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This 9 page Class Notes was uploaded by Ardith Gutmann on Saturday September 12, 2015. The Class Notes belongs to MAE 241 at West Virginia University taught by Konstantinos Sierros in Fall. Since its upload, it has received 18 views. For similar materials see /class/202650/mae-241-west-virginia-university in Mechanical Engineering at West Virginia University.
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Date Created: 09/12/15
1 5 Chapter 1 General Principles Mechanics a branch of the physical sciences concerned with the state of res or the motion of bodies subjected to the action of forces 11 Rigidbody mechanics the branch of mechanics that deals with bodies that don t deform 111Statics the branch of rigidbody mechanics which deals with equilibrium of bodies and in which bodies do not accelerate Fundamental Concepts 21 Basic Quantities in Statics 211Len th used to locate the position ofa point in space 212mconceived as a succession of events and not used in statics 213Mass a measure of the quantity of matter which is used to compare the action of one body with that of another 214 Force a push or pull exerted on one body by another body 22 ldealizations in Statics 221These are used to simplify the application of theory 222Particle ldealization 2221 The object must have a mass but a negligent size 2222 Allows you to avoid considering the object s geometry if the object is a small enough pa rt of the system 223Rigid Body ldealization 2231 The object is a combination of a large number of particles all at a fixed distance from one another 2232 The body s shape changes little enough when a load is applied that it can be ignored 2233 Allows you to skip considering the body s material 224Concentrated Force ldealization 2241 You can assume that a force acts at one point on the body 23 Newton s Laws of Motion 231First law The sum of all forces is equal to zero when a body doesn t accelerate 232Second law The net force on a body is equal to the body s mass times its acceleration 233Third law The force of one body on another is equal and opposite to the force of the second body on the first 24 Newton s Law of Gravitational Attraction 241Any two given bodies experience a mutual gravitational attraction toward the other 242Described by F Gm1m2Rquot2 2421 G is a gravitational constant 66710quot11 mquot3kgsquot2 2422 m1 and m2 are the masses of the bodies 2423 R is the radius between the two centers of mass 243Most of the time the only sizable gravitational force is between a body and the earth Units of Measurement 31 The four basic quantities are related by F ma meaning they aren t independent 32 Units cannot be selected arbitrarily You must stick to one system of units per problem 33 SI Units 331These are used internationally 332Length is in meters 333Time is in seconds 334Mass is in kilograms 4 335Force is in newtons 3351 1 N 1 kgmsquot2 336Acceleration due to gravity is 981 msquot2 34 US Customary 341Length is in feet 342Time is in seconds 343Force is in pounds 344Mass is in slugs 3441 1 slug 1lbsquot2ft 345Acceleration due to gravity is 322 ftsquot2 Common Unit Conversions 3511 b 4448 N 3521 slug 1459 kg 3531 ft 3048 m The International System of Units 41 Prefixes 411For large or small quantities a prefix can be added to a unit 4111 eg 4000000 N is 4000 kN or 4 MN meganewtons 4112 G giga 10quot9 4113 M mega 10quot6 4114 k kilo 10quot3 4115 m mii 10quot3 4116 mu micro 10quot6 4117 n nano 10quot9 42 Rules for Use 421Units are separated by dots when multiplied 4211 eg kgmsquot2 422The exponential power on a unit refers to the unit AND its prefix 4221 eg mmquot2mmquot2mmmm 423With kilo as an exception avoid using prefixes in the denominator of a fraction 4231 eg Do write kNm but not Nmm Numerical Calculations 51 Dimensional Homogeneity 511n an equation each term must be expressed in the same units You can t subtract meters from seconds 512You can use this to check that you re using the right equation for a problem The units won t match up if you re using the wrong function 52 Significant Figures 521The number of significant figures included in a final answer indicates the accuracy of the answer 522Use engineering notation to avoid confusion Pick the nearest multiple from 10quotn3 and continue much like scientific notation 53 Rounding Off Numbers 531Don t round up if the deciding digit is less than five Do round up otherwise 532f you re rounding to n significant figures consider the n1th digit when rounding 