Force is a Vector
A Force consists of:
• Magnitude
• Direction
– Line of Action
– Sense
• Point of application
– For a particle, all forces act at the same point
of application
• (contrast with a scalar, which has only a
magnitude)
Forces (Vectors) Add by the
Parallelogram Law
R  PQ
• Can solve:
– Graphically
– Using trigonometry
– Resolve into x and y components
• R
R is the resultant force acting
on particle A
Simple Example
Solve using Trigonometry
Law of sines
Law of cosines
Resolving Vectors into
x and y Components
Unit vectors:
F  Fx  Fy  Fx iˆ  Fy ˆj
Fx  F cos   F sin 
Fy  F sin   F cos 
F F  F F
2
x
tan  
Fy
Fx
2
y
φ

Resultant Force Acting
on a Particle
R  F
R  Rx iˆ  Ry ˆj
Rx   Fx ; Ry   Fy
Magnitude:
Direction:
R R R
2
x
tan  
Ry
Rx
2
y
Example
Rx  F1x  F2 x  0  F4 x
Ry  F1 y  F2 y  F3 y  F4 y
Forces in 3D
F  Fx  Fy  Fz  Fxiˆ  Fy ˆj  Fz kˆ
Fx  F cos x ; Fy  F cos y ; Fz  F cos z
Magnitude &
Direction in 3D
F  F ˆ
2
2
2
F

F

F

F

F
Magnitude:
x
y
z
Fy
Fx
Fz
; cos y  ; cos z 
Direction: cos  x 
F
F
F
ˆ  xiˆ   y ˆj  z kˆ  cos xiˆ  cos y ˆj  cos z kˆ
Determining Forces from Unit Vectors
N :  xN , y N , z N 
FMN  FMN ˆMN
ˆ
MN

MN
dy
MN
dz
M :  xM , yM , zM 
d
x
MN   xN  xM  iˆ   y N  yM  ˆj   z N  z M  kˆ
MN  d xiˆ  d y ˆj  d z kˆ
MN  d  d  d  d
2
x
2
y
2
z
dy
dx
dz
Fx  F ; Fy  F ; Fz  F
d
d
d
Equilibrium
• Using Components:
F  0
 F  0;  F
x
y
 0;
z
0
F1
• Graphically:
F  0  F  F  F
1
F
2
3
F2
F3
Equilibrium Problems
• In 2D, have 2 equations, so can solve for 2
unknowns
– Find magnitudes of two forces with known
directions
– Find magnitude and direction of one force,
knowing magnitude and direction of other
force(s)
• In 3D have 3 equations, so can solve for 3
unknowns
Rigid Bodies: Equivalent
Systems of Forces
Chapter 3
Rigid Body Motion
• Translation
– Caused by a Force
• Rotation
– Caused by a Moment
• A Force acting at a distance from a Point
– Several applied Moments:
• Determine magnitude and direction of single
resultant moment, which determines direction of
impending rotation.
Forces
• Principal of Transmissibility:
– Can slide a force along its line of action
Moment of a Force about a Point
• Moment Vector: M O  rO / A  F
– Magnitude: M O   r sin   F  dF
• Position Vector: rO / A
• Perpendicular
distance = d
Determining Moments in 2D
M B    rx Fy  ry Fx  kˆ
Determine + or - by the
right hand rule
For the picture here:
M B    rx Fy  ry Fx  kˆ
rB / A  rxiˆ  ry ˆj
rB / A   xA  xB  iˆ   y A  yB  ˆj
Determining Moments in 2D
• Right Hand Rule:
Determining Moments in 3D
• Moment Vector
• Position Vector:
M B  rB / A  F
rB / A   x A  xB  iˆ
  y A  yB  ˆj
  z A  z B  kˆ
Calculating a Vector Cross Product
MB  r  F
iˆ
M B  rx
Fx
4
ˆj
ry
Fy
5
kˆ
rz
Fz
6
M B  1  2  3   4  5  6 
iˆ
rx
Fx
ˆj
ry
Fy
1
2
3
M B   ry Fz  rz Fy  iˆ   rz Fx  rx Fz  ˆj   rx Fy  ry Fx  kˆ
Scalar Product
• Find the angle between two vectors,
knowing the components
P Q  PQ cos  PxQx  PyQy  PQ
z z
• Find the projection of a vector onto the line
of action of another vector
POL  ˆOL P
POL  x Px   y Py  z Pz
Moment of a Force, F,
About a Line, OL
MOL measures the tendency for the force, F, applied
at A to cause the rigid body to rotate about line OL

M OL  ˆOL M O  ˆOL r  F
x
y
z
M OL  rx
Fx
ry
Fy
rz
Fz
Called a “mixed triple product”

Moment of a Couple
• A Couple consists of two forces: F ,  F
– Same magnitude
– Parallel lines of action (but not co-linear)
– Opposite sense
• Moment caused by the couple:
F
M  r F
M   r sin   F  dF
Direction determined
by right hand rule
d
r
F
Moments Caused by Couples
• Moment vector caused by a couple is
sometimes called a “couple vector”
– Has components: M  M xiˆ  M y ˆj  M z kˆ
• Moment caused by a couple is a “free
vector”; you can place it at any point
Moments Caused by Couples
• Can add moments caused by two or more
couples
– In 2D: M   M 1  M 2
M 1  d1F1; M 2  d 2 F2
Determine + or - by the
right hand rule
Equivalent Couples
• Two couples are equivalent if they cause
the same moment:
Force-Couple System
• A force, F, acting at point A can be
replaced by the force, F, and a moment,
MO, acting at point O.
M O  rO / A  F
Create a Force-Couple System at a
Chosen Point
F
B
A
=
F
F
B
A
d
F
F
B
=
A
MA
MA = d F
Replace a Force-Couple System
with Just Forces
F
F
A
A
MA
C
F2
d2
=
C
F2
d2 F2 = MA
Reducing a System of Forces to a
Resultant Force-Couple System (at
a Chosen Point)
F1
r1
A
R
r2
r3
F3
F2
=
MA
R  F

MA   r F

Reduce a System of Forces to a
Single
Resultant
Force
F
1
r1
A
R
r2
R
B
F2
=
r3
R
MA
=
MA
R
F3
Using method
from prior slide
R
B
=
R
R
Ry
B
A
A
B
Rx
Rx
Ry
MA
R
R
dx
dx R = –MA