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## Chapter 6. Equilibrium of rigid bodies

A rigid body is said to in equilibrium if the resultant of all forces and all their moments taken about any and all points are zero.

Free body diagram (FBD): depiction of an object with ALL the external forces acting on it.

Reactions coming from the supports: usually a body is constrained against motions using supports. It is essentially to correctly estimate the number and type of reactions that a support can provide.

A slide-share presentation about equilibrium can help you to understand the key concepts of this chapter:

https://slideplayer.com/slide/4577628/

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## Chapter 4: Rigid bodies. (III) – Problems

ROTATIONAL DYNAMICS: EXERCISES

1. A mass M = 120 kg hangs from a beam which is held by a cable.
1. What is the tension T in the cable?
2. Does the wall exert any force on the beam? Which one?
2. A stone of mass 0.6 kg is forced to travel in a circle while it hangs from the ceiling, as shown in Figure. If the angle between the rope and the vertical is θ = 30º and the radius of the circle is r = 35 cm:
1. What is the speed of the stone?
2. What is the tension in the rope?
3. A wheel of radius r = 80 cm has moment of inertia 10 kg m2. It is rotating around its central axis propelled by a rocket attached to a point on its outer rim. The rocket is expelling gas
tangentially to the wheel, resulting in a constant force. Determine:

1. The magnitude of the equivalent force, if we know that the wheel, starting from rest, reaches an angular speed of 1 rev/s in 6 s.
2. The value of both tangential and normal acceleration in a point on the outer rim of the wheel.
3. The angle that the total acceleration forms with the radius at that point.
4. The time that the wheel takes to reach the same angular velocity, under the action of the same force, if we add a very thin ring of mass 5 kg around the outer rim.

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## Chapter 4: Rigid bodies. (III)

Chapter 4: RIGID BODIES III

1. ROTATION. CIRCULAR MOTION.
2. MOMENT OF A FORCE (TORQUE).
3. MOMENT ANGULAR CONSERVATION.

Massachusetts Institute of Technology have developed Open Course Wares (OCW) where you can follow a “Classical Mechanics course” by Internet.

One of the most important magnitudes in architecture is the moment of a force or torque. And this magnitude will be extremely important in Chapter 5: Equilibrium.

Linear momentum, i.e. p = m·v, has a rotational analogue that it is called angular momentum, L. For a symmetrical object rotating about a fixed axis through the centre of mass (CM), the angular momentum is L = I·ω where I is the moment of inertia and ω is the angular velocity about the axis of rotation. The SI units for L are kg·m^2/s, which has no special name.

We saw in Chapter 2: Dynamics laws and applications that Newton’s second law can be written more generally in terms of momentum ΣF = Δp/Δt. In a similar way, the rotational equivalent of Newton’s second law which is Στ = I·α, can also be written more generally in terms of angular momentum Στ = ΔL/Δt where Στ is the net torque acting to rotate the object and ΔL is the change in angular momentum in a interval time Δt.

Angular momentum is an important concept in physics because, under certain conditions, it is a conserved quantity. If Στ = ΔL/Δt on an object is zero then ΔL = 0, so L does not change. This is the law of conservation of angular momentum for a rotating object:

The total angular momentum of a rotating object remains constant if the net torque acting on it is zero.

• A video tutorial about rotation can be watched on the MIT-OCW webpage:

https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/week-10-rotational-motion/30.1-introduction-to-torque-and-rotational-dynamics

• A video tutorial about the moment of a force/torque can be watched on the MIT-OCW webpage:

https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/week-10-rotational-motion/30.4-torque

• A video tutorial about rotational dynamics can be watched on the MIT-OCW webpage:

https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/week-10-rotational-motion/31.1-relationship-between-torque-and-angular-acceleration

• A video tutorial about solved problems can be watched on the MIT-OCW webpages:

Worked example: moment-of-inertia-of-a-disc-from-a-falling-mass

Worked example: massive-pulley-problems

MisConceptual Questions

1. The symmetric simple truss is loaded as shown in Figure. Which force shown exerts the largest magnitude torque on the truss around point A? And around point B?
2. Calculate the net torque around point O due to the forces acting on the plate shown.

