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الاكواد والمواصفات الهندسه الميكانيكيه


Introduction to Mechanics & Symmetry


http://rapidshare.de/files/23005348/...__4AH.pdf.html


Illustrated Sourcebook of Mechanical Components


http://rapidshare.de/files/23001945/...__4AH.rar.html


Mechanisms & Mechanical Devices Sourcebook

http://rapidshare.de/files/23002869/...__4AH.rar.html


ASTM Standards series

ومن منتدى المهندس وهم عباره عن
Section 01 - Iron and Steel Products
Section 02 - Nonferrous Metal Products
Section 03 - Metals Test Methods and Analytical Procedures
Section 04 - Construction
Section 05 - Petroleum Products, Lubricants, and Fossil Fuels
Section 06 - Paints, Related Coatings, and Aromatics
Section 07 - Textiles
Section 08 - Plastics
Section 09 - Rubber
Section 10 - Electrical Insulation and Electronics
Section 11 - Water and Environmental Technology
Section 12 - Nuclear, Solar, and Geothermal Energy
Section 13 - Medical Devices and Services
Section 14 - General Methods and Instrumentation
Section 15 - General Products, Chemical Specialties, and End Use Products









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Fundamentals of Fluid Mechanics,4 Ed




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Stress resolution

Stress resolution
An element's stress state is essentially three-dimensional, generally with both normal and shear components in each of the three dimensions. The components are usually deduced by superposition of load building blocks as in the stress example of the previous section. The two- and one-dimensional cases illustrated here are particularisations of what is essentially three-dimensional.
The element may be rotated about all three axes into a unique principal orientation in which all shear stresses vanish. The corresponding normal stresses in this principal orientation are termed the principal stresses. A stress state is characterised most succintly by the principal stresses, say ( σ1, σ2, σ3 ), and failure theories - the next step in failure assessment - are expressed in terms of principal stresses. It is therefore necessary to examine how principal stresses are derived from Cartesian components.
Stress is a tensor entity, so complex tensor arithmetic must be applied in the three- dimensional case to evaluate stress components as the element rotates. We consider here only the simpler two dimensional case in which one direction is known to be principal, and resolution consists of element rotation only about that principal axis. Thus in the 2-dimensional sketch above, the z-axis is principal and x-y stress components vary as the element rotates about that axis. Fortunately the great majority of practical cases are two -dimensional since the relatively simple loading in conjunction with the natural choice of axes leads to one of the axes being principal automatically. For example there can be no stress on the free surface of a body, so the surface normal is an obvious choice for one of the three Cartesian axes - as there can be no shear on this surface, the axis is automatically a principal axis.
Although we consider only "two-dimensional stresses", it is important to remember always that stress states are essentially three dimensional.
In this course the positive convention adopted for the orientation of a plane (characterised by its normal) is shown at (a) below, and for normal and shear stress and strain at (b) - positive shear is counter-clockwise. The two sketches (c) illustrate the consistency of the positive senses for shear stresses and strains; the total shear strain (distortion) is γ = 2( γ/2).

Using these conventions, the plane stress state at 'A' in the example of the previous section is :
σx = -93, σy = 0, τxy = +34, ( τyx = -34) MPa
We now examine the variation of normal and shear stress components as the inclination of the face on which they occur changes.
Consider the elemental unit cube ( size 1x1x1 ) under the known positive 2-dimensional stress components shown in ( i) below - the third dimension (z) is principal.

Rotational equilibrium requires complementary shear, that is τyx must equal -τxy. This necessity has been incorporated into ( ii), from which it is apparent that the three components ( σx, σy, τxy ) together with the third principal are necessary to define the stress state. The three components are called the Cartesian triad.
We wish to evaluate - in terms of the Cartesian triad - the stress components ( σ, τ) in the general direction θ, so we consider force equilibrium of the wedge element ( iii), one of whose faces is inclined at θ. The height of the wedge remains 1 unit, however the dimension in the x-sense becomes 1.tanθ, and the length of the hypotenuse is 1.secθ.
The force components on each face of the wedge are the stress components multiplied by the face area - these are shown in ( iv). For force equilibrium of the wedge :
in the σ-sense :- σ.secθ - ( σx + τxy.tanθ ) cosθ - ( σy.tanθ + τxy ) sinθ = 0
in the τ-sense :- τ.secθ + ( σx + τxy.tanθ ) sinθ - ( σy.tanθ + τxy ) cosθ = 0
which on simplification give the required resolution equations :-
σ = 1/2 ( σx + σy ) + 1/2 ( σx - σy ) cos 2θ + τxy sin 2θ
τ = - 1/2 ( σx - σy ) sin 2θ + τxy cos 2θ

Similar equations but with different signs are encountered in the literature - sign differences arise from positive conventions other than the above.
The simultaneous occurence of sine and cosine terms in these equations makes it difficult to visualise how the resolved components ( σ, τ) vary as the direction θ changes. More easily interpreted equations result if the stress state is defined by the basic triad ( σm, σa, θp ) rather than by the Cartesian triad. θp is the inclination of the plane of the maximum principal stress with respect to the x-reference.
The basic triad is related formally to the Cartesian as follows :-
( 3a) σm = 1/2 ( σx + σy )
σa cos 2θp = 1/2 ( σx - σy ) that is σa = √[ {1/2 ( σx - σy ) }2 + τxy2 ]
σa sin 2θp = τxy θp = 1/2 arctan [ 2 τxy /( σx - σy ) ]
. . . . though these equations seldom need to be implemented.
Making these substitutions leads to resolution equations in the more meaningful form :-
( 4a) σ = σm + σa cos 2 ( θp - θ ) { nomenclature explanation }
τ = σa sin 2 ( θp - θ )
It is apparent that normal and shear stress components vary sinusoidally with direction θ (not unlike vector components) however the variation is second harmonic - that is stress components are the same along axes which lie at 180o to one another. The two sinusoids are of the same amplitude σa and out of phase by 45o.
σm is the constant component of the normal stress.
The principal and maximum shear stresses follow immediately as :-
( 5) σmax = σm + σa ( at θp )
σmin = σm - σa ( at θp - π/2 )
τmax = σa ( at θp - π/4 )

These relations are often expressed graphically via Mohr's stress circle, in which σm and σa represent the circle's centre location and radius respectively. The conventions require that angles on the circle, reckoned from the X-radius, are double the corresponding angles on the element (which are measured from the x-reference), and in the opposite sense. The reader should confirm that this construction satisfies equation ( 4a).


