Chapter 2 Resultant of Coplannar Force Systems Statics Chapter 2 Resultant of Coplannar Force Systems

2-1 Intro Two systems of forces = equivalent – if produce the same mechanical effect on rigid body Single force that is equivalent to a force system is called the resultant of the force system

2-2 Vector Representation Figure 2-1 represented graphically by a line segment AB with an arrowhead at one end – length of the line segment AB represents the magnitude of the force - direction is indicated by the angle from the reference axis. Equal Vectors – two vectors having same magnitude and same direction are said to be equal – figure 2-2 Negative Vector – two vectors having the same magnitude and opposite direction.

2-3 Resultant of Concurrent Forces Parallelogram Law – Figure 2-4 shows two vectors are added according to this law Triangle Rule – the sum of two vectors can also be determined by constructing one half of the parallelogram or triangle To find the vector sum , we first lay out p at A then lay out Q from the tip of P in a tip to tail fashion, the closing side of the triangle represents the sum of the two vectors Examples 2-1 Example 2-2 Example 2-3

2-4 Rectangular Components Any two or more forces whose resultant is equal to a force F are call the components of the force Two mutually perpendicular components are called the rectangular components Fx and Fy are the rectangular components of F in the x and y direction If the magnitude F and the direction angle of a force are known then from the right triangle Fx=Fcos @ and Fy = Fsin@ Counterclockwise measurement is regarded as positive, clockwise measurement as negative

2-4 Rectangular Components Magnitude and Direction – when the scalar components Fx and Fy of the force F are give , the magnitude of F may be determined from formula 2-3 page 47 And the reference angle from 2-4 page 47 Depending on which quadrant the force vector is in the direction angle in the standard position is First quadrant 0=@ Second quadrant 0= 180 -@ Third quadrant 0=180 + @ Fourth quadrant 0=360-@ Example 2-4 page 47 Example 2-5 page 48 Example 2-6 page 49 Example 2-7 page 50

2-5 Resultants by Rectangular Components The resultant of any number of concurrent coplanar forces can be determined by using their rectangular components Figure 2-9 R=F1 +F2+F3 Rx = F1x +F2x +f3x Rx = F1cos @1 + F2cos @2 + F3cos @3 Ry = F1y +F2y +f3y Rx = F1sin @1 + F2sin @2 + F3sin @3 Example 2-8 page 52 Example 2-9 page 53

2-6 Moment of Force Effects of a force – force tends to move a body along its line of action – it also tends to rotate a body about an axis The ability of a force to cause a body to rotate is measured by a quantity called the moment of the force. Wrench example – rotating moment also called the torque is produced by the applied force depends not only on the magnitude of the force but als on the perpendicular distance d from the center 0 of the bolt to the lne of action of the force Definition of Moment – two dimensional case – moment of a force about a point ie equal to the magnitude of the force multiplied by the perpendicular distance from 0 to the line of action of the force Formula 2-7 page 55 Units of moment are Ib* ft or Ib * in or N* m or kN * m

2-6 Moment of a Force Direction of Moments – example 2-13 P is CCW and Q is CW Summation of Moments – CCW will be considered positive CW will be considered negative. Example 2-10 page 56

2-7 Varignon’s Theorem States that the moment of a force about any point is equal to the sum of the moments produced by the components of the forces abut the same point The sum of the moments of the components must be the same as the moment of the force itself Formula 2-8 page 59 Principle of transmissibility The point of application of a force acting on a rigid body may be place anywhere along its line of action – moment arm is clearly independent of the point of application - therefore as long as the magnitude the direction and the line of actin of a force are defined the moment of a force about a given point may b determined by placing the force at any point along its line of actin. Example 2-11 Example 2-12

2-8 Couple Effect of a couple – two equal and opposite forces having parallel lines of actin form a couple – the sum of the moment of the two forces however is not zero – the effect of a couple acting on a rigid body therefore is to cause the rigid body to rotate about an axis perpendicular to the plane of the forces Moment of a couple – denoting the perpendicular distance between the two forces by d , the moment of a couple about an arbitrary point 0 is equal to formula 2-9 page 63 Since 0 is an arbitrary point – the moment of a couple about any point is equal to the magnitude of the forces times the perpendicular distance between the forces. Equivalent couples – two couples acting on the same plane or parallel planes are equivalent if they have the same moment acting in the same direction Addition of Couples – addition of two or more couples in a plane or parallel planes is the algebraic sum of their moments Example 2-13 page 63 Example 2-14 page 64

2-9 Replacing a Force with a Force couple system Two systems of forces are said to be equivalent if they produce the same mechanical effect on a rigid body – Equivalent force systems – are said to be equivalent if they have the same resultant force and the same resultant moment about the same point Force couple system – move a force to another point using the principle of transmissibility – we may add two equal and opposite forces without altering the mechanical effect of the original force – figure 2-17 page 66

2-10 Resultant of a nonconcurrent coplanar force system In this system there are no point of concurrency – so the locating of the line of action of the resultant is not immediately known Choose convenient x and y coordinate axes and then resolve each force into rectangular components – formula 2-11 Example 2-16 page 68 Example 2-17 page 69 Example 2-18 page 70

2-11 Resultant of Distributed Line Loads Distributed Load – occurs whenever the load applied to a body is not concentrated at a point – could be exerted along a line , over a area, or throughout an entire solid body Load intensity – distributed load along a line is characterized by a load intensity expressed as force per unit length. Examples 1000 Ib/ft , 1 kip/ft, 1 N/m or 1kn/m Uniform load – distributed load with a constant intensity is called uniform load – represented with a load diagram – shape like rectangular block Triangular load – distributed load whose intensity varies linearly from zero to a maximum intensity – represented as triangle Equivalent concentrated force – to determine the resultant of a force system – each distributed load may be replaced by its equivalent concentrated force Example 2-19 page 74 Example 2-20 page 75 Example 2-21 page 76