Statics of Particles

Contents Introduction Resultant of Two Forces Vectors Addition of Vectors Resultant of Several Concurrent Forces Sample Problem 2.1 Rectangular Components of a Force: Unit Vectors Addition of Forces by Summing Components Sample Problem 2.3 Equilibrium of a Particle Free-Body Diagrams Sample Problem 2.4 Sample Problem 2.6 Rectangular Components in Space

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.

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.

Vectors Vector: parameter possessing magnitude and direction which add according to the parallelogram law. Examples: displacements, velocities, accelerations. Scalar: parameter possessing magnitude but not direction. Examples: mass, volume, temperature 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.

Addition of Vectors Trapezoid rule for vector addition Triangle rule for vector addition B C Law of cosines, Law of sines, Vector addition is commutative, Vector subtraction

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, Multiplication of a vector by a scalar

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.

Problem 2.10 (also see Sample Problems 2.1-2.2 in the text) To steady a sign as it is being lowered, two cables are attached to the sign at A. Using trigonometry and knowing that the magnitude of P is 300N, determine: The required angle  if the resultant R of the two forces applied at A is to be vertical. The corresponding value of R.

Problem 2.19 Two structural members A and B are bolted to a bracket as shown. Knowing that both members are in compression and that the force is 30 kN in member A and 20 kN in member B, determine, using trigonometry, the magnitude and direction of the resultant of the forces applied to the bracket by members A and B.

Problem 2.8 The 50-lb force is to be resolved into components along lines a-a’ and b-b’. Using trigonometry, determine the angle  knowing that the component along b-b’ is 30 lb. What is the corresponding value of the component along a-a’?

Rectangular Components of a Force: Unit Vectors 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. 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

Addition of Forces by Summing Components Wish to find the resultant of 3 or more concurrent forces, Resolve each force into rectangular components The scalar components of the resultant are equal to the sum of the corresponding scalar components of the given forces. To find the resultant magnitude and direction,

Example Example: Express the force shown below using unit vectors.

Problem 2.33 (also see Sample Problem 2.3 in the text) Determine the resultant of the three forces of Prob. 2.22.

Example – using calculators Using calculators to find resultants Solutions to the problems shown above can be produced quickly using calculators that can perform operations using complex numbers (or numbers in polar and rectangular form). Polar numbers - can be used to represent forces in terms of their magnitude and angle Rectangular numbers - can be used to represent forces in terms of unit vectors Handout: See the handout entitled “Complex Numbers” which contains examples of representing forces in polar and rectangular form on various calculators. Example: Repeat the last example using the TI-85/86 or TI-89/92 calculator (Determine the resultant of the 3 forces on the hook below.)

Problem 2.34 Determine the resultant of the three forces of Prob. 2.23. Repeat the example above using the unitV[dx,dy] function on a calculator.

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 by three or more forces: graphical solution yields a closed polygon algebraic solution Particle acted upon by two forces: equal magnitude same line of action opposite sense

Free-Body Diagrams Free-Body Diagram: A sketch showing only the forces on the selected particle. Space Diagram: A sketch showing the physical conditions of the problem.

Problem 2.44 (also see Sample Problems 2.4-2.6 in the text) Knowing that  = 55, determine the tension in bar AC and in rope BC.

Problem 2.51 Two forces P and Q are applied as shown to an aircraft connection. Knowing that the connection is in equilibrium and that P = 400 lb and Q = 520 lb, determine the magnitudes of the forces exerted on the rods A and B.

Pulleys Pulleys Ideal pulleys simply change the direction of a force. The tension on each side of an ideal pulley is the same. The tension is the same everywhere in a given rope or cable if ideal pulleys are used. In a later chapter non-ideal pulleys are introduced (belt friction and bearing friction). 50 lb Vertical force Horizontal Example Determine the tension T required to support the 100 lb block shown below.

Pulleys Example: (Problem 6-68 in Statics, 9th Ed. by Hibbeler) Determine the force P needed to support the 100-lb weight. Each pulley has a weight of 10 lb. Also, what are the cord reactions at A and B?

Problem 2.70 A 350-lb load is supported by the rope-and-pulley arrangement shown. Knowing that  = 35, determine: The angle  The magnitude of the force P which should be exerted on the free end of the rope to maintain equilibrium. (Hint: The tension is the same on either side of an ideal pulley).

Rectangular Components in Space The vector is contained in the plane OBAC. Resolve into horizontal and vertical components. Resolve into rectangular components

Rectangular Components in Space With the angles between and the axes, is a unit vector along the line of action of and are the direction cosines for

Rectangular Components in Space Magnitude of a vector using x, y, and z coordinates: Show that (Equation 2.18) Also show that (Equation 2.20)

Rectangular Components in Space Direction of the force is defined by the location of two points,

Example Example: If F = 300i + 400j + 1200k lb: Find the unit vector  along the line of action of F Find the magnitude of F Express F in terms of |F| and  Find the angles that between F and the x, y, and z axes

Determining Resultants in Space Determining resultants using x, y, and z rectangular components Procedure: Express each force using unit vectors Add all x components for the total (resultant) x component, i.e., Rx = Fx Add all y components for the total (resultant) y component, i.e., Rx = Fy Add all z components for the total (resultant) z component, i.e., Rz = Fz Express the final result as:

Problem 2.93 Determine the magnitude and direction of the resultant of the two forces shown knowing that P = 500 lb and Q = 600 lb.

Equilibrium in Space Fx = 0 Fy = 0 Fz = 0 Equilibrium of a particle in space If an object in is equilibrium and if the problem is represented in three dimensions, then the relationship F = 0 can be expressed as: Fx = 0 Fy = 0 Fz = 0

Problem 2.103 Three cables are used to tether a balloon as shown. Determine the vertical force P exerted by the balloon at A knowing that the tension in cable AB is 60 lb.