Basic Probability

Chapter 1Basic Probability. 

This chapter is an introduction to the basic concepts of probability theory. 

Chance Events 

Randomness is all around us. Probability theory is the mathematical framework that allows us to analyze chance events in a logically sound manner. The probability of an event is a number indicating how likely that event will occur. This number is always between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. 

A classic example of a probabilistic experiment is a fair coin toss, in which the two possible outcomes are heads or tails. In this case, the probability of flipping a head or a tail is 1/2. In an actual series of coin tosses, we may get more or less than exactly 50% heads. But as the number of flips increases, the long-run frequency of heads is bound to get closer and closer to 50%. 

Flip the Coin 

Flip 100 times 

For an unfair or weighted coin, the two outcomes are not equally likely. You can change the weight or distribution of the coin by dragging the true probability bars (on the right in blue) up or down. If we assign numbers to the outcomes — say, 1 for heads, 0 for tails — then we have created the mathematical object known as a random variable. 

Expectation

The expectation of a random variable is a number that attempts to capture the center of that random variable's distribution. It can be interpreted as the long-run average of many independent samples from the given distribution. More precisely, it is defined as the probability-weighted sum of all possible values in the random variable's support, 

E[X]=∑x∈XxP(x) 

Consider the probabilistic experiment of rolling a fair die and watch as the running sample mean converges to the expectation of 3.5. 

Roll the Die 

Roll 100 times 

Change the distribution of the different faces of the die (thus making the die biased or "unfair") by adjusting the blue bars below and observe how this changes the expectation. 

Variance 

Whereas expectation provides a measure of centrality, the variance of a random variable quantifies the spread of that random variable's distribution. The variance is the average value of the squared difference between the random variable and its expectation, 

Var(X)=E[(X−E[X])2]

Draw cards randomly from a deck of ten cards. As you continue drawing cards, observe that the running average of squared differences (in green) begins to resemble the true variance (in blue).

Available at https://seeing-theory.brown.edu/basic-probability/index.html. Acceso el 1 de octubre, 2020.

READING: AutoCAD

The Importance of AutoCAD to a Mechanical Engineer

by Jeffrey Joyner

Mechanical engineering is a broad field that encompasses industry, business, medicine and even law. Planning and designing mechanical objects is the primary focus of mechanical engineers, whether they are producing engine and motor components or complete devices like refrigerators and robots. Design programs like AutoCAD help mechanical engineers do their jobs by helping them create preliminary designs and spot flaws before production, saving time and resources.

What is AutoCAD?

 CAD stands for "Computer Aided Design." AutoCAD is a line of two-dimensional and three-dimensional design software produced by the Autodesk company. It includes a powerful suite of features to improve workflow and create true-to-life maps, diagrams, structures and schematics. CAD software is equal part design and analysis. The design is needed to produce models and prepare component production, and the analysis helps calculate stress levels, the influence of forces and the influences of finite elements in a design. According to a General Electric survey, 60 percent of manufactured parts errors were related to incomplete, ambiguous or impossible drafts -- problems easily corrected with the support of software like AutoCAD.

Design Production and Troubleshooting

At the earliest stages of a design project, mechanical engineers can use AutoCAD to start sketching ideas and analyzing them to determine the best solution for a given problem. The software makes the process quick and easy; it eliminates the need to draw new blueprints for each version of an idea and simplifies redesigns. The software additionally helps interpret these designs, locating flaws, errors and inconsistencies the mechanical engineer might miss. Alternatively, the mechanical engineer can use AutoCAD to determine the source of a malfunction in an existing product by putting in the specifications and allowing the software to find the problem, allowing the engineer to be more efficient by going straight to the problem and finding a fix.

Simulations and Scenarios

One of the most useful functions of AutoCAD is its ability to provide a graphic simulation of how a constructed machine will function. Once a design prototype is complete in the software, AutoCAD can generate a simulated version of the prototype and show it in action. This is a function impossible for the mechanical engineer to otherwise reproduce without investing the time and resources in developing a real-life prototype. With the help of this simulation, mechanical engineers can determine if the machine will work as intended and make any necessary tweaks or fixes before it goes into production.