This example demonstrates typical stress resolution using the simple trigonometry of Mohr's circle rather than the formal resolution equations derived above. The example also shows clearly the variation of stress components with orientation, θ, as predicetd by ( 4a).


This script resolves the Cartesian components of a two-dimensional stress state into the principal components.

It is important to remember that a stress-strain state is always essentially three- dimensional, involving three principals. We have addressed only resolution in two dimensions to obtain a single Mohr's circle involving the principals in these two dimensions - however two other circles must exist relating these two principals with the third.
As noted above the third (eg. automatic) principal is usually deduced from the nature of the problem. Two common situations arise when the state is either one of :
plane stress :
If there is no stress orthogonal to the 1-2 resolution plane then σ3 = 0
plane strain :
If there is no strain orthogonal to the 1-2 resolution plane then ε3 = 0 and it follows from ( 2) that σ3 = ν ( σ1 + σ2 )




Fig F completes the Mohr's circles for the example above (Fig D) assuming that the element is loaded in plane stress, that is the three principals are ( -600, -100, 0 ) MPa.
Fig G illustrates the three principals and three Mohr's circles for a completely unrelated stress state where two-dimensional resolution happens to relate to the largest (3-2) circle and σ1 is the principal stress orthogonal to the resolution plane.

The outcome of stress resolution at an element must be a set of three principals - all three must be known before the element's safety can finally be assessed by application of an appropriate failure theory.
Strain resolution


Resolution of strain is generally unnecessary when assessing the safety of common engineering components, however the following description is given for completeness.
If a material behaves in a linear elastic manner then the directions of principal strains are identical to the directions of principal stresses, and all the preceding equations, and Mohr's circles, may be expressed in strain terms - provided that everywhere in the stress equations :
  • normal stress, σ, is replaced by normal strain, ε, and
  • shear stress, τ, is replaced by half the shear strain, ie. by γ/2.
The strain analogues of the foregoing are therefore :-
( 3b) εm = 1/2 ( εx + εy ) ; εa = 1/2 [ ( εx - εy )2 + γxy2 ]1/2 ; θp = 1/2 arctan [ γxy /( εx - εy ) ]
( 4b) ε = εm + εa cos 2 ( θp - θ ) ; γ/2 = εa sin 2 ( θp - θ )
Experimentally, surface stresses are found from strain gauges attached to accessible surfaces. It is useful therefore to be able to quickly interrelate stresses and strains for a plane stress state, for which, since the principal stress and strain directions coincide, it follows from ( 2) and ( 3) that :-
( 6) E εm = ( 1 - ν ) σm ; E εa = ( 1 + ν ) σa
which relate the centres and radii of the two circles. If these circles are drawn to scale, and it is required that their circumferences coincide for ease of drawing, then it may be shown that :
( 7) E $ε = ( 1 + ν ) $σ ; ( 1 + ν ) Cε = ( 1 - ν ) Cσ where
$ε is the scale of the strain circle ( strain units/mm ) $σ is the scale of the stress circle ( MPa/mm ) Cε is the distance (mm) of the strain circle centre from the shear axis Cσ is the distance (mm) of the stress circle centre from the shear axis
Although the use of scaled Mohr's circles is not necessarily advocated, it is strongly recommended that the circles are at least sketched free-hand as an aid to interpretation of the equations. It is important that skill is developed in visualising the interplay between components and principals.
The following example demonstrates application of strain-to-stress transformation.

Failure theories

Failure theories

We have seen that failure of a tensile member occurs when the stress caused by the actual load reaches the stress limit - the strength - of the member's material. Correlation of the actual stress with the maximum stress (strength) is straightforward in this case because they are both uniaxial. But how can we correlate the triaxial stress state in a component - whose material strength(s) is measured in uniaxial tests - to assess failure tendency?
Unfortunately there is at present no fundamental rationale for any such correlation.
We therefore postulate some attribute of the stress state as a descriptor of that state - an attribute such as the maximum stress or the specific energy - and then compare the values of this attribute for the given component triaxial state on the one hand and the uniaxial test state on the other. This postulate is the failure theory based upon the particular attribute selected; it is a useful theory only if its predictions are confirmed by experiment.
Currently no universal attribute has been identified which enables prediction of failure of both ductiles and brittles to an acceptable degree of accuracy. To appreciate why this is so, it is useful to recall the differences in behaviour of ductiles and brittles, as outlined in the sketch below.
  • Under normal stress, ductiles' compressive characteristics are approximately the same as their tensile characteristics (B,C below) whereas brittles are significantly weaker in tension - eg. Suc /Sut = 3.5 to 4 for the cast irons ( Suc and Sut are the compressive & tensile ultimates).
  • A ductile's shear strength is about half its tensile strength whereas shear and tensile strengths are about the same in the case of a brittle. Superposition of Mohr's circles of sizes corresponding to these proportions (D) suggests that a shear-based attribute is relevant to ductiles, whereas a normal-based attribute to brittles. (Note: the two superimposed circles correspond to completely different stress states - they have been superimposed only to highlight the circles' relative sizes.)




    This conclusion is reinforced when we consider the directions of the fracture surfaces under normal and shear loadings (E). Thus in tension, ductiles fail at 45o to the load sense, which is the direction of maximum shear. This same 45 o fracture sense is found with brittles under torsion, however in this case the Mohr's circle is centred on the origin thus indicating a maximum normal attribute for brittles.
  • A final qualitative argument involves strength under hydrostatic pressure, for which the three Mohr's circles degenerate to a single point (F). Brittles fail when the pressure corresponds approximately to the compressive strength, but it is difficult to assess when ductiles fail as the hydrostatic strength is so large - the absence of failure may be equated to the absence of shear.