533Exception If the digit is a five do whatever makes the previous digit even 54 Calculations 541When performing a sequence of calculations store the immediate results in your calculator and don t round off until you reach your final answer 542Most data in mechanics can reliably be rounded to three significant figures 6 General Procedure for Analysis 61 Solve problems to learn most effectively 62 Present solutions clearly and logically 63 Copy relevant data 64 Draw diagrams when helpful 65 Consider applicable principles and equations 66 Solve equations Round to three significant figures unless told otherwise 67 See if the answer makes sense 1 2 3 Chapter 2 Force Vectors Scalars and Vectors 11 Scalar any positive or negative physical quantity that can be completely specified by its magnitude 111eg length mass and time 12 Vector any physical quantity that requires both a magnitude and a direction for its complete description 121eg force position and moment 122Shown graphically by an arrow with the length representing the magnitude and the direction representing the direction of the vector The angle between the vector and its fixed axis defined the direction of its line of action The tip of the arrow indicates the sense of direction of the vector 123Denoted by drawing an arrow on top when writing or by bolding when typing Vector Operations 21 Multiplication and Division ofa Vector by a Scalar 211f the scalar is positive the vector s magnitude is increased by that factor but the direction stays the same 212f the scalar is negative the vector s magnitude is increased by that factor but the direction rotates by 180 degrees the opposite direction 22 Vector Addition 221All vector quantities obey the parallelogram law of addition 2211 Draw both vectors tail to tail 2212 Draw a translated copy of each vector at the others head 2213 The diagonal of the parallelogram is the resultant 222Can also use the triangle rule 2221 Put one vector s tail at the other vector s head 2222 The resultant is the line from the tail of the first to the head of the second 223Vector addition is commutative 2231 ABBA 224f two vectors are collinear have the same line of action you can just add the quantities 2241 ie If two vectors only have xcomponents you don t need to resolve them into component pieces to add them 23 Vector Subtraction 231 Think of it as a special case of addition 232Just multiply the vector being subtracted by the scalar 1 233Now you follow the same procedure as adding the two Vector Addition of Forces 31 Finding a Resultant Force 311You know two forces are acting on a body You want to find their combined force 312Use the triangle rule 313Use the law of sines or the law of cosines to find the magnitude and direction of the resultant force 32 Finding the Components of a Force 321You have a vector that lies between to axes You want to know it s components along each axis 322Draw a parallelogram from the tip of the vector and parallel to each axis 323Reduce it to a triangle by turning the sides of the parallelogram head to tail 324Use the law of sines to find the magnitudes of the components The directions are along the axes 33 Addition of Several Forces 331 You have more than two forces and want to find their resultant vector 332You can use 39 I I quot 39 of the r 0 law 333Find the parallelogram for the first two forces and find their resultant 334Use the resultant with the next force to make a new parallelogram and new resultant 335Repeat for all forces 336The first method can become difficult to calculate Instead use the rectangular component method in section 4 Addition of a System of Coplanar Forces 41 You can get rectangular components of a vector by resolving it along the x and yaxis 42 Scalar Notation 421The xcomponent Fx is Fcostheta 422The ycomponent Fy is Fsintheta 423Another method is to draw a smaller quotslopequot triangle proportional to the full triangle 4231 Then FxF ac where a is proportional to Fx and c is proportional to F 4232 Similarly FyF bc where b is proportional to Fy 43 Cartesian Vector Notation 431The Cartesian unit vectors are ihat and jhat ihat is one unit rightward jhat is one unit upward They are multiplied by scalars to communicate the desired vector 432They look like F Fx Fyj 44 Coplanar Force Resultants 441Using either scalar or Cartesian vector notation you can more easily add several vectors by simply adding their x and ycomponents since they are now collinear 4420nce you have the components of the resultant you can use Pythagorean