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Engineering Mechanics: Engineering Mechanics: Statics Statics
Chapter 2: Force Systems
ForceForce SystemsSystems
Part A: Two Dimensional Force Systems
ForceForce An action of one body on another Vector quantity
External and Internal forces
Mechanics of Rigid bodies: Principle of Transmissibility• Specify magnitude, direction, line of action• No need to specify point of application
Concurrent forces• Lines of action intersect at a point
Vector Components Vector Components A vector can be resolved into several vector components
Vector sum of the components must equal the original vector
Do not confused vector components with perpendicular projections
2D force systems•Most common 2D resolution of a force vector
•Express in terms of unit vectors ,
Rectangular ComponentsRectangular Components
F
x
y
i
xF
yF
j i j
ˆ ˆ
cos , sin x y x y
x y
F F F F i F j
F F F F
2 2x yF F F F
1tan y
x
F
F
Scalar components – can be positive and negative
2D Force Systems2D Force Systems Rectangular components are convenient for finding
the sum or resultant of two (or more) forces which are concurrent
R
1 2 1 1 2 2
1 2 1 2
ˆ ˆ ˆ ˆ ( ) ( )
ˆ ˆ = ( ) ( )
x y x y
x x y y
R F F F i F j F i F j
F F i F F j
Actual problems do not come with reference axes. Choose the most convenient
one!
Example 2.1Example 2.1
The link is subjected to two forces F1 and F2. Determine the magnitude and direction of the resultant force.
2 2236.8 582.8
629 N
RF N N
1 582.8tan236.8
67.9
NN
Solution
Example 2/1 (p. 29) Example 2/1 (p. 29)
Determine the x and y scalar components of each of the three forces
Unit vectors
• = Unit vector in direction of
cos direction cosinex
x
V
V
Rectangular componentsRectangular components
V
n
x
y
i
xV
yV
j
ˆ ˆˆ ˆ
ˆ ˆ cos cos
x y yx
x y
V i V j VVVn i j
V V V V
i j
n
V
x
y
2 2cos cos 1x y
The line of action of the 34-kN force runs through the points A and B as shown in the figure.
(a) Determine the x and y scalar component of F.
(b) Write F in vector form.
Problem 2/4Problem 2/4
MomentMoment In addition to tendency to move a body
in the direction of its application, a force tends to rotate a body about an axis.
The axis is any line which neither intersects nor is parallel to the line of action
This rotational tendency is known as the moment M of the force Proportional to force F and the
perpendicular distance from the axis to the line of action of the force d
The magnitude of M is M = Fd
MomentMoment The moment is a vector M perpendicular
to the plane of the body. Sense of M is determined by the right-
hand rule Direction of the thumb = arrowhead Fingers curled in the direction of the
rotational tendency
In a given plane (2D),we may speak of moment about a point which means moment with respect to an axis normal to the plane and passing through the point.
+, - signs are used for moment directions – must be consistent throughout the problem!
MomentMoment A vector approach for moment
calculations is proper for 3D problems. Moment of F about point A maybe
represented by the cross-product
where r = a position vector from point A to any point on the line of action of F
M = r x F
M = Fr sin = Fd
Example 2/5 (p. 40)Example 2/5 (p. 40)
Calculate the magnitude of the moment about the base point O of the 600-N force by using both scalar and vector approaches.