Video lecture: Rotational dynamic example

Problem set number 5

1. A wheel of radius r = 80 cm has moment of inertia 10 kg·m^2. It is rotating around its central axis propelled by a rocket attached to a point on its outer rim. The rocket is expelling gas
tangentially to the wheel, resulting in a constant force. Determine:

1. The magnitude of the equivalent force, if we know that the wheel, starting from rest, reaches an angular speed of 1 rev/s in 6 s.
2. The value of both tangential and normal acceleration in a point on the outer rim of the wheel.
3. The angle that the total acceleration forms with the radius at that point.
4. The time that the wheel takes to reach the same angular velocity, under the action of the same force, if we add a very thin ring of mass 5 kg around the outer rim.

If you have some doubts, you can watch next video related to rotational dynamic (Professor Michel van Biezen):

If you were satisfied with this example, you can check more video lectures on the webpage:

http://ilectureonline.com

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## Practise 3: Moment of inertia of plane surfaces

Please, first of all review following materials you can download by UACloud:

1. presentations AP1_chap00, AP2_chap01 and
2. template where you can check how to write laboratory reports “Sample_Physics_Lab_Report”.

You should finish second practise related to “Hooke’s law” and submit it as soon as possible by “Evaluation tools“.

PRACTISE 3: MOMENT OF INERTIA OF PLANE SURFACES

NOTE: I will upload practise guide today (in Spanish).

1. AIMS:
1. Build any irregular shape surface which value of the moment of inertia with respect an axis (X or Y) is known.
2. Determine its centre of gravity experimentally.
3. Write your lab report explaining all processes you did, results in tables and calculations.
2. PROCEDURE:
1. The moment of inertia (with respect rectangular axis X or Y through one side) should be I = 0.010 kg/m^2 with an uncertainty of 10%.
2. Area should be composite by three regular areas at least, but not equals, i.e. a rectangle, a triangle and a semicircle but not two triangles and a rectangle.
3. Moreover, an empty area will be required in the composite figure (here you can repeat previous shapes).
3. EXAMPLES:NOTE: You can do it by group of two people and take into account you must not present the same area, i.e. exactly the same measurements or shapes.

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## The laboratory report

This entry help you to prepare laboratory reports for all general science and engineering at PUC-EPS-UA. It describes the structure of a good laboratory report, outlines the different sections of the report, and explains the need for each of them. It also introduces some standard conventions and rules for writing reports of professional quality.

Laboratory reports will be graded not only for technical content but also for writing and style. The quality of your written report will strongly affect your grade for the course. Students are required to follow the general rules in this entry and the specific format instructions given to them by their laboratory instructor.

You can read a complete guide to laboratory report writing here or here (see below a copy) and an example here. Just you need to adapt this rules to our course.

# General Outline of a Laboratory Report

Scientific writing is just as important as scientific investigation or experimenting. Although the major part of scientific investigation takes place in the laboratory–connecting equipment together, repairing, obtaining supplies and samples, checking each apparatus for consistency, calibration, and finally data collection by running the experiment—a great deal of time is spent to present the results in a concise, objective, critical and conclusive format called laboratory report (similar to research paper). Therefore, a well-organized laboratory report is much more effective and influential than one without a structure. There is no short list of instructions for writing a good laboratory report. You may have only one chance to influence your reader. While ineffective writing can turn off the readers, a well-written laboratory report can have impacts on your reputation, chance of employment or promotion. You may also draw the attention of the scientific community to your work and retain them as your readers.

Sections of a laboratory report:

A laboratory report usually have several sections identified by titles. A typical report would include such sections as TITLE, INTRODUCTION, PROCEDURE, RESULTS, and DISCUSSION/CONCLUSION. If you are using a computer to type your work, section headings should be in boldface.