s theorem to find the magnitude and the inverse tangent to find the angle Draw a sketch to make sure the angle makes sense Cartesian Vectors 51 RightHanded Coordinate System 511A rectangular coordinate system has its zaxis pointing in the direction of the right hand when the fingers are curled about the axisfrom the xaxis to the yaxis See page 43 for a picture 52 Rectangular Components of a Vector 521A vector can have three rectangular components depending on how it is oriented 522Using vector addition A Ax Ay Az 53 Cartesian Unit Vectors 531When you have a Zaxis you add a khat to serve as its unit vector 54 Cartesian Vector Representation 541A vector A in Cartesian vector form is A Ax Ayj Azk 542Using this to separate the magnitudes from the directions simplifies vector algebra operations in three dimensions 55 Magnitude of a Cartesian Vector 551To find the magnitude use a 3D version of Pythagorean s theorem 552A sqrtAxquot2 Ayquot2 AZquot2 56 Direction of a Cartesian Vector 561The coordinate direction angles are alpha beta and gamma They are measured between the tail of A and the positive x y and z axes 562Alpha beta and gamma are always between 0 and 180 degrees 563You must consider the projection of the vector onto each axis see page 45 for a picture 564AxA cosapha 565AyA cosbeta 566AzA cosgamma 567Find the angles by using inverse cosine on each 568You can use a unit vector uA in the direction of A to obtain the cosines 5681 uA AA AxiA AyjA AzkA 5682 A is the square root of the sum of its squared components 5683 uA cosaphai cosbetaj cosgammak 5684 Since uA is equal to 1 cosapha 2 cosbetaquot2 cosgammaquot2 1 56841 This lets you find the third cosine when you only know the first two 6 Addition of Cartesian Vectors 61 R A B Ax Bxi Ay Byj Az Bzk 7 Position Vectors 71 X Y and Z coordinates 711Use a righthanded coordinate system as discussed above 712The positive 2 axis points upward so it measures heightaltitude 713The x and yaxes lie of the horizontal plane 714Points in space are located relative to the origin 72 Position Vector 721A position vector r is defined as a fixed vector which locates a point in space relative to another point 722f r extends from the origin to Pxyz then r xi yj 2k 723Place the separate vectors headtotail to get to the head of the resultant vector 724f the position vector is directed from point A to point B you get rA rAB rB 725R XlBl XBlli lel lellJ39 23 ZlAllk 726You subtract the coordinates of the tail of vector A from the head of vector B 8 Force Vector Directed Along a Line 81 The resultant force will have the same direction as its position vector 82 The position vector can be turned into a unit vector through the equation u rr 83 From here F FXBl XlAlli lel lellJ39 23 ZlAllkllsqrtllxlBl XAllquot2 lel lellAZ 23 ZlAllAZl 831See page 59 for a more neat presentation 9 Dot Product 91 The dot product makes it easier to find the angle between two lines or the components of a force parallel and perpendicular to a line in 3D 92 The dot product ofA and B is written A B A dot B and is equal to ABcostheta where theta is the angle between them and is between 0 and 180 degrees 93 The dot product is always a scalar not a vector 94 Laws of Operation 941Commutative law AB BA 942Multiplication by a scalar aAB aAB AaB 943Distributive law AB D AB AD 95 Cartesian Vector Formulation 951A unit vector multiplied by itself is 1 A unit vector multiplied by a different unit vector is 0 952AB AxBx AyBy AzBz 953To determine the dot product of two vectors simply multiply their corresponding components and sum the products algebraically 954The result will be a positive or negative scalar 96 Applications 961Find the angle formed between two vectors or intersecting lines 9611 Theta cosquot1ABAB 962Find the components ofa vector parallel and perpendicular to a line 9621 Aa Acostheta Aua where a is the line whose parallel and perpendicular components we re seeking 9622 Aa Aaua 9623 For the perpendicular component theta cosquot1AuAA and Aperpendicular Asintheta or if Aa is known Aperpendicular sqrtAquot2 Aaquot2 1 Iquot Chapter 3 Equilibrium ofa Particle Condition for the Equilibrium ofa Particle 11 A particle is in equilibrium when it maintains a constant velocity including a velocity of 0 12 Often static equilibrium means the object is at rest 13 You can test equilibrium by checking Newton s first law of motion 1310bjects in equilibrium have a net force of zero The FreeBody Diagram 21 To apply equilibrium equations you must account for all the forces