The upshot of these observations is the adoption of the failure theories indicated - using
  • shear-based attributes for ductiles
  • normal-based attributes for brittles
These theories are examined below.

We refer to the equivalent stress σe for a ductile material only, as the uniaxial stress which has the same failure tendency as a given triaxial state - on the basis of whichever attribute is relevant. The safety factor then follows from ( 1a) as n = S/σe.
Distortion Energy Theory ( aka. von Mises Theory )
It may be shown that the specific distortion (or shear strain) energy for a linear material under the triaxial state (σ1, σ2, σ3 ) is proportional to [ ( σ1 - σ2 )2 + ( σ2 - σ3 )2 + ( σ3 - σ1 )2 ].
For a uniaxial stress ( σe, 0, 0 ), or merely σe, this expression becomes 2σe2. So σe is equivalent to the triaxial system - ie. gives rise to the same distortion energy - if :-

( 8) 2σe2 = ( σ1 - σ2 )2 + ( σ2 - σ3 )2 + ( σ3 - σ1 )2
The equivalent stress is often referred to as the von Mises stress, after the propounder.
Maximum Shear Stress Theory ( aka. Tresca Theory )
This theory postulates that if a uniaxial stress σe, and a triaxial stress state give rise to the same maximum shear stress, then the failure tendencies of the two states are identical; that is, σe is equivalent to the triaxial state. So, for the same diameter of the largest Mohr's circle :-
( 9) σe = σmax - σmin ; where
σmax = max (σ1, σ2, σ3 )
σmin = min (σ1, σ2, σ3 )
The intermediate principal - which happens to be σ2 in the sketch - has no effect upon failure tendency according to this theory.
The failure theory appears as a series of lines at 45o when the minimum principal is plotted against the maximum principal, as shown below. Each point on this graph corresponds to a unique "largest" Mohr's circle and hence to an infinite number of stress states when the intermediate principal is considered. The 45o lines are each associated with a particular value of stress difference, ie. of equivalent stress from ( 9), and hence of safety factor. No point can appear to the left of the equality line on this graph, as this would imply a reversal of the minimum / maximum roles. The equivalent stress varies from zero on the equality line, through to the strength, S, on the failure locus (n=1).
The safety factor may be interpreted graphically as the ratio of intercepts on the axis of maximum principal, n = S/σe. The safe band extends to infinity in both directions as foreshadowed by the hydrostatic results.
Pure shear is represented by the line through the origin, perpendicular to the equality line, cutting the failure locus at a shear strength Ss which is half the tensile strength. This tallies with experimental findings as noted above.
Either of the foregoing theories may be applied to ductiles: distortion energy represents experimental results somewhat better than does maximum shear stress. The small numerical difference between the two theories may be appreciated by defining the intermediate principal's value in relation to σmax, σmin by the parameter λ ( 0 ≤ λ ≤ 1 ) as shown in the sketch, substituting into ( 8) and invoking ( 9) to give :
σe DE / σe MSS = √ ( 1 - λ + λ2 ) ≤ 1 which is plotted here.
There is no numerical difference between the equivalent stresses predicted by the theories when the intermediate coincides with either extreme; the maximum difference is 13.4% when the intermediate lies midway between the extremes. The maximum shear stress theory is always conservative.
Both theories have their drawbacks in use - the distortion energy theory introduces non- linearities into otherwise linear problems, whereas the maximum shear stress theory requires the ordering of the principals - which is easy enough when these are enumerated but most awkward when the principals are expressed algebraically, as ordering leads to multiple possible expressions for the equivalent stress. The particularisation of the equations to plane stress is left to the reader ( see tutorial problem #16 ).
Modified Mohr Theory
This theory derives from the maximum normal stress criterion which historically was the first failure theory proposed for brittles, and is similar to the maximum shear stress theory in that the intermediate principal plays no part in the failure process. The concept of equivalent stress is not applicable to brittles owing to their differing tensile and compressive characteristics (250 MPa tensile has a much greater failure tendency than 250 MPa compressive) so the safety factor must be obtained from the principals and strengths directly, either graphically or numerically.
The maximum normal stress criterion failure locus is shown in σmax / σmin space at (a) below. The locus is defined by the two lines σmax = St the tensile ultimate, and σmin = - Sc the compressive ultimate; the two lines meet in the corner x. The pure shear line cuts the failure locus at the shear ultimate Ss - so this theory has Ss = St which we have seen is typical for brittles.
Above the line from origin through corner x, failure is dictated by tensile considerations; to the left of this line the compressive mechanism is more critical - so, to cater for both tension and compression :-
( i) 1/n = maximum ( σmax/St, - σmin/Sc )

Experimental failure points are shown; the theory is evidently dangerous to use for design in the fourth quadrant since points which the theory gives as safe lie outside experimental failure results.
Mohr's internal friction theory (b) overcompensates as it is unduly conservative in the fourth quadrant and unrealistic in prediciting a shear strength substantially less than the tensile, however the modified Mohr theory represented by the three linear segments of (c) describes experiments well. The third segment is defined empirically to pass through the points ( St, - St ) and ( 0, - Sc ), thus ensuring equal tensile and shear strengths; the addition of this third segment to ( i) leads to :-
( 10) 1/n = maximum ( σmax/St , - σmin/Sc , σmax/St - ( σmax+ σmin)/Sc )

The modified Mohr theory will be used in this course for brittle materials.




EXAMPLE
A component is made from a brittle material of tensile and compressive strengths 200 and 600 MPa respectively.
The table illustrates how the safety factor is evaluated at each of three different points (a), (b) and (c) on the component, the stress state at each point being defined by the three given principals which are tabulated. principals, MPa σmax σmin 1/n from ( 10) a. ( 100, 75, 50) 100 50 max( 1/2, -1/12, 1/4) = 1/2 b. ( -200, 10, 50) 50 -200 max( 1/4, 1/3, 1/2) = 1/2 c. ( -200, -300, -100) -100 -300 max( -1/2, 1/2, 1/6) = 1/2 The selected points all happen to lie on then =

2 locus as illustrated.




