectors in the Cartesian plane. Luckily, you can accomplish your Cartesian vector creations easily with the handy vector formulas in this list:

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A unified analytic solution approach to both static bending and free vibration problems of rectangular thin plates is demonstrated in this paper, with focus on the application to corner-supported plates. The solution procedure is based on a novel symplectic superposition method, which transforms the problems into the Hamiltonian system and yields accurate enough results via step-by-step rigorous derivation. The main advantage of the developed approach is its wide applicability since no trial solutions are needed in the analysis, which is completely different from the other methods. Numerical examples for both static bending and free vibration plates are presented to validate the developed analytic solutions and to offer new numerical results. The approach is expected to serve as a benchmark analytic approach due to its effectiveness and accuracy.

Static bending and free vibration problems of thin plates are two types of fundamental issues in mechanical and civil engineering as well as in applied mathematics, with extensive applications such as floor slabs for buildings, bridge decks and flat panels for aircrafts. In view of their importance, the problems have received considerable attention. Since the governing equations as well as boundary conditions for thin plates have been established long ago, the main focus has been on the solutions, which has brought in a variety of solution methods for various plates. Most of these methods are approximate/numerical ones such as the finite difference method1,2, the finite strip method3,4, the finite element method (FEM)5,6, the boundary element method7,8, the differential quadrature method9,10, the discrete singular convolution method11,12,13,14, the meshless method15,16,17, the collocation method18,19,20, the Illyushin approximation method21,22, the Rayleigh-Ritz method and Galerkin method23.

In comparison with the prosperity of approximate/numerical methods, analytic methods are scarce for both static bending and free vibration problems of rectangular thin plates. The reason is that the governing partial differential equation for the problems is very difficult to solve analytically except the cases of plates with two opposite edges simply supported, which have the classical Lvy-type semi-inverse solutions. For the plates without two opposite edges simply supported, there exist several representative analytic methods such as the semi-inverse superposition method24,25, series method23, integral transform method26 and symplectic elasticity method27,28,29,30,31,32.

It should be noted that many of previous analytic methods are only suitable for one type of static bending and free vibration problems. In this paper, a unified analytic solution approach to static bending and free vibration problems of rectangular thin plates is developed. The approach is implemented in the symplectic space within the framework of the Hamiltonian system. Superposition of two fundamental problems, which are solved analytically, is applied. Therefore, it is referred to as the symplectic superposition approach33. It was first proposed to solve the static bending problems34,35 and was successfully extended to free vibration problems recently36. We thus find a way to analytically solve both static bending and free vibration problems in a unified procedure. When the static bending solutions are obtain by the current approach, the free vibration solutions can be readily obtained without extra methodological effort.

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