MchT 111 - Mechanics For Technology: Statics

INTRODUCTION

Mechanics is the physical science that deals with the behavior of bodies under the influence of forces.  Mechanics can be divided into 3 categories: 1.)  mechanics of rigid bodies,  2.)  mechanics of deformable bodies, and 3.)  mechanics of fluids.  This course is a beginning course in statics, which is a portion of the mechanics of rigid bodies.

Mechanics of Rigid Bodies:  This course deals solely with the mechanics of rigid bodies.  A rigid body is a body which does not deform under the influence of forces.  In all real applications, there is always deformation, however, many stuctures exhibit very small deformations under normal loading conditions, and rigid body mechanics can be used with sufficient accuracy in those cases.  Also, the principles of rigid body mechanics are some of the building blocks needed for the mechanics of deformable bodies.

The mechanics of rigid bodies is sub-divided into two areas, statics and dynamics, with dynamics being further subdivided into kinematics and kinetics.  Statics is the study of bodies in equilibrium.  This means there are no unbalanced forces on the body, thus the body is either at rest or moving at a uniform velocity.  Dynamics is the study of bodies which are not in equilibrium, thus there is acceleration.  Kinematics is the study of the motion of a body, without regard for how the motion is produced.  This is sometimes called the "geometry of motion".  Kinematic principles are often applied to the analysis of machine members to determine positions, velocities, or accelerations at various parts of the machines' operation.  Kinetics is the study of the forces which cause motion, or the forces which result from motion.

Mechanics of Deformable Bodies:  The mechanics of deformable bodies deals with how forces are distributed inside bodies, and with the deformations caused by these internal force distributions.  These internal force produce "stresses" in the body, which could ultimately result in the failure of the material itself.  Principles of rigid body mechanics often provide the beginning steps in analyzing these internal stresses, and resulting deformations.  These will be studied in courses called Strength of Materials or Mechanics of Materials.

Mechanics of Fluids:  The mechanics of fluids is the branch of mechanics that deals with liquids or gases.  Fluids are commonly used in engineering applications.  They can be classified as incompressible, or compressible.  While all real fluids are compressible to some degree, most liquids can be analyzed as incompressible in many engineering applications.  Applications of fluid mechanics abound, from hydraulics and general flow in pipes to air flow in ducts to advanced applications in turbines and aerospace.  The study of the mechanics of fluids will be studied in courses called Fluid Mechanics, Compressible Flow, Hydraulics, and others.

History of Mechanics:  The basic principles of statics were developed very early.  The fundamentals of levers, inclined planes, and other principles were needed by early civilizations to construct huge structures such as the pyramids.  Below is a timeline giving important milestones in the development of mechanics.

 400 BC Archytus of Tarentum - Theory of Pulleys 287-212 BC Archimedes - Lever equilibrium, buoyancy principle 1452-1519 Leonardo da Vinci - Equilibrium, concept of moments 1473-1543 Copernicus - Proposed that the earth revolves around the sun 1548-1620 Stevinus - Inclined planes, parallelogram law for addition of forces 1564-1642 Stevinus, Galileo - Virtual work principles 1564-1642 Galileo - Dynamics of pendulums, falling bodies 1629-1695 Huygens - Accurate measurement o fthe acceleration due to gravity 1642-1727 Newton - Law of universal gravitation, laws of motion 1654-1722 Varignon - Work with moment and force relationships 1667-1748 Bernoulli - Application of virtual work to equilibrium 1707-1793 Euler - Rigid body systems, moments of inertia 1717-1783 D'Alembert - Concept of inertia force 1736-1813 Lagrange - Formalized generalized equations of motion 1792-1843 Coriolis - Work with moving frames of reference 1858-1947 Planck - Quantum mechanics 1879-1955 Einstein - Theory of relativity

Fundamental Quantities:  There are four fundamental quantities in mechanics, length, time, mass, and force.  Length, time, and mass are known as absolute quantities, and are independent of each other.  Force is not an absolute quantity since it is related to mass, and changes in velocity.

Length:  Length is the quantity used to describe the position of a point in space relative to another point.  This distance is described in terms of a standard unit of length.  The universally accepted standard unit for length is the meter.  This distance standard has been refined over the years.  Originally, it was one ten-millionth of the earth's quadrant, not an easy measurement to make.  In 1889, the meter was defined as the distance between two finely inscribed lines on a platinum-iridium bar which was held to specific environmental conditions.  This definition held until October 14, 1960 when the distance was redefined as 1,650,763.73 wavelengths of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton-86 atom in a vacuum.  This produced very small errors in the measurements of the speed of light, so on October 20, 1983, the meter was redefined to be  the length of the path traveled by light in a vacuum during 1/299,792,458 of a second.  The relationship between the meter and the inch has been defined as: 1 in. = 2.54 cm (exactly).