Work & Energy

Objectives

Here are the objectives for today's lesson.
Before you begin to study the lesson, take a few minutes to read the objectives and the study questions for this lesson.
Look for key words and ideas as you read. Be sure to read these objectives in the study guide and refer to them as you study the lesson.
Focusing on the learning objectives will help you to study, to understand the important concepts and to synthesize.
Compare the objectives with the study questions for the lesson to be sure that you have the concepts under control.
1. Define the physical concept of work and apply it to various situations.
2. Distinguish between work and power.
3. Define kinetic energy and its variables.
4. Define potential energy and its variables.
5. State the work energy theorem and define its applications.
6. Define the characteristics of conservative and non conservative systems
7. Apply conservation of energy to various physical situations


Questions

1. Distinguish between work and power.
2. Define work as it is used in physical science and give an example of work.
3. Is work being done when a heavy weight is held motionless overhead? Explain.
4. How is power different from work or energy?
5. When we say that work is done by or against some agent, what do we mean? Name some "agents'>
6. What do we mean by conservative forces? Cite some examples.
7. What is the work/energy theorem?
8. How is energy different from momentum and inertia?
9. Define energy?
10. What do we mean when we say, "The concept of energy would be useless if it were not conserved?"
11. Compare kinetic and potential energy using the appropriate physical quantities?
12. How do we know whether or not an object has energy?
13. Discuss conservation of energy using the pendulum as an example.
14. What do we mean when we speak of a "conservative system?"
15. How do we know that energy is conserved when an object falls under the influence of gravity in the absence of air friction?


1. Introduction

In this lesson we will introduce the concept of work as the result of forces which cause motion. We will see that the result of these forces depends on the situation, whether the forces are applied horizontally or vertically and the degree to which friction interferes with the motion.
We will distinguish energy from momentum as we see that energy is what something acquires as the result of work being done, sometimes. With a precise physical definition of work and a concept of two different types of mechanical energy, we will explore the "sometimes".
The work/energy theorem shows a definite and specific relationship between work and energy which is useful in a bewildering variety of situations.
Near the end of the lesson we will examine the distinction between conservative and nonconservative forces and the apparent loss of energy in the latter.
Finally we will revisit the swinging balls and see that the missing requirement is the conservation of energy.
1.1. Newton's Laws are correct but inadequate for certain kinds of processes and interactions
1.1.1. momentum describes collisions, but not completely
1.1.2. heat can be integrated into and explained by mechanical theory

1.1.2.1. studied since Aristotle
1.1.2.2. little understanding of principles
1.1.2.3. nothing new added until 19th century using Newtonian paradigm
1.1.2.4. concept of particles in motion will unite mechanics and atomic theory


1.2. History is complex
1.2.1. No single individual is responsible for theory
1.2.1.1. developed over 150 year period

1.2.2. Historical accident that Newton developed laws in terms of kinematics
1.2.2.1. Newton's efforts were directed towards explaining planetary motion
1.2.2.2. Newton's Laws can be derived from conservation laws


1.2.3. Conservation of momentum proposed by Hooke, Wren, Huygens
1.2.3.1. contemporaries of Newton
1.2.3.2. Newton's work provided framework (paradigm) used to formally derive concept later


1.2.4. Galileo
1.2.4.1. speed depends on height
1.2.4.2. rolls up to same height as rolled from
1.2.4.3. inertia is special case of energy conservation


1.2.5. Others
1.2.5.1. Newton defined "vis insita" or innate force of matter
1.2.5.2. Huygens: mv2 is conserved in certain collisions
1.2.5.3. Leibnitz called it "vis viva" (living force)
1.2.5.4. later combined with Newton's Laws and Galileo's Kinematic equations


2. Energy

Energy is an abstract concept. Whether or not such a thing as energy "exists" outside the mind is one we will not address. We will note that it is a very useful concept, it provides the needed explanation for interactions such as the swinging ball, no violation of the conservation concept has never been observed, it leads to understanding other phenomena linked to heat which we will study in later programs.
Does it exist? Who cares. It is useful and consistent in describing certain interactions. It fits all of the definitions of a good theory as stated in Program 9.
Energy can be recognized and quantified by its effect on matter, and in fact is only useful because it is conserved. Not only that, but the concept, or something like it seems to be necessary to understand the universe.
Although the concepts of work and energy were developed from Newton's laws, it is apparent that the Laws can be derived from the definition of work and energy. From the conservation law even Galileo's kinematic relationships can be derived. It is apparent that the two methods are equivalent, but different, ways of looking at the same quantities of distance, acceleration, velocity, force, mass, and time.
For practical purposes, we use whichever method suits itself to the problem at hand. For theoretical understanding we observe the mathematical equivalence and move on to higher order abstractions.
2.1. Concept is abstract
2.2. something with energy can exert force or move things
2.3. ENERGY cannot be seen or measured directly, but amount can be calculated
2.4. ENERGY is recognized by its effect on matter

2.4.1. measured by work done
2.4.2. work has precise physical definition


2.5. ENERGY is a useful concept only because of conservation
2.5.1. represents stored work
2.5.2. could not define unless conserved
2.5.3. exists in many different forms
2.5.4. objects exchange energy through forces when interacting
2.5.5. amount remains constant during change


2.6. ENERGY is necessary to understand universe
2.6.1. more laws allow increased prediction
2.6.1.1. mathematical law = relationship
2.6.1.2. relationship = equation
2.6.1.3. more equations => more variables can be incorporated into systems
2.6.1.4. more variables => more complex and therefore more realistic situations


2.6.2. is especially powerful when combined with momentum conservation

3. Work

Work is defined simply as a force which is applied through a displacement, or distance. This is one of the simplest definitions in all of physics, but also one of the most useful. We will see the result, or effect of work later in the lesson when we consider the relationships between work and energy.
Work is done against some agent, such as gravity, the stiffness of a spring, inertia, or friction. It is independent of time, meaning that there are no restrictions on how fast or slow the work is done.
3.1. Force times distance (Fd)
3.1.1. can double work by doubling either force or distance
3.1.2. precise and limited definition

3.1.2.1. note contrast with everyday usage
3.1.2.2. reason for definition is to relate to energy
3.1.2.3. must be consistent to be meaningful


3.1.3. work is only done if motion is involved
3.1.3.1. no work is done by stationary force
3.1.3.2. component of force in direction of motion does work


3.1.4. 1 joule = 1 Newton x 1 meter
3.1.4.1. 1 J = 1 N*M

3.2. Work is done by or against some agent
3.2.1. inertia, gravity, friction, elasticity, electric force, magnetic force, etc.