acting on an object 22 Think of the object as a particle that is isolated from its surroundings You can use this to create a freebody diagram to help solve a problem 23 Freebody diagram a drawing that shows a particle with all the forces that act on it 24 Springs 241A linearly elastic spring that doesn t deform and is of length L supports a particle the length of the spring will change in direction proportion to the force F acting on it 242Spring constan a characteristic that defines that elasticity of a spring Also called stiffness and referred to as k 243F ks describes the magnitude of force exerted on a linearly elastic spring with stiffness k and change in distance s from its unloaded state 2431 s Lloaded Lunloaded 2432 A positive s represents an elongation and means the force pulls on the spring A negative s represents a shortening and means the force pushes on the spring 25 Cables and Pulleys 251Al cables are assumed to have negligible weight and are assumed not to stretch 252A cable can only support a tensionpulling force 253The tension force found in a continuous cable pulled over a frictionless pulley must have constant magnitude to stay in equilibrium Therefore for any angle theta a cable is subjected to a constant tension T throughout its length 26 Drawing a FreeBody Diagram 261Draw an outlined shape of the particle 262Show all forces acting on the particle Do NOT include forces the particle is exerting on other objects 2621 Active force a force that tends to set a particle in motion 2622 Reactive force a force that is the result of constraints or supports that tend to prevent motion 263dentify each force Label known values with magnitude and direction Use letters to represent unknown forces 3 Coplanar Force Systems 31 In the xy plane all forces on a particle can be broken into their ihat and jhat components and then summed to determine whether or not the particle is in equilibrium 32 Fnet xihat Fnet yjhat 0 33 For the above to be true Fnet x and Fnet y must both equal zero 34 You can use Fnet x 0 and Fnet y 0 to solve two unknown angles or magnitudes in a problem 35 When summing the x and yforces you first determine which direction is the positive x direction and which is the positive ydirection Then you add forces by making those in the positive direction positive numbers and those in the negative direction negative numbers P 36 fa force s magnitude is unknown you can make a best guess about its direction You ll know your guess was wrong if your solution is a negative scalar 37 Steps for Coplanar Force Equilibrium Problems 371Start with a freebody diagram 3711 Establish x and y axes in a suitable direction Generally positive xaxis to your right and positive yaxis pointing up works well The most important part is to stick to the axes you chose throughout the problem 3712 Label all the known and unknown force magnitudes and directions on the diagram 3713 Make an assumption for the direction of a force with an unknown magnitude 372Use the equations of equilibrium 3721 Use the equations Fnet x 0 and Fnet y 0 3722 Consider components pointed along a positive axis to be positive and components pointed along a negative axis to be negative 3723 If there are more than two unknown quantities and there is a spring use the equation F ks 3724 The magnitude of a force is always positive If you get a negative result its sense of direction is reversed from what you assumed when you drew the freebody diagram ThreeDimensional Force Systems 41 In a threedimensional force system you can resolve forces into i j and k components to create the equilibrium requirement of Fnet xi Fnet yj Fnet zk 0 42 For this equation to be true the following must all be true 421Fnet x 0 422Fnet y 0 423Fnet Z 0 43 The algebraic sum of the components of all forces acting on the particle along each coordinate axis must be zero 44 This set allows you to solve for up to three unknowns 45 Procedure 451FreeBody Diagram 4511 Establish x y and z axes in a suitable orientation 4512 Label known and unknown force magnitudes and directions 4513 Assume a sense for each unknown force 452Equations of Equilibrium 4521 Use scalar equations of equilibrium 45211 Fnet x 0 45212 Fnet y 0 45213 Fnet Z 0 4522 If the geometry is difficult first express each force as a Cartesian vector and substitute the vectors into Fnet 0 setting i j and k equal to zero 4523 If a solution for a force is negative the sense is simply the reverse of what you assumed
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