ME 2151 ENGINEERING MECHANICS

ME 2151

ENGINEERING MECHANICSIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0UNIT-IBASICS & STATICS OF PARTICLES INTRODUCTIONIntroduction Units and Dimensions Laws of Mechanics Lames theorem,Parallelogram and triangular Law of forces Vectors Vectorial representation of forces and moments Vector operations: additions, subtraction, dot product, cross product Coplanar Forces Resolution and Composition of forces Equilibrium of a particle Forces in space Equilibrium of a particle in space Equivalent systems of forces Principle of transmissibility Single equivalent force.IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0COURSE GOALSThis course has two specific goals:

(i) To introduce students to basic concepts of force, couples and moments in two and three dimensions.

(ii) To develop analytical skills relevant to the areas mentioned in (i) above.

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0COURSE OBJECTIVESUpon successful completion of this course, students should be able to:(i)Determine the resultant of coplanar and space force systems.

(ii)Determine the centroid and center of mass of plane areas and volumes.

(iii)Distinguish between concurrent, coplanar and space force systems

(iv)Draw free body diagrams.

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0COURSE OBJECTIVES CONTD.(v)Analyze the reactions and pin forces induces in coplanar and space systems using equilibrium equations and free body diagrams.

(vi)Determine friction forces and their influence upon the equilibrium of a system.

(vii) Apply sound analytical techniques and logical procedures in the solution of engineering problems.

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0INTRODUCTION-UNITS & DIMENSIONSUnits is a standard name for measuring the dimensionsCommen system of units isF.P.S (foot,pound,second)C.G.S (Centimetere-gram-second)M.K.S (Metre-kilogram-second)S.I (international system of units)IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0LAWS OF MECHANICS

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0LAMES THEOREMIf three forces acting at a point are in equilibrium each force will proportional to the sine of angle between the other forces

ABCIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0QPRParalleologram LawConstruct a Parm. with two Forces as Parts. The resultant of the forces is the diagonal.

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0TRIANGLE LAWIf two forces are acting simultaneously on a particle and can be represent by the two sides of triangle taken in order then the third side represents the resultantPQR = P + QQ + P = P + Q. This is the cummutative law of vector additionIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0ExampleTwo structural members B and C are bolted to bracket A. Knowing that both members are in tension and that P = 30 kN and Q = 20 kN, determine the magnitude and direction of the resultant force exerted on the bracket.

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0Solution

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0VECTORSVector is a mathematical representation of force in its magnitude and direction.It is also representation of magnitude and direction of displacement,velocity, acceleration,etcIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0VECTORIAL REPRESENTATION OF FORCES & MOMENTS

Vector components may be expressed as products of the unit vectors with the scalar magnitudes of the vector components.

Fx and Fy are referred to as the scalar components of

May resolve a force vector into perpendicular components so that the resulting parallelogram is a rectangle. are referred to as rectangular vector components and

Define perpendicular unit vectors which are parallel to the x and y axes.IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0VECTOR OPERATIONSUsing the Parallelogram Law, Construct a Pram with two Forces as Parts. The resultant of the forces is the diagonal.Addition,subtraction,dot product, cross product

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0Addition of Vectors

Trapezoid rule for vector additionTriangle rule for vector addition

BBCC

Law of cosines,Law of sines,

Vector addition is commutative,

Vector subtractionIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0Vector Addition Contd.Triangle Rule: Draw the first Vector. Join the tail of the Second to the head of the First and then join the head of the third to the tail of the first force to get the resultant force, RIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0VECTOR SUBTRACTIONSubtraction of vector is defined as addition of the corresponding negative vectors The difference can be found by adding the vector P and the negative vector QIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0 Vector Subtraction

P - Q = P + (- Q)

PQ P-QP -QQPP - QParm. RuleTriangle RuleIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0DOT PRODUCT & CROSS PRODUCTDot or scalar product of two vectors P and Q is a scalar quantity and is defined as the product of the magnitude of the two vectors and the cosine of their included angle Cross product of two vectors P and Q is a scalar quantity and is defined as the product of the magnitude of the two vectors and the sine of their included angleIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0COPLANAR FORCESCoplanar forces: set of forces which all pass through the same point.

A set of concurrent forces applied to a particle may be replaced by a single resultant force which is the vector sum of the applied forces.

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0RECTANGULAR COMPONENTS OF FORCE CONTD.In many problems, it is desirable to resolve force F into two perpendicular components in the x and y directions. Fx and Fy are called rectangular vector components. In two-dimensions, the cartesian unit vectors i and j are used to designate the directions of x and y axes.Fx = Fx i and Fy = Fy ji.e. F = Fx i + Fy jFx and Fy are scalar components of F

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0RECTANGULAR COMPONENTS OF FORCE CONTD.

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0RESOLUTION & COMPOSITION OF FORCES

Concurrent forces: set of forces which all pass through the same point.

A set of concurrent forces applied to a particle may be replaced by a single resultant force which is the vector sum of the applied forces.

Vector force components: two or more force vectors which, together, have the same effect as a single force vector.IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0Equilibrium of a ParticleWhen the resultant of all forces acting on a particle is zero, the particle is in equilibrium.

Particle acted upon by two forces:equal magnitudesame line of actionopposite sense

Particle acted upon by three or more forces:graphical solution yields a closed polygonalgebraic solution

Newtons First Law: If the resultant force on a particle is zero, the particle will remain at rest or will continue at constant speed in a straight line.IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0Forces in SpaceRectangular ComponentsFyFxFzjikF

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0EQUILIBRIUM OF PARTICAL IN SPACE

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0Forces in Space Contd.

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0EQUIVALENT SYSTEMS OF FORCESFor equilibrium: Fx = 0 and F y = 0.Note: Considering Newtons first law of motion, equilibrium can mean that the particle is either at rest or moving in a straight line at constant speed.

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0Determine the resultant of the three forces below.

25o45o350 N800 N600 N60oyxIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0

25o45o350 N800 N600 N60oyxIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0ExampleA hoist trolley is subjected to the three forces shown. Knowing that = 40o , determine (a) the magnitude of force, P for which the resultant of the three forces is vertical (b) the corresponding magnitude of the resultant.

1000 NP2000 N

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0Solution

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0PRINCIPLE OF TRANSMISSIBILITYThe state of rest or motion of a rigid body is unaltered if a force acting on the body ifs replaced by another force of the same magnitude and direction but acting any where on the body along the line of action of the replaced forceIFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0SINGLE EQUIVALENT FORCE

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0

IFETCE/MECH/U-II/KAMALANATHAN.R/I YR/II SEM/ME 2151/EM/UNIT-1/PPT/VER1.0