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 of Two Forces Vectors Ad of Vectors Resultant of Several Concurrent Forces Sample Problem.1 Sample Problem. Rectangular Components of a Force: Unit Vectors Ad of Forces b Summing Components Sample Problem.3 Equilibrium of a Particle Free-Bod Diagrams Sample Problem.4 Sample Problem.6 Rectangular Components in Space Sample Problem.7 -
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 impl a restriction to miniscule bodies. Rather, the stud is restricted to analses in which the size and shape of the bodies is not significant so that all forces ma be assumed to be applied at a single point. - 3
Resultant of Two Forces force: action of one bod on another; characterized b its point of application, magnitude, line of action, and sense. Eperimental evidence shows that the combined effect of two forces ma be represented b 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 quantit. - 4
Vectors Vector: parameters possessing magnitude and direction which add according to the parallelogram law. Eamples: displacements, velocities, accelerations. Scalar: parameters possessing magnitude but not direction. Eamples: mass, volume, temperature Vector classifications: - Fied or bound vectors have well defined points of application that cannot be changed without affecting an analsis. - Free vectors ma be freel moved in space without changing their effect on an analsis. - Sliding vectors ma be applied anwhere along their line of action without affecting an analsis. Equal vectors have the same magnitude and direction. Negative vector of a given vector has the same magnitude and the opposite direction. - 5
Ad of Vectors Trapezoid rule for vector ad Triangle rule for vector ad B C C Law of cosines, R P Q R P Q PQ cos B B Law of sines, sin A sin B Q R sin C A Vector ad is commutative, P Q Q P Vector subtraction - 6
Ad of Vectors Ad of three or more vectors through repeated application of the triangle rule The polgon rule for the ad of three or more vectors. Vector ad is associative, P Q S P Q S P Q S Multiplication of a vector b a scalar - 7
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 ma be replaced b 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 Problem.1 SOLUTION: The two forces act on a bolt at A. Determine their resultant. Graphical solution - construct a parallelogram with sides in the same direction as P and Q and lengths in proportion. Graphicall evaluate the resultant which is equivalent in direction and proportional in magnitude to the the diagonal. Trigonometric solution - use the triangle rule for vector ad in conjunction with the law of cosines and law of sines to find the resultant. - 9
Sample Problem.1 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
Sample Problem.1 Trigonometric solution - Appl the triangle rule. From the Law of Cosines, R P sin A Q sin A Q PQ cos B 40N 60N 40N60Ncos155 R 97.73N From the Law of Sines, sin B R Q sin B R sin155 A 15.04 0 A 35. 04 60N 97.73N - 11
Sample Problem. A barge is pulled b two tugboats. If the resultant of the forces eerted b the tugboats is 5000 N directed along the ais of the barge, determine a) the tension in each of the ropes for = 45 o, b) the value of for which the tension in rope is a minimum. SOLUTION: Find a graphical solution b appling the Parallelogram Rule for vector ad. The parallelogram has sides in the directions of the two ropes and a diagonal in the direction of the barge ais and length proportional to 5000 N. Find a trigonometric solution b appling the Triangle Rule for vector ad. With the magnitude and direction of the resultant known and the directions of the other two sides parallel to the ropes given, appl the Law of Sines to find the rope tensions. The angle for minimum tension in rope is determined b appling the Triangle Rule and observing the effect of variations in. - 1
Sample Problem. Graphical solution - Parallelogram Rule with known resultant direction and magnitude, known directions for sides. T1 3700 N T 600 N Trigonometric solution - Triangle Rule with Law of Sines T 1 T 5000 N sin 45 sin30 sin105 T1 3660 N T 590 N - 13
Sample Problem. The angle for minimum tension in rope is determined b appling the Triangle Rule and observing the effect of variations in. The minimum tension in rope occurs when T 1 and T are perpendicular. T T1 5000 Nsin 30 5000 Ncos 30 T 500 N T 1 4330 N 90 30 60-14
Rectangular Components of a Force: Unit Vectors Ma resolve a force vector into perpendicular components so that the resulting parallelogram is a rectangle. F and F are referred to as rectangular vector components and F F F Define perpendicular unit vectors i and j which are parallel to the and aes. Vector components ma be epressed as products of the unit vectors with the scalar magnitudes of the vector components. F F i F j F and F are referred to as the scalar components of F - 15
ighth - 16 Ad of Forces b Summing Components S Q P R Wish to find 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 R i R Resolve each force into rectangular components F S Q P R The scalar components of the resultant are equal to the sum of the corresponding scalar components of the given forces. F S Q P R R R R R R 1 tan To find the resultant magnitude and direction,