Title:

The title can usually draw attention of the reader to your work. It should clearly represent the work presented. If the purpose of the experiment is to measure the gravitational acceleration of the earth using pendulum as the experimental apparatus, the title should be like “ Measurement of the Gravitational Acceleration Using Simple Pendulum”. Avoid “The” as the first word in the title for it will lead to misleading searches when one uses the database.

Introduction:

State the purpose of the experiment in general terms. For example, “ It is possible to measure the gravitational acceleration using the oscillations of a simple pendulum.”

Review the existing information or the theory. Reader will look for some reminder of the basic information relating to this particular area. This can be done by giving him/her a brief summary of the existing state of knowledge. We can also include a summary of earlier work with proper references.

Supply a paragraph or two about how the basic information, such as an equation representing the behavior of a model (theory), can be used to make measurements.

Procedure:

Indicate what parameter or properties of the system you are measuring. Usually you change a parameter of the system (such as changing the temperature, independent variable), and measure its effect (such as the length of a metal rod, dependent variable).

Specify such measurement details as the type of standard or instrument used to make the measurement (for example, meter stick or vernier calliper, etc). Give the instrument uncertainties. For example, if we are using a meter stick, we can say, “ the length of the rod is measured using a laboratory meter stick accurate to within 1 cm. You may also give, if necessary, an apparatus diagram.

Results:

• Provide tables showing your measurement with units.
• Describe the uncertainties: standard, instrument, random errors
• Provide graphs. Graphs should be neat, clear, and include the axis label and units.
• Computation of the final answer: slope calculation, averages, and standard deviations all in proper significant figures.

Discussions/Conclusions:

• Present your findings from the experiment.
• Evaluate the outcome objectively, taking a candid and unbiased point of view. Suppose that the outcome is not close to what you expected. Even then, after checking your results, give reasons why you believe that outcome is not consistent with the expected. Make it plain, simple. Make factual statements such as “graph 1 shows a linear variation of velocity with time”.
• State the discrepancies between the experimental results and the model (theory), and discuss the sources of the differences in terms of the errors by offering logical inferences.
• Suggest improvements

Although these do not make an exhaustive list of do’s or don’ts, they nevertheless offer a framework around which one can write an effective report. In our experiment, some of the items indicated under each section may not be needed. I will give you more feedback in class. I expect that, the lab reports, either typed or handwritten, should be neat, clear, and organized. Points will be deducted for these, as well as for missing units and failing to follow the outline (i.e. title, introduction, procedure, results, conclusion) given above.

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## Physics laboratory tutorial

We can say that physics is the science of the measure. Unfortunately, this means we have to learn the error analysis treatment of our data. You can download all the documents from our campus virtual at University of Alicante (in Spanish or Catalan/Valencian). Nevertheless, it could be interesting to read the Physics Laboratory Tutorial from the Columbia University as well.

We did video laboratory experiments in physics (in Spanish) you can watch by this link.

or watching this YouTube channel at University of Alicante link (in Spanish too).

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## Chapter 11. Structural members: beams

Chapter 11. INTERNAL FORCES IN STRUCTURAL MEMBERS: ISOSTATIC PLANE BEAMS

11.1 Isostatic beams. Introduction.

11.2 Reactions at supports.

11.3 Types of loads on beams.

11.4 Internal forces in beams. Sign convention.

11.5 Loads, shears and axial forces.

11.6 Bending moments.

11.7 Graphical analysis of a beam.

11.8 Elastic curve of a beam.

You should learn to solve any isostatic plane beam after the theoretical lectures, the problem sessions and the homework you did during this part of the subject. Chapter objectives can be  summarized as follows:

• To show how to use the method of sections to determine the internal loadings in a member.
• To generate this procedure by formulating equations that can be plotted so that they describe the internal shear and moment throughout a member.
• To draw the shear and moment diagrams for any isostatic plane beam.

The next video explains how to solve a beam using the method of sections (Spanish from a Professor at the Universitat Politècnica de València).

https://media.upv.es/#/portal/video/74c39eb3-0e90-504d-80f2-9664a5f4b92a

Bibliography

Rodes Roca, J. J., Durá Doménech, A. i Vera Guarinos, J., Fonaments físics de les construccions arquitectòniques (Publicacions de la Universitat d’Alacant, Alacant, 2011). Capítol 12.

