ME 141. Engineering Mechanics

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of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.

1 ME 141 Engineering Mechanics Lecture : Statics of particles Ahma Shahei Shakil Lecturer, Dept. of Mechanical Engg, BUET sshakil@me.buet.ac.b, shakil6791@gmail.com Website: teacher.buet.ac.b/sshakil Courtes: Vector Mechanics for Engineers, Beer an Johnston

2 esultant of Two orces force: action of one bo on another; characterize b its point of application, magnitue, line of action, an sense. Eperimental evience shows that the combine effect of two forces ma be represente b a single resultant force. The resultant is equivalent to the iagonal of a parallelogram which contains the two forces in ajacent legs. orce is a vector quantit.

3 Vectors Vector: parameters possessing magnitue an irection which a accoring to the parallelogram law. Eamples: isplacements, velocities, accelerations. Scalar: parameters possessing magnitue but not irection. Eamples: mass, volume, temperature Vector classifications: - ie or boun vectors have well efine points of application that cannot be change without affecting an analsis. - ree vectors ma be freel move in space without changing their effect on an analsis. - Sliing vectors ma be applie anwhere along their line of action without affecting an analsis. Equal vectors have the same magnitue an irection. Negative vector of a given vector has the same magnitue an the opposite irection. 3

4 Aition of Vectors Trapezoi rule for vector aition Triangle rule for vector aition B C C Law of cosines, P Q P Q PQ cos B B Law of sines, sin A sin B Q sin C A Vector aition is commutative, P Q Q P 4

5 Aition of Vectors The polgon rule for the aition of three or more vectors. Vector subtraction 5

6 esultant an components of forces Concurrent forces: set of forces which all pass through the same point. A set of concurrent forces applie to a particle ma be replace b a single resultant force which is the vector sum of the applie forces. Vector force components: two or more force vectors which, together, have the same effect as a single force vector. 6

7 Prob#.1 (Beer) Graphical solution - A parallelogram with sies equal to P an Q is rawn to scale. The magnitue an irection of the resultant or of the iagonal to the parallelogram are measure, The two forces act on a bolt at A. Determine their resultant. 98 N 35 7

8 Prob#.1 (Beer) Graphical solution - A triangle is rawn with P an Q hea-to-tail an to scale. The magnitue an irection of the resultant or of the thir sie of the triangle are measure, 98 N 35 rom the Law of Sines, sin A Q sin B A A Trigonometric solution - Appl the triangle rule. rom the Law of Cosines, or,sin A sin Q B P Q PQ cos B 40N 60N 40N60Ncos N sin N 97.73N

9 Prob #. (Beer) A barge is pulle b two tugboats. If the resultant of the forces eerte b the tugboats is 5000 lbf irecte along the ais of the barge, etermine Graphical solution - Parallelogram ule with known resultant irection an magnitue, known irections for sies. a) the tension in each of the ropes for = 45 o T 3700 lbf T 1 600lbf 9

10 Prob #. (Beer) Trigonometric solution - Triangle ule with Law of Sines T 1 T 5000 lbf sin 45 sin 30 sin105 T 3660 lbf T 1 590lbf 10

11 ectangular components of a force Ma resolve a force vector into perpenicular components so that the resulting parallelogram is a rectangle. an are referre to as rectangular vector components an Define perpenicular unit vectors i an j which are parallel to the an aes. Vector components ma be epresse as proucts of the unit vectors with the scalar magnitues of the vector components. i an are referre to as the scalar components of j 11

Aition of forces b summing components 1 S Q P Wish to fin the resultant of 3 or more concurrent forces, j S Q P i S Q P j S i S j Q i Q j P i P j i esolve each force into rectangular

12 Aition of forces b summing components 1 S Q P Wish to fin the resultant of 3 or more concurrent forces, j S Q P i S Q P j S i S j Q i Q j P i P j i esolve each force into rectangular components S Q P The scalar components of the resultant are equal to the sum of the corresponing scalar components of the given forces. S Q P 1 tan To fin the resultant magnitue an irection,

13 Problem.3 (Beer) our forces act on bolt A as shown. Determine the resultant of the force on the bolt. 13

Solution esolve each force into rectangular components. force 1 3 4 mag 150 80 110 100 comp 19.9 7.4 0 96.6 comp 75.0 75. 110.0 5.9 199.1 14.

14 Solution esolve each force into rectangular components. force mag comp comp Determine the components of the resultant b aing the corresponing force components. Calculate the magnitue an irection N tan 199.1N 199.6N

15 Equilibrium of a particle When the resultant of all forces acting on a particle is zero, the particle is in equilibrium. Newton s irst Law: If the resultant force on a particle is zero, the particle will remain at rest or will continue at constant spee in a straight line. Particle acte upon b two forces: - equal magnitue - same line of action - opposite sense Particle acte upon b three or more forces: - graphical solution iels a close polgon - algebraic solution

16 ree-bo Diagrams Space Diagram: A sketch showing the phsical conitions of the problem. ree-bo Diagram: A sketch showing onl the forces on the selecte particle. 16

17 Problem.6 It is esire to etermine the rag force at a given spee on a prototpe sailboat hull. A moel is place in a test channel an three cables are use to align its bow on the channel centerline. or a given spee, the tension is 40 lb in cable AB an 60 lb in cable AE. Determine the rag force eerte on the hull an the tension in cable AC. 17

18 Solution rom the free bo iagram of the hull 7 ft tan ft ft tan 4 ft Now, conition for equilibrium, T T T 0 AB AC AE D 18

19 Solution esolving the vector equilibrium equation into two component equations, T AB 40 lbsin 60.6 i 40 lb lbi lb j TAC TAC sin 0.56i TAC cos 0.56 j 0.351TAC i TAC j T 60 lbi i D D cos 60.6 j T AC T AC D i 60 j 19

20 Solution T T AC AC D i 60 j This equation is satisfie onl if each component of the resultant is equal to zero T T 60 AC AC D T AC D 4.9 lb lb 0

21 ectangular Components in Space The vector is esolve into containe in the plane OBAC. horizontal an vertical components. h cos sin h esolve into rectangular components h cos sin h sin cos sin sin 1

22 ectangular Components in Space With the angles between an the aes, cos cos z cos z i j zk cos i cos j cos zk cos i cos j cos zk is a unit vector along the line of action of an cos, cos are the irection cosines for, an cos z

23 3 ectangular Components in Space Direction of the force is efine b the location of two points, 1 1 1,, an,, z N z M k j i z z k j i N M z z z z z 1,, an vector joining 1 1 1

24 Prob#.111 (Beer) A transmission tower is hel b three gu wires attache to a pin at A an anchore b bolts at B, C, an D. If the tension in wire AB is 630 lb, etermine the vertical force P eerte b the tower on the pin at A. 4

STATICS. Statics of Particles VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.

STATICS. Statics of Particles VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr. Eighth E CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. Statics of Particles Lecture Notes: J. Walt Oler Teas Tech Universit Contents Introduction Resultant