Quality Assurance and Control

Thanks to the analysis components of AutoCAD, mechanical engineers can simulate a variety of environments and stresses upon a prototype. This allows them to determine the functionality of a part or machine in extreme environments or under high-stress conditions difficult to test outside simulation. These simulations also provide a demonstration of a prototype's expected performance over time, allowing accurate estimates to be made on a machine's functional life span before requiring maintenance or replacement. AutoCAD allows mechanical engineers to produce useful specifications and give clients exactly what they need in an efficient time frame.

Available at: https://work.chron.com/importance-autocad-mechanical-engineer-26569.html [Acceso el 6 de octubre, 2019]

                                                                    Image from Google images.

READING: The Pros and Cons of Diesel Engines

The Pros and Cons of Diesel Engines

By Deanna Sclar

If you’re considering buying a new automobile, compare the pros and cons of diesel-powered vehicles. Consider these facts to help you decide between an engine powered by diesel fuel and a gasoline-powered one:

• PRO: Diesels get great mileage. They typically deliver 25 to 30 percent better fuel economy than similarly performing gasoline engines. Diesels also can deliver as much or more fuel economy than traditional gasoline-electric hybrids, depending on the models involved and whatever rapidly developing automotive technology achieves.
• CON: Although diesel fuel used to be cheaper than gasoline, it now often costs the same amount or more. Diesel fuel is also used for commercial trucks, home and industrial generators, and heating oil, so as demand for diesel passenger vehicles grows, the price of diesel fuel is likely to continue to rise because of competition from those other users.

Even if the price goes up, diesel fuel would have to be 25 to 30 percent more expensive than gas to erase the cost advantage of a diesel engine’s greater fuel efficiency.

• PRO: Diesel fuel is one of the most efficient and energy dense fuels available today. Because it contains more usable energy than gasoline, it delivers better fuel economy.
• CON: Although diesel fuel is considered more efficient because it converts heat into energy rather than sending the heat out the tailpipe as gas-powered vehicles do, it doesn’t result in flashy high-speed performance. In some ways, a gasoline-powered engine is like a racehorse — high-strung, fiery, and fast — whereas a diesel engine is more like a workhorse — slower, stronger, and more enduring.
• PRO: Diesels have no spark plugs or distributors. Therefore, they never need ignition tune-ups.
• CON: Diesels still need regular maintenance to keep them running. You have to change the oil and the air, oil, and fuel filters. Cleaner diesel fuels no longer require you to bleed excess water out of the system, but many vehicles still have water separators that need to be emptied manually.
• PRO: Diesel engines are built more ruggedly to withstand the rigors of higher compression. Consequently, they usually go much longer than gas-powered vehicles before they require major repairs. Mercedes-Benz holds the longevity record with several vehicles clocking more than 900,000 miles on their original engines! You may not want to hang onto the same vehicle for 900,000 miles, but longevity and dependability like that can sure help with trade-in and resale values.
• CON: If you neglect the maintenance and the fuel injection system breaks down, you may have to pay a diesel mechanic more money to get things unsnaggled than you would to repair a gasoline system because diesel engines are more technologically advanced.
• PRO: Because of the way it burns fuel, a diesel engine provides far more torque to the driveshaft than does a gasoline engine. As a result, most modern diesel passenger cars are much faster from a standing start than their gas-powered counterparts. What’s more, diesel-powered trucks, SUVs, and cars also can out-tow gas-powered vehicles while still delivering that improved fuel economy.

Available at: https://www.dummies.com/home-garden/car-repair/diesel-engines/the-pros-and-cons-of-diesel-engines/ [Acceso el 6 de octubre, 2019]

IMPERATIVE - Differential Carrier Installation

Differential Installation Instructions

Please read completely before beginning.

Disassembly

Make sure that you have all the parts and tools you will need. The extent of disassembly depends on the job being done and the inspection findings. Lift the vehicle using an appropriate lift or a jack and safe jack stands. Always make certain that the vehicle is safely supported before working underneath. 

Unbolt the driveshaft from the yoke. 

Remove the differential cover or unbolt the third member. Let the oil drain into a suitable container. 

Please recycle your waste oil. 