Rodes Roca, J. J., Exercicis i problemes dels fonaments físics d’arquitectura. I. Vectors lliscants i geometria de masses (ECU, Alacant, 2009)

Rodes Roca, J. J. i Durá Doménech, A., Exercicis i problemes dels fonaments físics d’arquitectura. II. Estàtica aplicada a les estructures (Col·lecció Joan Fuster 154, Universitat d’Alacant, 2013)

Tipler, P. A. i Mosca, G., Física per a la ciència i la tecnologia, Volum 1 (Reverté, Barcelona, 2010). Capítols 1 i 12.

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## Chapter 10. Multi-member structures: trusses

Chapter 10. INTERNAL FORCES IN STRUCTURAL MEMBERS: PLANE TRUSSES.

10.1 Plane trusses. Introduction.

10.2 Assumptions made in truss analysis.

10.3 Isostatic and hyperstatic systems.

10.4 Method of joints.

10.5 Method of Maxwell-Cremona.

10.6 Method of Ritter or sections.

You should learn to solve any plane truss after the theoretical lectures, the problem sessions and the homework you did during this part of the subject.

Trusses from Amr Hamed

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## Why do you have to know magnitudes and units so well?

Usually the students thought that this aspect of physics, in particular, and science, in general, is not important. However, it is not the same mistakes in an exercise on paper than on a real problem of engineering or science.

To give a value to this entry, we attached some news and the causes of the accidents that have occurred for reasons that should never happen.

1. The NASA’s mission Mars Climate Orbiter (September 1999), had a part of the engineering team working in English units (feet, inches and pounds) and another one working in the decimal metric system. Incredible but true and below are some links for verification:
1. Official site for the mission here.
2. CNN news related to this fact, here and here.
3. News on the Washington Post.
4. Or on the BBC.
2. On July 23, 1983, Air Canada Flight 143, a Boeing 767-233 jet, ran out of fuel at an altitude of 41,000 feet (12,000 m), about halfway through its flight originating in Montreal to Edmonton. Fuel loading was miscalculated due to a misunderstanding of the recently adopted metric system kp which replaced the imperial system pounds.
1. CBC digital archives.
2. Another reference in a classroom of mathematics, here.
3. News on the New York Times.
3. The Tokyo Disneyland’s Space Mountain:
4. Can you find more examples?
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## Rules or common sense?

The best way to learn physics is practise, practise, and practise. You need to check whether you understood all the concepts or not, therefore, try to think logically and correlate your questions with real life situations. I saw in this web-page some ideas for learning physics and other technical courses. In summary, these are some points to take it into account:

1. Never miss a class. Ever. Although you do not believe it, you can learn physics with lectures.
2. Never fail to do every problem of every assignment.
3. If you are required to hand in problem solutions, do the problem twice. The first version should go in your own notebook, along with all the failed attempts. The second should be a copy to hand in.
4. Always prepare for each class. That means have a look at what is coming up in the text or notes after you have done the assignments. Check the guide of the subject.
5. Write out your work for every problem clearly. Show every step, even if your phone mobile has 1 Gb of memory.
6. Do not ever try to erase your mistakes, just cross out with a single line.
7. Always draw a picture for each problem and label it clearly.
8. To study for tests, do problems. Write down any formulas each time you use them and you will know them by heart without any further effort.
9. Always ask for help, but make sure that you have done your part before you go to the teacher. This means that you must work out the offending problem neatly up to the point where you lose the trail.
10. All that really ever works is to review and to practise solving problems.
11. Learn to draw a good graph, properly labelled and scaled.
12. Always do your own work, especially in laboratory settings. That means preparing your own report on your own, even if the data was collected by someone else.
13. Always prepare for the laboratory: know what you are going to do and how you are going to do it.
14. Last but not least, it is important in your career to demonstrate your integrity as a student and as a person. A reputation for honesty will serve you far better than any course grade. It is incredible, isn’t?!!!!