东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A02 Statics of Particles

Statics of Particles

Statics of Particles

Contents·Introduction（绪论）·ResultantofTwoForces（两个力的合力）·Vectors（矢量）·AdditionofVectors（矢量加和）·Resultantof SeveralConcurrentForces（多个力的合力）·Rectangular Components ofa Force:Unit Vectors（力的直角坐标分量：单位矢量）·Addition ofForcesbySummingComponents（通过分量相加求合力）·EquilibriumofaParticle（质点平衡条件）·Free-BodyDiagrams（受力简图）2

Contents • Introduction（绪论） • Resultant of Two Forces（两个力的合力） • Vectors（矢量） • Addition of Vectors（矢量加和） • Resultant of Several Concurrent Forces（多个力的合力） • Rectangular Components of a Force: Unit Vectors（力的直角坐标 分量：单位矢量） • Addition of Forces by Summing Components（通过分量相加求 合力） • Equilibrium of a Particle（质点平衡条件） • Free-Body Diagrams（受力简图） 2

Introduction: The objective for the current chapter is to investigate the effects of forceson particles:- replacing multiple forces acting on a particle with a singleequivalent or resultant force,- relations between forces acting on a particle that is in astate of equilibrium.. The focus on particles does not imply a restriction to miniscule bodiesRather, the study is restricted to analyses in which the size and shape ofthe bodies is not significant so that all forces may be assumed to beapplied at a single point.3

Introduction • The objective for the current chapter is to investigate the effects of forces on particles: - replacing multiple forces acting on a particle with a single equivalent or resultant force, - relations between forces acting on a particle that is in a state of equilibrium. • The focus on particles does not imply a restriction to miniscule bodies. Rather, the study is restricted to analyses in which the size and shape of the bodies is not significant so that all forces may be assumed to be applied at a single point. 3

Resultant of Two Forces:force: action of one body on another:characterized by its point of application,101bmagnitude, line of action, and sense.30°A. Experimental evidence shows that thecombined effect of twoforces may berepresented by a single resultant force. The resultant is equivalent to the diagonal ofRa parallelogram which contains the twoforces in adjacent legsA. Force is a vector quantity4

Resultant of Two Forces • force: action of one body on another; characterized by its point of application, magnitude, line of action, and sense. • Experimental evidence shows that the combined effect of two forces may be represented by a single resultant force. • The resultant is equivalent to the diagonal of a parallelogram which contains the two forces in adjacent legs. • Force is a vector quantity. 4

VectorsVector: parameter possessing magnitude and directionwhich add according to the parallelogram law. Examples:displacements, velocities, accelerations.+(: Scalar: parameter possessing magnitude but not directionQExamples: mass, volume, temperature.Vectorclassifications:-Fixed orbound vectors have well defined points ofapplication that cannot be changed without affectingan analysis.-Freevectorsmaybefreelymoved inspacewithoutchanging their effect on an analysis- Sliding vectors may be applied anywhere along theirline of action without affecting an analysisEqual vectors have the same magnitude and direction: Negative vector of a given vector has the same magnitudeand the opposite direction5

Vectors • Vector: parameter possessing magnitude and direction which add according to the parallelogram law. Examples: displacements, velocities, accelerations. • Vector classifications: - Fixed or bound vectors have well defined points of application that cannot be changed without affecting an analysis. - Free vectors may be freely moved in space without changing their effect on an analysis. - Sliding vectors may be applied anywhere along their line of action without affecting an analysis. • Equal vectors have the same magnitude and direction. • Negative vector of a given vector has the same magnitude and the opposite direction. • Scalar: parameter possessing magnitude but not direction. Examples: mass, volume, temperature 5

Addition of VectorsTrapezoidruleforvectoraddition?Triangle rule for vector additionP+OA? Law of cosines,QR2 = P2 +Q?-2PQcos BBR=P+QOQLawof sinesQP+ACsin BsinCsin A(a)PRQBQAG(b). Vector addition is commutative,P+0=0+P-.Vectorsubtraction(b)(a)6

Addition of Vectors • Trapezoid rule for vector addition • Triangle rule for vector addition B B C C R P Q R P Q PQ B         2 cos 2 2 2 • Law of cosines, • Law of sines, sin sin sin ABC Q R P   • Vector addition is commutative, P Q Q P        • Vector subtraction 6

Addition of Vectors.Addition of three or more vectors throughrepeated application of the triangle ruleQ+SP+Q+S. The polygon rule for the addition of three ormorevectors.Vector addition is associative,P+Q+SP+Q+S=(P+@)+S=P+@+S1.5P. Multiplicationof a vectorby a scalar2P7

Addition of Vectors • Addition of three or more vectors through repeated application of the triangle rule • The polygon rule for the addition of three or more vectors. • Vector addition is associative, P Q S P Q S P Q S                  • Multiplication of a vector by a scalar 7

Resultant of Several Concurrent ForcesConcurrent forces: set of forces which allpass through the same point.Asetofconcurrentforcesappliedtoaparticle may bereplaced by a singleresultantforcewhichisthevector sumofthe(b)(a)applied forces.·Vectorforce components:two or more forcevectors which, together, have the same effectasasingleforcevector(b)(c)8

Resultant of Several Concurrent 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. 8

Sample ProblemQ=60NSOLUTION:25°: Graphical solution - construct aP=40Nparallelogram with sides in the same1200direction as P and Q and lengths inAproportion. Graphically evaluate theresultant which is equivalent in directionand proportional in magnitude to the thediagonal.The two forces act on a bolt at A.: Trigonometric solution - use the triangleDeterminetheirresultantrule for vector addition in conjunctionwiththelawof cosinesandlaw ofsinesto find the resultant.9

Sample Problem The two forces act on a bolt at A. Determine their resultant. SOLUTION: • Graphical solution - construct a parallelogram with sides in the same direction as P and Q and lengths in proportion. Graphically evaluate the resultant which is equivalent in direction and proportional in magnitude to the the diagonal. • Trigonometric solution - use the triangle rule for vector addition in conjunction with the law of cosines and law of sines to find the resultant. 9

R. Graphical solution - A parallelogram with sidesequal to P and Q is drawn to scale. Themagnitudeanddirectionoftheresultantorofthe diagonal to the parallelogram are measured.OR=98Nα=35°P.Graphical solution -A triangle is drawn with Pand Qhead-to-tail and to scale.ThemagnitudeRanddirectionoftheresultantorofthethirdsideofthetrianglearemeasuredR=98Nα=35°P10

• Graphical solution - A parallelogram with sides equal to P and Q is drawn to scale. The magnitude and direction of the resultant or of the diagonal to the parallelogram are measured, R  98 N   35 • Graphical solution - A triangle is drawn with P and Q head-to-tail and to scale. The magnitude and direction of the resultant or of the third side of the triangle are measured, R  98 N   35 10