Time:  Time is the interval between two events.  The generally accepted standard unit for time is the second.  A second was originally defined as 1/86,400 of the average period of revolution of the earth on its axis.  In 1956, the definition of a second was refined to be 1/31,556,925.9747 of the time needed for the earth to orbit the sun in the year 1900.  Obviously, this definition could cause measurement problems.  Therefore, on October 13,1967 the second was redefined to be "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the fundamental state of the atom of cesium 133"

Mass:  Mass is a property of matter.  Mass can be considered to be the amount of matter contained in a body.  The mass of a body determines both the action of gravity on the body, and the resistance to changes in motion.  This resistance to changes in motion is referred to as inertia, which is a result of the mass of a body.  The internationally accepted unit of mass is the kilogram, which is defined as the mass of the International Prototype Kilogram, a platinum-iridium mass stored near Paris, France.

Force:  Force is a derived unit, but a very important unit in the study of mechanics.  Force is often defined as the action of one body on another.  Force may or may not be the result of direct contact between bodies.  Gravitational, and electromagnetic forces are examples of forces which result from actions other than direct contact.  Forces have both magnitude and direction, and are therefore vectors, a concept which will be discussed later.  Force interactions always occur in equal but opposite pairs.  There are two principle effects of forces, they tend to change the motion of a system, and they tend to deform a system.  If the body neither changes its motion nor deforms, then other resisting forces must be developed on the body.  These resisting forces will be of primary concern in out study of statics.  The basic unit of force is the Newton in the SI system and the pound in the English system.  These units will be discussed in more detail later.

Newton's Laws:  Newton's three laws provide the foundation for the study of mechanics.  These laws are:

 First Law:  A body at rest will remain at rest, and a body in motion will remain at a uniform speed in a straight line, unless it is acted on by an imbalanced force. Second Law:  If an imbalanced force is applied to a body, the body will accelerate in the direction of the imbalance, with a magnitude proportional to the imbalance. Third Law:  For every action, there is an equal but opposite reaction.

These three laws will be applied frequently as this course develops.

Mass vs. Weight:  As stated above, mass is a fundamental quantity of matter.  It is independent of location and surroundings.  The weight of a body is the force exerted on the body due to gravitational attraction of the Earth, or some other massive body such as a planet or the moon.  The weight is therefore not independent of location, in fact, it depends very much on location.  The relationship between mass and weight can be expressed as:

W=mg           (1-1)

 Where: W is weight m is mass g is acceleration due to gravity Approximate values for g on earth are: g = 9.807 m/s2 (SI units) g = 32.17 ft/s2 (English units)

Units of Measure:  Two different systems of units will be used in this course, the International System of Units, or SI, and the English Engineering System, sometimes referred to as US Customary Units.  As stated above, the four fundamental units that are of concern in mechanics are mass, length, time, and force.  Looking at equation 1-1, it can be seen that these four units cannot be independently defined.  Any three of the four can be defined, and the fourth with necessarily be a derived unit.

International System of Units (SI):  In the SI system, the three fundamental quantities are mass, length, and time.  The units are kilograms (kg) for mass, meter (m) for length, and second (s) for time.  All of these units are defined as stated above.  All of these units are independent of location, and therefore this system is referred to as an absolute system of units.  In this system, force is a derived unit.  The force unit is called a newton (N), and is defined as the force required to accelerate a mass of 1 kg at a rate of 1 meter/sec.  So, we can write:

 1 N = (1 kg)(1 m/s2)  or 1 N = 1 kg.m/s2 (1-2)

The weight of an object is the gravitational force which is exerted on that object which causes it to accelerate downward at the acceleration due to gravity, or g.  So, we can write for the weight of a 1 kg mass:

 W = mg W = (1 kg)(9.807 m/s2) W = 9.807 N (1-3)

So, the weight of a 1 kg mass is 9.807 N.  This is often rounded off to 9.81, and that is the value that will be used from now on.

In the SI system, orders of magnitude of a unit are given by the prefix which is used.  The table below lists many of the common prefixes.

 Multiplier Prefix Symbol 109 giga G 106 mega M 103 kilo k 10-2 centi c 10-3 milli m 10-6 micro µ 10-9 nano n 10-12 pico p

To use these prefixes, the prefix is placed directly in front of the symbol for the basic unit.  For example, a millimeter is 10-3 meters, and the symbol is mm. Common units in mechanics using prefixes are shown in table 1-2.