Problem 2/43 Problem 2/43
(a) Calculate the moment of the 90-N force about point O for the condition = 15º. (b) Determine the value of for which the moment about O is (b.1) zero (b.2) a maximum
CoupleCouple Moment produced by two equal, opposite,
and noncollinear forces = couple
Moment of a couple has the same value for all moment center
Vector approach
Couple M is a free vector
M = F(a+d) – Fa = Fd
M = rA x F + rB x (-F) = (rA - rB) x F = r x F
CoupleCouple Equivalent couples
Change of values F and d Force in different directions but parallel plane Product Fd remains the same
Force-Couple SystemsForce-Couple Systems Replacement of a force by a force and a couple Force F is replaced by a parallel force F and a
counterclockwise couple Fd
Example Replace the force by an equivalent system at point O
Also, reverse the problem by the replacement of a force and a couple by a single force
Problem 2/67Problem 2/67
The wrench is subjected to the 200-N force and the force P as shown. If the equivalent of the two forces is a for R at O and a couple expressed as the vector M = 20 kN.m, determine the vector expressions for P and R
ResultantsResultants The simplest force combination which can
replace the original forces without changing the external effect on the rigid body
Resultant = a force-couple system
1 2 3
2 2
-1
, , ( ) ( )
= tan
x x y y x y
y
x
R F F F F
R F R F R F F
R
R
ResultantsResultants Choose a reference point (point O) and
move all forces to that point Add all forces at O to form the resultant
force R and add all moment to form the resultant couple MO
Find the line of action of R by requiring R to have a moment of MO
( )
= O
O
R F
M M Fd
Rd M
Problem 2/79Problem 2/79
Replace the three forces acting on the bent pipe by a single equivalent force R. Specify the distance x from point O to the point on the x-axis through which the line of action of R passes.
ForceForce SystemsSystems
Part B: Three Dimensional Force Systems
Rectangular components in 3D
•Express in terms of unit vectors , ,
• cosx, cosy , cosz are the direction cosines
• cosx = l, cosy = m, cos z= n
Three-Dimensional Force Three-Dimensional Force SystemSystem
ˆ ˆ ˆ x y zF F i F j Fk
2 2 2x y zF F F F
i j k
cos , cos , cosx x y y z zF F F F F F
ˆ ˆ ˆ ( )F F li mj nk
Rectangular components in 3D
• If the coordinates of points A and B on the line of action are known,
• If two angles and which orient the line of action of the force are known,
Three-Dimensional Force Three-Dimensional Force SystemSystem
2 1 2 1 2 1
2 2 22 1 2 1 2 1
ˆ ˆ ˆ( ) ( ) ( )
( ) ( ) ( )F
x x i y y j z z kABF Fn F F
AB x x y y z z
cos , sin
cos cos , cos sinxy z
x y
F F F F
F F F F
Problem 2/98Problem 2/98 The cable exerts a tension of 2 kN on the fixed bracket at
A. Write the vector expression for the tension T.
Dot product
Orthogonal projection of Fcos of F in the direction of Q Orthogonal projection of Qcos of Q in the direction of F
We can express Fx = Fcosx of the force F as Fx =
If the projection of F in the n-direction is
Three-Dimensional Force Three-Dimensional Force SystemSystem
cosP Q PQ
F i
F n
ExampleExample Find the projection of T along the line OA
Moment of force F about the axis through point O is
r runs from O to any point on the line of action of F Point O and force F establish a plane A The vector Mo is normal to the plane in the direction
established by the right-hand rule
Evaluating the cross product
Moment and CoupleMoment and Couple
MO = r x F
ˆ ˆ ˆ
O x y z
x y z
i j k
M r r r
F F F
Moment about an arbitrary axis
known as triple scalar product (see appendix C/7)
The triple scalar product may be represented by the determinant
where l, m, n are the direction cosines of the unit vector n
Moment and CoupleMoment and Couple
( )M r F n n
x y z
x y z
r r r
M M F F F
l m n
A tension T of magniture 10 kN is applied to the cable attached to the top A of the rigid mast and secured to the ground at B. Determine the moment Mz of T about the z-axis passing through the base O.
Sample Problem 2/10 Sample Problem 2/10
A force system can be reduced to a resultant force and a resultant couple
ResultantsResultants
1 2 3
1 2 3 ( )
R F F F F
M M M M r F
Any general force systems can be represented by a wrench
Wrench ResultantsWrench Resultants
Replace the two forces and single couple by an equivalent force-couple system at point A
Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts
Problem 2/143Problem 2/143
Special cases• Concurrent forces – no moments about point of
concurrency• Coplanar forces – 2D• Parallel forces (not in the same plane) – magnitude of
resultant = algebraic sum of the forces• Wrench resultant – resultant couple M is parallel to the
resultant force R• Example of positive wrench = screw driver
ResultantsResultants
Replace the resultant of the force system acting on the pipe assembly by a single force R at A and a couple M
Determine the wrench resultant and the coordinate in the xy plane through which the resultant force of the wrench acts
Problem 2/142Problem 2/142