3.3. Work is independent of time

4. Power

Power and work are often confused with one another or used interchangeably. The difference is one of time. The rate at which work is done is power. Intuitively, the faster a weight is lifted overhead the more power is consumed. Not so obvious is that the force applied and the work done do not depend on the speed in any way.
Power is also an electrical term, providing an important connection between the mechanical world and the electrical world. That we can describe electricity in mechanical terms gives us faith that our techniques are indeed universal and apply to more than just a small sample of the world.
4.1. Rate at which work is done
4.2. 1 watt = 1 joule per second

4.2.1. 1 W = 1 J/s

4.3. also electrical term
4.3.1. 1 watt = 1 volt x 1 ampere
4.3.1.1. 1 kilowatt = 1000 watts
4.3.1.2. 1 kilowatt hour = 1000 watts for one hour


4.3.1.2.1. 1000 J/s x 3600 s = 3,600,000 J

4.3.2. shows correspondence between mechanical and electrical systems
4.3.3. Newtonian paradigm extends into study of electricity


5. Kinetic Energy

Kinetic energy is the energy of motion. It is the result of work being done against inertia.
We will begin studying this and other forms of mechanical energy in the ideal state, that is in the absence of friction. Recall that Galileo used a similar idealization to arrive at the principle of inertia, so we are justified in thinking in "frictionless" terms as long as we don't forget that it is really there in all real situations.
It is easy to see how to calculate the kinetic energy possessed by a moving mass by combining Newton's second law with Galileo's kinematics.
We should begin to see that kinetic energy is one way of storing work in the form of motion. Although kinetic energy looks intuitively like momentum, it is not the same. Yes, both momentum and kinetic energy involve mass and velocity, but the relationships are different and they are really quite different thing. One important difference is that, while there is only one form of momentum, there are many forms of energy. This means that, although one moving object may transfer its momentum to another, the momentum cannot change form in the same object. Energy can do that, and it leads to all kinds of interesting results logically and mathematically.
5.1. energy of motion
5.1.1. Kinetic energy equals one half m v squared

5.2. work done against inertia
5.3. moving object is capable of exerting force

5.3.1. force is required to give motion to the cart
5.3.2. amount of force depends on speed and mass
5.3.3. distance over which force can be applied depends on the magnitude of force and the amount of work done on the cart


5.4. derivation (from Galileo's kinematics and Newton's laws)

5.4.1. assuming no friction
5.4.2. Do not panic if you don't get this


This is included only to show that the "formula" for kinetic energy is not arbitrary. It is derived from the question: what is the result of applying a certain force horizontally over a given distance if the object in question gains speed on a level surface according to Newton's laws and Galileo's kinematics?
5.5. in words

An object of a given mass will acquire a certain velocity when accelerated by a given force for some specified distance. When the force is no longer applied, the object behaves according to the first law, maintaining a constant speed in a straight line until another force acts to stop it. The stopping force may involve a different combination of force and distance than that required to give it its velocity in the first place.
The work done against the inertia of the cart is stored in the motion of the object, to be used at will, exerting an unspecified force for some unspecified distance as long as the product of force and distance does not exceed the amount of work done on the cart in the first place.
5.6. Kinetic energy is a way of storing work in the form of motion
5.6.1. not just force, but a certain relationship between force and distance
5.6.2. not the same as momentum

5.6.2.1. hard to see difference since both are involved in any change in motion
5.6.2.2. inertia (mass) in motion possesses both kinetic energy and momentum
5.6.2.3. Newton's vis in sita has two components
5.6.2.4. momentum is acquired by impulse (Ft)

5.6.2.4.1. second law: impulse = change in momentum

5.6.2.5. kinetic energy is acquired by work (Fd)
5.6.2.6. kinetic energy divided by momentum equals velocity


5.7. Kinetic Energy is only one form of energy
5.7.1. there is only one form of momentum
5.7.1.1. important difference between the two concepts
5.7.1.2. impulse changes momentum


5.7.2. Work can be done without changing kinetic energy
5.7.2.1. lifting against gravity
5.7.2.2. stretching or compressing spring
5.7.2.3. pushing or pulling against electric or magnetic force
5.7.2.4. sliding at constant speed against friction


5.7.3. Something changes when work is done
5.7.3.1. if not kinetic energy then what?

6. Potential Energy

Potential energy results when work is done against certain kinds of forces which are known as "conservative" forces. Gravity, elasticity, electric forces and magnetic forces are examples of conservative forces.
We define gravitational potential energy as the stored work done against a conservative force. Usually this is related to the change of location of one object in relation to the force required to move it a certain distance.
With this definition is it easy to note than an object acquires gravitational potential energy when a force (equal to its weight) is used to lift it to a certain height. It then possesses something it didn't have before, that is the ability to do work.
For example if a weight is allowed to settle slowly on the diameter of a wheel, the wheel can turn and do work. This is the principle of the water mill which has been used for centuries to grind grain.
The potential energy of the weight can be changed to kinetic energy is the weight is allowed to freefall. AS IT LOSES POTENTIAL ENERGY (gets closer to the ground) IT GAINS KINETIC ENERGY (gains speed in freefall). It is obvious in a quantitative sense that this is true. Even more interesting is that it can be demonstrated that the LOSS OF POTENTIAL ENERGY IS EXACTLY EQUAL TO THE GAIN OF KINETIC ENERGY. This is once again assuming the frictionless case, but we already know that the concept of freefall generalizes to the case of no air friction.