 Unit Symbol Equivalence kilometer km 1 km = 1000 m centimeter cm 1 cm = .01 m millimeter mm 1 mm = .001 m gram g 1000 g = 1 kg kilonewton kN 1 kN = 1000 N

English Engineering Units (US Customary Units):  The fundamental quantities in this system are weight, length, and time.  Since weight is included, this system depends on the gravitational attraction at the location of interest. Therefore, this is sometimes refered to as a gravitational system of units.  The units are pounds (lb) for weight, foot (ft) for length, and second (s) for time.  The second was defined previously, but the pound and the foot need to be defined.

The pound is defined as the weight of a standard platinum mass kept at the Bureau of Standards, measured at sea level and at a latitude of 45o.  This mass is 0.45359243 kg.

The foot is defined as 0.3048 meters.

With these fundamental units defined, it is now possible to define a unit of mass in this system.  There are two units of mass used in the English system.  One is called the pound mass (lbm) and the other is called the slug.  It can be confusing at times, but hopefully this will clear some of the confusion up.

The   lbm is the mass which weighs 1 pound force (lbf) at sea level.  This is frequently used in various commercial applications, but when these units are used in equation 1-1, the units are not totally consistent, and a conversion factor needs to be applied.  Therefore, this unit is not often used in engineering applications.

The slug is the more common unit for engineering applications.  It is defined as the mass which would be accelerated 1 ft/s2 by a force of 1 lb.

 1 slug = 1 lb.s2/ ft (1-4)

The weight of an object is the gravitational force which is exerted on that object which causes it to accelerate downward at the acceleration due to gravity, or g.  So, we can write for the weight of a 1 kg mass:

 m = W / g m = (1 lb)(32.2 ft / s2) m = 1/32.2 slugs (1-5)

So, the mass of a 1 pound weight is 1/32.2 slugs or .0311 slugs.

Table 1-3 gives common quantities that will be used in mechanics, and factors for converting between SI and English system units.

 English to SI SI to English Quantity To Convert Multiply By To Obtain To Convert Multiply By To Obtain Length inches 2.540 centimeters centimeters .3937 inches inches 2.540 x 10-2 meters meters 39.37 inches feet 0.3048 meters meters 3.281 feet miles 1.609 kilometers kilometers .6214 miles Mass slug 14.59 kilogram kilogram .06854 slug Force pound 4.448 newton newton .2248 pound Velocity ft / s 0.3048 m / s m / s 3.281 ft / s in / s 2.540 x 10-2 m / s m / s 39.37 in / s mph 0.4470 m / s m / s 2.237 mph Acceleration ft / s2 0.3048 m / s2 m / s2 3.281 ft / s2 in / s2 2.540 x 10-2 m / s2 m / s2 39.37 in / s2 Pressure psi 6.895 kPa kPa .1450 psi Torque ft . lb 1.356 N . m N . m .7376 ft . lb Work ft . lb 1.356 Joule Joule .7376 ft . lb Power ft . lb / s 1.356 Watt Watt .7376 ft . lb / s hp 745.7 Watt Kilowatt 1.341 hp

For other conversions, you might want to try one of these links: CONVERSION 1 or CONVERSION 2.

Dimensional Homogeneity:  It is important for all units in an equation to be compatible in order to obtain the correct solution.  This is called dimensional homogeneity.  For example, in equation 1-1, the unit of weight might be a newton, mass might be in kilograms, and the acceleration due to gravity might be in m/s2in that case, all of the units would be compatible, and a correct answer would be the result. But what if the mass were given in grams instead of kilograms. Grams is not a unit which is compatible with the equation unless an appropriate conversion factor is used. It is very important to check every equation for compatible unites before proceeding to the solution.

Significant Figures:  Rules abound for rounding off engineering calculations.  The accuracy of any calculation depends on many factors.  Obviously, the accuracy of the calculation method itself plays a large role in the final result.  The accuracy of the initial data is also very important.  There are methods used for determining the possible error present in a calculation.  The widespread use of hand held calculators gives the false impression that results can be reliable to many significant figures.  However, in this course, there is no way of knowing how accurate the given data is, so the assumption will be that the given data is accurate enough to allow for three to four significant figures in the final solution.  Intermediate results should be worked out to five to six significant figures.

Idealizations:  Real mechanical systems are very complex.  It is common practice to simplify systems for computational purposes.  The simpler the model of the system becomes, the greater the chance for variations between the model results and the real life results.  For a first course in statics, there are several idealizations that are made which have proven to be very valid in a wide range of applications.  Three of them will be discussed here, and others will be discussed as the need arises.

Rigid Body:  Any real body will undergo a deformation when subjected to a load.  It might bend, or crush, or twist, or any number of possible distortions may take place.  Sometimes the deformations are very small.  To take it to a ridiculous extreme, a calculation could be made to determine how much a bridge would deform if a fly landed on it.  Obviously, it would be an extremely small amount, and negligible in the overall design of the bridge.  So even though the bridge would have some sort of deformation, it could be considered not to have any at all for that case.  Many important things can be learned by performing calculations based on the assumption of no deformation.  When this assumption is made, the resulting object is said to be a rigid body.  In this course, all bodies will be assumed rigid, and deformations will not be considered.

