CHAPTER ENT 151 STATICS Lecture Notes: Azizul bin Mohamad KUKUM Statics of Particles Contents Introduction Resultant of Two Forces Vectors Addition of Vectors Resultant of Several Concurrent Forces Sample Problem.1 Rectangular Components of a Force: Unit Vectors Addition 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 - Introduction Resultant of Two Forces The obective 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. 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 adacent legs. Force is a vector quantit. - 3-4
Vectors Vector: parameter possessing magnitude and direction which add according to the parallelogram law. Eamples: displacements, velocities, accelerations. Scalar: parameter 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 Addition of Vectors C B C B Trapezoid rule for vector addition Triangle rule for vector addition Law of cosines, R P + Q PQ cos B R P + Q Law of sines, sin A sin B sin C Q R A Vector addition is commutative, P + Q Q + P Vector subtraction - 6 Addition of Vectors Addition of three or more vectors through repeated application of the triangle rule The polgon rule for the addition of three or more vectors. Vector addition is associative, P + Q + S P + Q + S P + Q + S ( ) ( ) 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. Multiplication of a vector b a scalar - 7-8
Sample Problem.1 Sample Problem.1 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 addition in conunction with the law of cosines and law of sines to find the resultant. 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 α 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 α 35-9 - 10 Sample Problem.1 Trigonometric solution - Appl the triangle rule. From the Law of Cosines, R P + Q PQ cos B ( 40N) + ( 60N) ( 40N)( 60N) cos155 R 97.73N From the Law of Sines, sin A sin B Q R sin A sin B Q R sin155 A 15.04 α 0 + A α 35. 04 60N 97.73N Sample Problem. A barge is pulled b two tugboats. If the resultant of the forces eerted b the tugboats is 5000 lbf directed along the ais of the barge, determine a) the tension in each of the ropes for α 45 o, b) the value of α for which the tension in rope is a minimum. Find a graphical solution b appling the Parallelogram Rule for vector addition. 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 lbf. Find a trigonometric solution b appling the Triangle Rule for vector addition. With the magnitude and direction of the resultant known and the directions of the other two sides parallel to the ropes, 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 α. - 11-1
Sample Problem. Sample Problem. The angle for minimum tension in rope is determined b appling the Triangle Rule and observing the effect of variations in α. Graphical solution - Parallelogram Rule with known resultant direction and magnitude, known directions for sides. T 18.5kN T 13.0kN 1 The minimum tension in rope occurs when T 1 and T are perpendicular. ( ) T 5kN sin 30 T 1.5kN Trigonometric solution - Triangle Rule with Law of Sines T1 T 5kN sin 45 sin 30 sin105 ( ) T1 5kN cos 30 T1 1.65kN α 90 30 α 60 T1 18.30kN T 1.94kN - 13-14 Rectangular Components of a Force: Unit Vectors Vector components ma be epressed as products of the unit vectors with the scalar magnitudes of the vector components. F Fi + F F and F are referred to as the scalar components of F 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 which are parallel to the and aes. - 15 Addition of Forces b Summing Components Wish to find the resultant of 3 or more concurrent forces, R P + Q + S Resolve each force into rectangular components Ri + R P i + P + Qi + Q + Si + S ( P + Q + S ) i + ( P + Q + S ) The scalar components of the resultant are equal to the sum of the corresponding scalar components of the given forces. R P + Q + S R P + Q + S F F To find the resultant magnitude and direction, 1 R R R + R θ tan R - 16
Sample Problem.3 Four forces act on bolt A as shown. Determine the resultant of the force on the bolt. Resolve each force into rectangular components. Determine the components of the resultant b adding the corresponding force components. Calculate the magnitude and direction of the resultant. - 17 Sample Problem.3 Resolve each force into rectangular components. force F1 F F3 F 4 mag 150 80 110 100 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 199.1 + 14.3 R 199.6N 14.3N tan α α 4. 1 199.1N - 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. Free-Bod Diagrams 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 Space Diagram: A sketch showing the phsical conditions of the problem. Free-Bod Diagram: A sketch showing onl the forces on the selected particle. - 0
Sample Problem.4 Sample Problem.4 In a ship-unloading operation, a 3500-lb 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? Construct a free-bod diagram for the particle at the unction of the rope and cable. Appl the conditions for equilibrium b creating a closed polgon from the forces applied to the particle. Appl trigonometric relations to determine the unknown force magnitudes. Construct a free-bod diagram for the particle at A. Appl the conditions for equilibrium. Solve for the unknown force magnitudes. TAB TAB T AC 15.6kN sin10 sin sin 58 TAC 15.9kN 0.64kN - 1 - Sample Problem.6 Sample Problem.6 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 lb in cable AB and 60 lb 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 condition 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. Choosing the hull as the free bod, draw a free-bod diagram..13m tanα 1.75 1.m α 60.6 0.46 m tan β 0.375 1. m β 0.56 Epress the condition 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 - 3-4
Sample Problem.6 Resolve the vector equilibrium equation into two component equations. Solve for the two unknown cable tensions. TAB ( 180N) sin 60.6 i + ( 180N) cos 60.6 ( 156.3N) i + ( 89.3N) TAC TAC sin 0.56 i + TAC cos 0.56 0.351 TAC i + 0.9363TAC T ( 70N) i F F i D D Sample Problem.6 R 0 ( 34.73 + 0.351TAC + FD ) i + ( 19.84 + 0.9363T 60) AC This equation is satisfied onl if each component of the resultant is equal to zero ( ) ( ) F 0 0 34.73+ 0.351 T + F AC D F 0 0 19.84 + 0.9363T 70N AC R 0 + + 19.84 0.9363T 60 ( 34.73 0.351 TAC FD ) i + ( + ) AC TAC + 193N F + 88.5N D - 5-6