6.1. work done against conservative forces
6.1.1. gravity, springs, electricity, magnetism

6.2. Gravitational Potential Energy

6.2.1. this equation simply shows that work done equals energy gained
6.2.2. E = mgh

6.2.2.1. mg is the weight of a given mass and represents the force necessary to lift it
6.2.2.2. h is the vertical distance through which the upwards force is applied to counter the weight


6.2.3. work done against gravity is stored in position of object

6.3. an object in freefall will acquire kinetic energy as it loses potential energy
6.3.1. can be shown that amount of kinetic energy gained exactly equals amount of potential energy lost (in the absence of friction)



Conservation of Energy

Kinematic Equations

Note that we have used the equivalent symbols for acceleration ("a" for the general case, "g" for gravitational) and distance ("x" for the general case, "h" for "height") but the two relationships are identical. So we must conclude from this either:



Newton's laws and Galileo's kinematic equations are BOTH INCORRECT.
OR
Energy is CONSERVED in falling objects.
Either we give up on Galileo's definitions of motion AND Newton's laws, or we accept that energy is conserved in falling objects. Since we don't want to reject the work of Galileo and Newton, we choose to state that ENERGY IS CONSERVED IN FALLING OBJECTS.
7. Work/Energy Theorem

The work/energy theorem is a simple statement which relates work to energy in a simple way. The work/energy theorem simply states that the total amount of work done is equal to all changes in energy. This is easy when the only two forms of energy are potential and kinetic. It is even easy to include real-life friction in this statement. When doing so, we can simply see friction as "eating" some of the energy and therefore reducing the amount available to be converted from potential to kinetic or vise-versa.
It is important to not that it is CHANGES in energy which are significant, not the absolute amount of energy possessed by an object. By CHANGES we mean "gains or losses". Here we see that the work energy theorem is telling us that for every gain or loss in energy the re is a complimentary loss or gain somewhere else, or else work is done as a result.
7.1. work done = total change in energy
7.1.1. (read "the change in energy equals work done"). Work done results in a change in energy somewhere in the system and a change in energy requires work to be done by or against some agent..
7.1.2. work done causes an change in the total energy of all kinds
7.1.3. theorem defines an equation which accounts for all types of energy
7.2. changes or differences in energy are important, not the absolute amount
7.2.1. speed doesn't kill, it's the sudden stop (rapid change in energy)
7.2.2. being on top of a tall building (having lots of potential energy) won't hurt you unless you fall
7.2.3. initial and final states are important

7.2.3.1. initial state need not be at rest
7.2.3.1.1. work is done in changing speed from 0 to 30 mph
7.2.3.1.2. work is also done in changing speed from 30 to 60 mph, but not the same amount as from 0 to 30


7.2.3.2. rock hits windshield vs. windshield hits rock: equivalent
7.2.3.3. carrying a box up one flight of stairs requires the same amount of energy regardless of which floor it started from


8. Conservative Systems:

A conservative system is an idealized system in which no work is done. Specifically it is a system in which the total energy change is zero.
Remember that energy can be transferred between objects, but can also be transformed within a single object. Although there are no truly conservative systems, many systems in nature approximate the conservative system closely enough to deserve consideration.
We want to look at two different kinds of conservative systems, those in which energy is transformed (changed from one form to another) in a single object, and those in which energy is transferred (from one object to another.)
8.1. defined as a system in which the total energy change is zero
8.1.1. (read as "delta" E equals zero meaning "the change in E is zero)
8.1.2. loss of one form of energy results in a gain in energy somewhere else within the system

8.1.2.1. different form of energy
8.1.2.2. energy given to another object


8.2. non collisions

8.2.1. falling objects without friction
8.2.1.1. loss of potential energy equals gain in kinetic energy
8.2.1.2. the sum of kinetic and potential energy, called total mechanical energy, remains constant
8.2.1.3. mgh = 1/2 mv2 (see table above)


8.2.2. pendulum

The pendulum may be visualized as a mechanical system which continually converts between kinetic and potential energy. In the ideal case no energy is lost and so at any given time the total energy is the sum of the kinetic and potential energy. It should be obvious that the pendulum gains kinetic energy (related to speed) as it loses potential energy (related to height) and vice-versa. At its highest point all the energy is in the form of potential while at its lowest point all the pendulum's energy is in the form of kinetic.

8.2.2.1. work is not path dependent

In the absence of friction no work is done in moving an object horizontally. The only work that is done is related to the change in potential energy. Path independence means that we can move an object such as the pendulum in a series of short horizontal steps (as in the "infinitely small" horizontal movements of the circle) or as a single horizontal displacement followed by a single vertical displacement.



In the diagram at left no work is done moving an object along a horizontal direction when there is no friction (recall Galileo's principle of inertia. No force is required to keep an object moving. A small amount of work is necessary to start it moving and an equal amount is "given back" when it is stopped.)

Whether the motion is circular (as with the pendulum), up a series of steps, or in one horizontal movement followed by lifting the height h, the work done is the same to raise the object to a height h.

This is what we mean by "Path Independence".
8.2.2.1.1. work done is the same for a given displacement in the absence of friction

8.2.2.2. strobe photo of pendulum

8.2.2.2.1. rises to same height on either side
8.2.2.2.2. like Galileo's ball on the incline


8.2.2.3. graph of energy of pendulum

8.2.3. others
8.2.3.1. mass/spring
8.2.3.2. ball in valley
8.2.3.3. planet in orbit
8.2.3.4. electron in atom
8.2.3.5. simple machines

8.2.3.5.1. lever, inclined plane, hydraulic jack

8.3. collisions

8.3.1. kinetic energy conserved only in elastic collisions
8.3.1.1. elastic collision <==> kinetic energy conserved
8.3.1.2. inelastic collisions involve frictional "losses"
8.3.1.3. certain collisions are nearly elastic
8.3.1.4. steel ball colliding with steel ball for example


8.3.2. can use with conservation of momentum to predict final state of motion

9. Nonconservative systems:

All real systems are actually nonconservative, that is they are systems where mechanical energy does not remain constant. In these systems energy is added from outside the system or escapes from the system. Mechanical energy refers to kinetic and potential energy specifically.
In real mechanical systems there is friction, which creates forces. Those friction forces, when applied to moving objects, amount to small amounts of work being done at the expense of the mechanical energy of the system.
At first glance this would appear to violate the conservation principle, but it doesn't. In every case where energy seems to disappear, the missing energy can be located by looking outside the system. In other words, if we look at the surroundings in which the systems exists, we will find something there which has gained or lost energy.
From within the mechanical system it appears that energy slowly leaks away. Be sure to study the graphs in this section and compare them with the graphs of conservative systems. In that comparison you will begin to see the genius in Galileo's method of isolating the main features while eliminating the complications such as friction.
We are getting a little ahead of the story and we will see in later programs how the concept of energy conservation was exonerated by James Joule.
9.1. A system where mechanical energy does not remain constant
9.1.1. requires that energy be added from or lost to outside of system

9.2. Real mechanical systems have friction
9.2.1. where does energy go?
9.2.1.1. Could it be energy is not really conserved?