Sample Problem.3 SOLUTION: Resolve each force into rectangular components. Determine the components of the resultant b adding the corresponding force components. Four forces act on bolt A as shown. Determine the resultant of the force on the bolt. Calculate the magnitude and direction of the resultant. - 17
Sample Problem.3 SOLUTION: Resolve each force into rectangular components. force F1 F F3 F 4 mag 150 80 110 100 R 199.1 14.3 14.3N tan 199.1N comp 19.9 7.4 0 96.6 comp 75.0 75. 110.0 5.9 R 199.1 14. 3 R Determine the components of the resultant b adding the corresponding force components. Calculate the magnitude and direction. R 199.6N 4. 1-18
Equilibrium of a Particle When the resultant of all forces acting on a particle is zero, the particle is in equilibrium. Newton s 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. Particle acted upon b two forces: - equal magnitude - same line of action - opposite sense Particle acted upon b three or more forces: - graphical solution ields a closed polgon - algebraic solution R F 0 F 0 F 0-19
Free-Bod Diagrams Space Diagram: A sketch showing the phsical cons of the problem. Free-Bod Diagram: A sketch showing onl the forces on the selected particle. - 0
Sample Problem.4 SOLUTION: Construct a free-bod diagram for the particle at the junction of the rope and cable. Appl the cons for equilibrium b creating a closed polgon from the forces applied to the particle. In a ship-unloading operation, a 3500-N automobile is supported b a cable. A rope is tied to the cable and pulled to center the automobile over its intended position. What is the tension in the rope? Appl trigonometric relations to determine the unknown force magnitudes. - 1
Sample Problem.4 SOLUTION: Construct a free-bod diagram for the particle at A. Appl the cons for equilibrium. Solve for the unknown force magnitudes. T AB T AC 3500 N sin10 sin sin 58 T AB 3570 N T AC 144 N -
Sample Problem.6 SOLUTION: It is desired to determine the drag force at a given speed on a prototpe sailboat hull. A model is placed in a test channel and three cables are used to align its bow on the channel centerline. For a given speed, the tension is 40 N in cable AB and 60 N in cable AE. Determine the drag force eerted on the hull and the tension in cable AC. Choosing the hull as the free bod, draw a free-bod diagram. Epress the con for equilibrium for the hull b writing that the sum of all forces must be zero. Resolve the vector equilibrium equation into two component equations. Solve for the two unknown cable tensions. - 3
Sample Problem.6 SOLUTION: Choosing the hull as the free bod, draw a free-bod diagram. 7 m tan 1.75 4 m 60.5 1.5 m tan 4 m 0.56 0.375 Epress the con for equilibrium for the hull b writing that the sum of all forces must be zero. R T T T F 0 AB AC AE D - 4
Sample Problem.6 Resolve the vector equilibrium equation into two component equations. Solve for the two unknown cable tensions. TAB 40 Nsin 60.6 i 40 Ncos60.6 34.73 Ni 19.84 Nj TAC TAC sin 0.56i TAC cos 0.56 j 0.351TAC i 0.9363TAC j T 60 Ni F F i D D j R 0 34.73 0.351T 19.84 0.9363T AC AC FD i 60 j - 5
Sample Problem.6 R 0 34.73 0.351T 19.84 0.9363T AC AC FD i 60 j This equation is satisfied onl if each component of the resultant is equal to zero. 0 0 34.73 0.351T F F 0 0 19.84 0.9363T 60 T F AC D F 4.9 N 19.66 N AC AC D - 6
Rectangular Components in Space The vector F is contained in the plane OBAC. Resolve F into horizontal and vertical components. F F h F cos F sin Resolve F h into rectangular components. F Fh cos F sin cos F F h sin F sin sin - 7
Rectangular Components in Space With the angles between F and the aes, F F cos F F cos Fz F cos z F Fi F j Fzk Fcos i cos j cos zk F cos i cos j cos zk is a unit vector along the line of action of F and cos, cos are the direction cosines for F, and cos z - 8
ighth - 9 Rectangular Components in Space Direction of the force is defined b the location of two points, 1 1 1,, and,, z N z M d Fd F d Fd F d Fd F k d j d i d d F F z z d d d k d j d i d N M d z z z z z 1 and vector joining 1 1 1
Sample Problem.7 SOLUTION: Based on the relative locations of the points A and B, determine the unit vector pointing from A towards B. Appl the unit vector to determine the components of the force acting on A. The tension in the gu wire is 500 N. Determine: a) components F, F, F z of the force acting on the bolt at A, b) the angles,, z defining the direction of the force Noting that the components of the unit vector are the direction cosines for the vector, calculate the corresponding angles. - 30
Sample Problem.7 SOLUTION: Determine the unit vector pointing from A towards B. AB 40 m i 80 m j 30m k AB 94.3 m 40 m 80m 30m 40 80 30 i j k 94.3 94.3 94.3 0.44i 0.848 j 0.318k Determine the components of the force. F F 500 N 0.44i 0.848 j 0.318k 1060 N i 10 N j 795 N k - 31
Sample Problem.7 Noting that the components of the unit vector are the direction cosines for the vector, calculate the corresponding angles. cos i cos j cos zk 0.44i 0.848 j 0.318k z 115.1 3.0 71.5-3