Remove c-clip axles by removing the differential cross pin bolt and cross pin shaft, pushing the axles in and pulling the c-clips. Full float axles are unbolted at the hubs. 

Punch both carrier caps with identification marks so that you will be able to re-install them on the same side and in the same direction. Most carriers can be pried out of the housing with a pry bar. Further disassembly depends on the job being done. If you’re changing the ring and pinion or the pinion bearings, remove the pinion nut with an air gun while holding the yoke, or use a long breaker bar and brace the yoke (bolt it to a long board) so that it can’t move. 

Knock the pinion gear out to the rear with a brass punch, taking care not to damage the threads. 

Keep track of the location and thickness of all of the original shims. Pinion bearings must be pressed off. Carrier bearings can be pulled using a bearing puller. Internal parts (inside the carrier) can be removed as necessary.

                                      Source: http://www.differentials.com/technical-help/installation-instructions

                                                          

PRACTICA: TRADUCCION: ROTATION, TORQUES

Rotational kinetic energy

Let us picture a rigid body turning with angular velocity ω, like the Earth in this picture. Mentally,

let's divide it up into a collection of small masses. With respect to the axis of rotation, a single mass m at radius r is travelling at speed of 

                v = rω
 (You may revise circular motion at this point.) Its kinetic energy is ½ mv². So let's imagine the dividing the object up into many masses m_i at distances r_ifrom the axis. Each has v_i = r_iω, where ω has the same value for all the masses because the object is (by assumption) rigid. So the total rotational kinetic energy is

K_rot= Σ K_i= Σ ½ m_ir_i²ω²

where the summation is over all of the i. ½ ω² is a common factor in every term of the sum, so

K_rot= ½(Σ m_ir_i²)ω² = ½ Iω² where

I = Σ m_ir_i²is the moment of inertia.  

This is the result for a collection of discrete masses, m_i. For a continuous body, we should normally divide it up into small elements of volume, dV. (You can revise calculus.) From the definition of density ρ, each has mass

dm = ρdV.

Instead of an ordinary summation, we do an integral (the equivalent of summation for very small divisions), and we have

K_rot = ½(∫ dm.r²)ω² = ½ Iω² where

I = ∫ r².dm is the moment of inertia for a continous body

and where the integration is over the whole volume occupied by the rigid body in question.

 Rotational kinematics

As mentioned in the multimedia tutorial, there are very strong analogies between linear and rotational kinematics. If s is the distance of the arc travelled along a circle of radius r, then angular displacement θ is just s/r. Angular velocity ω = dθ/dt = (ds/dt)/r = v/r. Angular acceleration α = dω/dt = (dv/dt)/r = a/r.

So, as shown in the diagram below, the analogies of the linear quantities s, v and a are θ, ω and α, which we obtain by dividing the linear quantities by r.

The graphs above show displacement, velocity and acceleration for linear motion with constant acceleration (at left) and for circular motion with constant angular acceleration. Just for practice, let's derive the new equations (and revise the kinematics section if this looks difficult!) If we consider motion with constant acceleration, and remember that α = dω/dt, we have

ω = ∫ α dt = αt + ω₀

And from ω = dθ/dt, we can integrate again to get:

θ = ∫ ω dt = ½αt² + ωt + θ₀

From the two equations above, we can eliminate t to get

ω² − ω₀² = 2α(θ − θ₀).

So we have equations completely analogous to those of linear kinematics:

ω = ω₀ + αt and θ = θ₀ + ω₀t + ½αt² and ω² − ω₀² = 2α(θ − θ₀)
v = v₀ + at and s = s₀ + v₀t + ½at² and v² − v₀² = 2a(s − s₀).

Torque: dependence on displacement, force and angle

Forces cause accelerations. To make something turn, we apply a torque. We shall define if first, and then explain why this definition is logical. Later we shall see the complete analogy with Newton's laws for linear motion.

The torque τ is defined by

τ = rX F

where force F acts at a point displaced by r from the axis. The magnitude of the torque is given by

τ = r F sin θ

where θ is the angle between r and F.

(You may need to look at the cross product section of the  support page on vectors.)