9.2.2. Energy is still conserved if system is enlarged to include surroundings
9.2.3. like a bank account

9.2.3.1. transfer of money between savings and checking does not affect the total balance
9.2.3.2. deposits and withdrawals affect balance

9.2.3.2.1. this can be accounted for

9.2.3.3. interest also affects balance
9.2.3.3.1. books will not balance if it is ignored
9.2.3.3.2. money is not really disappearing
9.2.3.3.3. system must be expanded in order to account for "loss"


9.3. Energy slowly "leaks away" from mechanical system
9.3.1. graph with low friction

9.3.2. graph with high friction

9.3.3. Energy can be transformed back and forth between many different types
9.3.3.1. not at 100% efficiency
9.3.3.2. efficiency is ratio of work done to energy input
9.3.3.3. some is lost to the system in each transfer
9.3.3.4. still accountable, but no longer in the form of mechanical energy


10. Swinging Balls Revisited

Now it is time to revisit the swinging balls and see what role conservation of energy plays in describing their motion. It is clear that energy is conserved when the number of balls out equals the number of balls in. Recall that the balls are all of identical mass.
10.1. Conservation of Energy


We can reconsider those combination of balls in and out which satisfied the conservation of momentum requirement. You will recall that there are many combinations of number of balls and speed of balls which still conserves momentum. In general conservation of momentum is satisfied whenever the velocity is inversely proportional to the number of balls, like this:

Don't panic. This just says what we saw with the balls in program 18. As long as the speed of the balls is reduced by the same factor as the number of ball increases momentum is conserved.
The same is not true for energy. In fact there are no other combinations of speeds and number of balls which conserve energy. The only condition that conserves both energy and momentum is the one that happen reliably time after time. The number of balls out equals the number of balls in.
When you study the illustrations of the situations which do not conserve energy, pay special attention to the squared proportion in the kinetic energy expression. It is that proportion which prevents the balls from coming out in various combinations. Here's why:

Squaring a quantity is not the same as squaring a number. Try it. For example

because when you square the quantity 2v the two becomes a four.

Now the two on the left side and the four on the right side don't cancel out anymore and so we see that it is not an equation. 10.2. Other possibilities which do not conserve energy
10.2.1. the video program shows the graphics which are the same ones used for conservation of momentum in program 18. Do not panic here. Take some time to calculate the kinetic energy for each ball before and after the collisions. If you need numbers, use m = 1 and v = 2. Try to convince yourself that energy is only conserved if the number and speeds of balls in exactly equals the number and speeds of balls out.

10.2.2. two balls in, one ball out at twice the speed

10.2.3. two balls in, four balls out at half the speed

10.3. In fact there are no other combinations which conserve energy
10.3.1. THE ONLY COMBINATION WHICH CONSERVES BOTH ENERGY AND MOMENTUM IS:
10.3.1.1. THE NUMBER AND VELOCITIES OF BALLS OUT IS EQUAL TO THE NUMBER AND VELOCITIES OF BALLS IN

الحرارة

الحرارة من أهم أنواع الطاقة. وعندما نفكر في الحرارة نفكر عادة في الإحساس الذي تجعلنا الحرارة نحس به. فعلى سبيل المثال، في اليوم شديد الحرارة، ربما تجعلنا نحس بالضيق وعدم الراحة. ولكن أهمية الحرارة في حياتنا تتجاوز بكثير مجرد الشعور الذي تجعلنا نحس به

ما الحرارة
الحرارة شكل من أشكال الطاقة. ولا يمكن رؤية الحرارة أو الطاقة ولكن يمكن رؤية الأثر الذي يحدثانه. فمثلاً، ينتج عن احتراق الوقود في محركات الطائرة النفاثة غازات ساخنة تتمدد فتوفر القدرة اللازمة لتحريك الطائرة

لا يعلم أحد الحد الأقصى الذي يُمكن أن ترتفع إليه درجات الحرارة. لكن درجة الحرارة داخل أسخن النجوم تُقدَّر بملايين الدرجات. أما أقلّ درجة حرارة يمكن (نظريًا) الوصول إليها، وتُسمّى بالصفر المطلق، فهي - 273,15°م.
عند درجة الصفر المطلق، لا تحتوي الأجسام على طاقة حرارية أبدًا. ولم يتمكن الفيزيائيون حتى الآن من تبريد أيّ جسم من الأجسام إلى درجة الصفر المطلق، لذا فإن أيّ جسم ـ بما في ذلك أبرد الأجسام ـ يحتوي على بعض الطاقة الحرارية
استخدامات الحراره
نستخدم الحرارة في منازلنا في مجالات شتى؛ إذ نستخدمها في تدفئة المنازل وطبخ الطعام وتسخين الماء وتجفيف الملابس بعد غسلها، كما أن الحرارة هي التي تجعل المصابيح الكهربائية تضيء.
أما مجالات استخدام الحرارة في الصناعة فتكاد لا تحصر. فنحن نستخدمها في فصل الفلزات من خاماتها وفي تكرير البترول الخام. ونستخدمها في صهر الفلزات وتشكيلها وقطعها وتغليفها وتقويتها وضمّها بعضها لبعضً. ونستخدم الحرارة أيضًا في صناعة أو تحضير الأغذية والزجاج والورق والمنسوجات وعدّة منتجات أخرى.
نستخدم الحرارة أيضًا في تشغيل معداتنا الآلية؛ فالحرارة التي تتولّد من الوقود المحترق في محركات كل من الطائرات والسيارات والصواريخ والسفن توفر القدرة اللازمة لتحريك هذه الآليات. وكذلك تجعل الحرارة التوربينات الضخمة تدور وتولد الكهرباء التي تزودنا بالإضاءة والقدرة اللازمة لتشغيل كلّ أنواع الأجهزة، من مشحذة أقلام الرصاص الكهربائية إلى القاطرة الكهربائية.
مصادر الحرارة
مصدر الحرارة هو أي شيء يُعطي حرارة. تصدر الحرارة التي نستخدمها، أو التي تؤثر على الحياة والأحداث على ظهـر الأرض، من ستة مصـادر رئيسيـة هي: 1- الشمس و2- الأرض و3- التفاعلات الكيميائية و4- الطاقة النووية و5- الاحتكاك و6- الكهرباء.
الشمس مصدرنا الحراريّ الأهم و حرارتها ناتجه عن احتراق كتلتها نتيجة تفاعلات فيها

