1

Engineering Mechanics: Statics

Chapter 2: Force SystemsPart A: Two Dimensional Force Systems

Force

� Force = 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

2

2D Force Systems

� Rectangular components are convenient for finding the sum or resultant of two (or more) forces which are concurrent

Rv

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

= + = + + +

+ + +

v v v

Actual problems do not come with reference axes. Choose the most convenient one!

Moment

� 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 momentM 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

3

Moment

� 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!

Moment

� 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 α = Fd

4

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 2/50

(a) Calculate the moment of the 90-N force about point O for the condition θ = 15º.

(b) Determine the value of θ for which the

moment about O is (b.1) zero (b.2) a maximum

5

Couple

� 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

Couple

� Equivalent couples

� Change of values F and d

� Force in different directions but parallel plane

� Product Fd remains the same

6

Force-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 2/76 (modified)

The device shown is a part of an automobile seat-back-release mechanism.

The part is subjected to the 4-N force exerted at A and a 300-N-mm restoring moment exerted by a hidden torsional spring.

Find an equivalent force-couple system at point O of the 4-N force

7

Resultants

� 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

Rθ

= + + + = Σ

= Σ = Σ = Σ + Σ

v v v v v

K

Resultants

� 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 coupleMO

� 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

= Σ

= Σ = Σ

v v

8

Problem 2/87

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.

Problem 2/76

The device shown is a part of an automobile seat-back-release mechanism.

The part is subjected to the 4-N force exerted at A and a 300-N-mm restoring moment exerted by a hidden torsional spring.

Determine the y-intercept of the line of action of the single equivalent force.

9

Force Systems

Part B: Three Dimensional Force Systems

� Rectangular components in 3D

•Express in terms of unit vectors , ,

•cosθx, cosθy , cosθz are the direction cosines

•cosθx = l, cosθy = m, cosθ z= n

Three-Dimensional Force System

ˆ ˆ ˆ x y zF F i F j F k= + +v

2 2 2

x y zF F F F= + +

i j k

cos , cos , cosx x y y z zF F F F F Fθ θ θ= = =

ˆ ˆ ˆ ( )F F li mj nk= + +v

10

� Rectangular components in 3D

• If the coordinates of points A and B on the line of action are known,

• If two angles θ and φ which orient the line of action of the force are known,

Three-Dimensional Force System

2 1 2 1 2 1

2 2 2

2 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

− + − + −= = =

− + − + −

vv v

cos , sin

cos cos , cos sin

xy z

x y

F F F F

F F F F

φ φ

φ θ φ θ

= =

= =

Problem 2/98

� The cable exerts a tension of 2 kN on the fixed bracket at A.

Write the vector expression for the tension T.

11

� Dot product

� Orthogonal projection of Fcosα of F in the direction of Q

� Orthogonal projection of Qcosα of Q in the direction of F

� We can express Fx = Fcosθx of the force F as Fx =

� If the projection of F in the n-direction is

Three-Dimensional Force System

cosP Q PQ α⋅ =vv

F i⋅vv

F n⋅v v

Example

� Find the projection of T along the line OA

12

� 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 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 vectorn

Moment and Couple

( )M r F n nλ = × ⋅v vv v v

x y z

x y z

r r r

M M F F F

l m n

λ λ= =

v

13

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

� A force system can be reduced to a resultant force and a resultant couple

Resultants

1 2 3

1 2 3 ( )

R F F F F

M M M M r F

= + + = ∑

= + + + = ∑ ×

v v v v v

L

v v v v vvL

14

� The motor mounted on the bracket is acted on by its 160-N weight, and its shaft resists the 120-N thrust and 25-N.m couple applied to it. Determine the resultant of the force system shown in terms of a force R at A and a couple M.

Problem 2/154

� Any general force systems can be represented by a wrench

Wrench Resultants

15

� 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 2/143

� 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

Resultants

16

� 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 2/151