We shall discuss the magnitude first, then the direction.

The photos at right show three ways of using a spanner. In the first pair, we compare a small value of r (small torque) with a large r and large τ. In the second, we compare θ = zero and θ = 90°. In the former case, the torque is zero. From experience, you know that you need large r, θ = 90° and large F to obtain the maximum torque.

                                       Disponible en: http://www.animations.physics.unsw.edu.au/jw/rotation.htm

ACTIVITIES: READING COMPREHENSION (SIMPLE PRESENT) - CLASSICAL MECHANICS

CLASSICAL MECHANICS

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology.

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Besides this, many specializations within the subject deal with gases,liquids, solids, and other specific sub-topics. Classical mechanics provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to introduce the other major sub-field of mechanics, quantum mechanics, which reconciles the macroscopic laws of physics with the atomic nature of matter and handles the wave-particle duality of atoms andmolecules. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity.General relativity unifies special relativity with Newton's law of universal gravitation, allowing physicists to handle gravitation at a deeper level.

REALICE LAS SIGUIENTES ACTIVIDADES:

RESPONDA:
What is classical mechanics?

_______________________________________

Is the study of bodies motion an old study? 

_______________________________________

ELIJA LA OPCION CORRECTA:

La mecánica clásica describe 

·          - objetos pequeños

·          - objetos grandes

La mecánica clásica se ocupa de describir objetos como *

·          - cohetes espaciales

·          - la astronomía

La mecánica clásica proporciona

·          - resultados muy apropiados

·          - resultados muy exactos

La mécanica cuántica se ocupa de

·          - objetos microscópicos

·          - objetos macrocópicos

COMPLETE LAS SIGUIENTES AFIRMACIONES:  

·         La mecánica cuántica armoniza ...

·         Esta subdisciplina maneja ...

·         En el caso de objetos que alcanzan la velocidad de la luz ...

·         La ley de gravedad permite a los físicos ... 

PRACTICE: READING COMPREHENSION: SIMPLE PRESENT "FLUID MECHANICS"

Fluid mechanics

Relationship to continuum mechanics

Fluid mechanics is a sub discipline of continuum mechanics, as illustrated in the following table.

Continuum mechanics The study of the physics of continuous materials	Solid mechanics The study of the physics of continuous materials with a defined rest shape.	Elasticity Describes materials that return to their rest shape after an applied stress.
		Plasticity Describes materials that permanently deform after a sufficient applied stress.	Rheology The study of materials with both solid and fluid characteristics.
	Fluid mechanics The study of the physics of continuous materials which take the shape of their container.	Non-Newtonian fluids
		Newtonian fluids

In a mechanical view, a fluid is a substance that does not support shear stress; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress.

Assumptions

Like any mathematical model of the real world, fluid mechanics makes some basic assumptions about the materials being studied. These assumptions are turned into equations that must be satisfied if the assumptions are to be held true. For example, consider an incompressible fluid in three dimensions. The assumption that mass is conserved means that for any fixed closed surface (such as a sphere) the rate of mass passing from outside to inside the surface must be the same as rate of mass passing the other way. (Alternatively, the mass inside remains constant, as does the mass outside). This can be turned into an integral equation over the surface.

Fluid mechanics assumes that every fluid obeys the following:

·         Conservation of mass

·         Conservation of energy

·         Conservation of momentum

·         The continuum hypothesis, detailed below.

Further, it is often useful (at subsonic conditions) to assume a fluid is incompressible – that is, the density of the fluid does not change.

Similarly, it can sometimes be assumed that the viscosity of the fluid is zero. Gases can often be assumed to be inviscid. If a fluid is viscous, and its flow contained in some way (e.g. in a pipe), then the flow at the boundary must have zero velocity. For a viscous fluid, if the boundary is not porous, the shear forces between the fluid and the boundary results also in a zero velocity for the fluid at the boundary. This is called the no-slip condition. For a porous media otherwise, in the frontier of the containing vessel, the slip condition is not zero velocity, and the fluid has a discontinuous velocity field between the free fluid and the fluid in the porous media.

REALICE LA PRACTICA ENTRANDO A LOS COMENTARIOS