الأرض تحتوي على كميات كبيرة من الحرارة على أعماق بعيدة بباطنها. ويتسرَّب جزء من هذه الحرارة إلى السطح عندما يثور بركان

التفاعلات الكيميائية يمكن أن تُنتج الحرارة بعدّة طُرق. ويُسمَّى التفاعل الكيميائيّ الذي تتّحد فيه مادة ما مع الأكسجين الأكسدة. وتنتج الأكسدة السريعة الحرارة بسرعة تكفي لإشعال اللّهب. وعندما يحترق الفحم أو الخشب أو الغاز الطبيعي أو أيّ وقود آخر، تتحد بعض المواد الموجودة في ذاك الوقود مع أكسجين الهواء فتكوّن مركّبات أخرى. ويُنتج هذا التفاعل الكيميائي، الذي يُعرف بالاحتراق، حرارة ونارً
وفي كل الكائنات الحية، يتحول الطعام إلى حرارة، بالإضافة إلى طاقة وأنسجة حيّة عن طريق عملية التفاعل الحيوي، والتي تُسمى أيضًا الأيض. والأيض سلسلة تفاعلات كيميائية معقدة متوالية تقوم بها الخلايا الحية.

الاحتكاك عندما يحتك جسم بجسم آخر تنتج حرارة. ويمثل الاحتكاك في معظم الأحيان مصدر حرارة غير مرغوب فيه لأنه ربما يُتلف الأشياء. فمثلاً الحرارة التي تنتج في أية آلة عندما تحتك أجزاؤها بعضها ببعض ربما تؤدي إلى تآكل هذه الأجزاء. ولذا يوضع زيت التشحيم بين أجزاء الآليات المتحركة المتلامسة، وينقص زيت التشحيم فاعلية الاحتكاك وبالتالي يقلّل توليد الحرارة
الكهرباء يولد انسياب الكهرباء خلال الفلزات والسبائك وسائر الموصّلات (مواد تحمل أو توصل التيار الكهربائي) حرارة. ويستعمل الناس هذه الحرارة في تشغيل العديد من الأجهزة. ومن هذه الأجهزة المحرقات الكهربائية والأفران الكهربائية، وأجهزة التجفيف، والتدفئة ومحمصات الخبز الكهربائية، والكاويات الكهربائية
العزل الحراري
هو طريقة للتحكُّم في تحرك الحرارة بحبسها داخل أو خارج مكان ما. فمثلاً، تُعزل المباني السكنية حراريًا لتحبس الحرارة داخلها في فصل الشتاء وخارجها في فصل الصيف. ويستخدم الناس ثلاث طرق للعزل الحراري لأن الحرارة تنتقل بإحدى ثلاث طرق مختلفة.
وهناك مواد معينة، كالخشب والبلاستيك، عوازل جيدة ضد انتقال الحرارة بالتوصيل. ولهذا السبب تصنع مقابض العديد من أواني المطبخ الفلزية من هذه المواد. وتسخن هذه الأواني الفلزية بسرعة بالتوصيل ولكن تبقى مقابضها باردة.

توظيف الحرارة

تحويل الحرارة إلى حركة. توجد علاقة بين الطاقة الميكانيكية والطاقة الحرارية. فمثلاً، تتحول الطاقة الميكانيكية إلى حرارة بوساطة الاحتكاك بين الأجزاء المتحركة لأي آلة. ويمكن، في المقابل، تحويل الطاقة الحرارية إلى طاقة ميكانيكية في المحركات الحرارية.
ويمكن تقسيم المحركات الحرارية إلى مجموعتين: 1- محركات الاحتراق الخارجي و2-محركات الاحتراق الداخلي. وتُنْتَج الحرارة اللازمة لتشغيل محركات الاحتراق الخارجي خارج هذه المحركات. وتتضمن هذه المحركات التوربينات (العنفات) الغازية والبخارية والمحركات البخارية الترددية. أما محركات الاحتراق الداخلي، فإنها تنتج حرارة تشغيلها من الوقود المحترق بداخلها. وتتضمن هذه المحركات محركات الديزل والمحركات التي تُدار بالبنزين ومحركات الطائرة النفاثة ومحركات الصواريخ.
ويمثل التوربين البخاري مثالاً جيدًا لمحركات الاحتراق الخارجي. هنا، تحوِّل الحرارة الصادرة من وقود محترق أو مفاعل نووي الماء في الغلاية إلى بخار. وينقل البخار خلال أنابيب إلى التوربين الذي يحتوي على سلسلة من عجلات ذات زعانف معدنية مثبتة بعمود. ويتمدّد البخار ذو درجة الحرارة المرتفعة عندما يندفع خلال التوربين وبالتالي يدفع الزعانف ويجعلها تدور هي والعمود. وتكون درجة حرارة البخار الخارج من التوربين أقل بكثير من درجة حرارة البخار الداخل. ويمكن للعمود الدوّار في هذا المحرك، أن يدير مولدًا كهربائيًا أو يحرك المروحة التي تدفع سفينة أو أن يعمل عملاً آخر مفيدًا.


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