Particle: A particle is an object that has mass, but no size.  This is a nice definition, but not real practical.  However, for the purposes of many calculations, objects of all sizes can be modelled as particles.  The object would be assumed to have all of its mass centered at the center of mass.  This assumption will be made often throughout this course.  In order to decide if an object can be modelled as a particle, it is important to look at several factors.  One of them is deformation.  If a real object will be significantly deformed by the application of forces, then the particle model will probably not work.  Another case which may exclude the use of a particle model would be moving parts in a system.  If moving parts happen to keep the center of mass in the same place, but have an effect on the response to forces, then a particle model would not work.  These kinds of decisions will not be made in this course, but the student should be aware that they may have to be made in real life.

Point Force:  Whenever a force is applied to a body, it must be distributed over some finite area.  If a force, even a very small force, is concentrated on one point, that point would be under an infinite stress, and would fail. However, the concept of a point force is very useful in mechanics.  When a force is concentrated on a small area, it can be considered, for purposes of statics, to be concentrated at a point.  The amount of error introduced by this assumption is very small, and the smaller the actual area of contact, the smaller the error. In this course, the concept of a point force will be used extensively.

Problem Solving Technique: A first course in statics is often one of the the first technical courses taken by a technology student.  It is important to develop a good problem solving technique early, which will carry through into other similar courses, and beyond.  Many procedures and techniques are given in textbooks, and each appears to be a little different than the others, but each has the same goal in mind.  To develop a procedure for an orderly approach to solving problems, which will result in fewer errors, a better understanding of the problem, neat and organized solutions, and easier troubleshooting to find errors that may exist.  The method outlined here is an example of such a systematic approach.  There are no claims that it is the best, or the only method, but the important thing is to get into the habit of following a technique such as this.

1.) Define the problem
2.) Collect information
3.) Generate a "plan of attack"
4.) Apply the appropriate principles and equations
5.) Solve
6.) Verify the solution

Define the Problem:  Defining the problem may involve several steps.  The first is to read the problem carefully. Make sure you understand what it is the problem is asking for.  Whenever possible, write down your understanding of the problem, but as a minimum, have a clear problem definition in mind before attacking the solution.  Identify what result is  requested.

Collect Information: Here is where you start writing down information.  Make all appropriate sketches, neatly and clearly.  Make sure all given information is either shown on the sketches, or is listed separately.  ALL given information should be written down.  Do not just rely on reading from the book.  Make a list of all of the unknowns. This will help direct your thinking towards an efficient solution to the problem.  Make a list of any assumptions as appropriate.

Generate a "Plan of Attack": Study the problem, and determine what theories are required for solution. Consider all of the relevant formulas, and the constraints and limitations of those formulas.  Decide if the best approach is to proceed with a hand calculation, or to use a computer assisted approach.  Factors such as how often the calculation must be done, complexity of the calculation, required accuracy, and software available to you should be considered.  Most problems in this course will be done using hand calculations, so that decision will be made for you, but once that is decided, it is still important to have a clear view of where you are headed with the calculation.  This becomes more important as the complexity of the problems increase. It is often worth the time to make an informed decision on the theory or approach to use.  A bad decision can cost you a great deal of time as you get into the solution itself.

Apply the Appropriate Principles and Equations:  Write down the equations to be used in symbol form.  Make substitutions into the equations only after you are confident that all of the correct equations have been selected.

Solve:  This is obviously a very important step in the process.  It is very important to have the proper mathematical background to be able to correctly implement the solution.  A strong ability in algebra, trigonometry, and geometry will be required to finally arrive at the correct solution.  It is very surprising how often this step causes the most problems.  Solving of the equations must be done carefully, systematically, and checked thoroughly.

Verify the Solution:  After solving the equations, the process is not over.  The question "does the answer make sense?" should always be asked after the solution is complete.  Often by looking at a problem you can guess at an appropriate range for the answer.  If your solution doesn't make sense compared to your guess, then it is the first hint that something may be wrong.  The importance of getting the correct answer is often overlooked in when a lot of partial credit is given for proper set up of problems.  But consider this, if an engineer does everything right, but hits the wrong button on his calculator and is off by a factor of 10 while designing a bridge, do you want to be the first person to drive across?  Be diligent to verify your solutions. You can never be absolutely sure you are correct, but the more times you question your results, the more likely you are of having the right solution.

Following these steps, or a similar procedure, will not only increase your chances of arriving at the correct solution, but will also make it much easier to go back and find errors.  If steps are skipped, and things are not written down, then it is much more difficult for you or someone else to follow your calculations in the future.