mechanics vector force

1. 1.1 MECHANICS
 Body of Knowledge which
Deals with the Study and
Prediction of the State of Rest
or Motion of Particles and
Bodies under the action of
Forces

2. PARTS OF MECHANICS

3. 1.2 STATICS
 Statics Deals With the Equilibrium
of Bodies, That Is Those That Are
Either at Rest or Move With a
Constant Velocity.
 Dynamics Is Concerned With the
Accelerated Motion of Bodies and
Will Be Dealt in the Next Semester.

4. ME13A: ENGINEERING
STATICS
CHAPTER TWO:
STATICS OF PARTICLES

5. A particle has a mass but a size that can
be neglected.
When a body is idealised as a particle,
the principles of mechanics reduce to a
simplified form, since the geometry of
the body will not be concerned in the
analysis of the problem.
2.1 PARTICLE

6. PARTICLE CONTINUED
 All the forces acting on a
body will be assumed to be
applied at the same point,
that is the forces are
assumed concurrent.

7. 2.2 FORCE ON A PARTICLE
 A Force is a Vector quantity and must
have Magnitude, Direction and Point of
action.

F
P

8. Force on a Particle Contd.
 Note: Point P is the point of action of
force and and are directions. To
notify that F is a vector, it is printed in
bold as in the text book.
 Its magnitude is denoted as |F| or
simply F.


9. Force on a Particle Contd.
 There can be many forces acting on a
particle.
 The resultant of a system of forces
on a particle is the single force
which has the same effect as the
system of forces. The resultant of
two forces can be found using the
paralleolegram law.

10. 2.2.VECTOR OPERATIONS
 2.3.1 EQUAL VECTORS
Two vectors are equal if they are equal
in magnitude and act in the same
direction.
pP
Q

11. Equal Vectors Contd.
 Forces equal in Magnitude can act in
opposite Directions
S
R

12. Q
P
R
2.3.2 Vector Addition
Using the Paralleologram Law, Construct a
Parm. with two Forces as Parts. The
resultant of the forces is the diagonal.

13. Vector Addition Contd.
 Triangle Rule: Draw the first Vector. Join
the tail of the Second to the head of the
First and then join the head of the third to
the tail of the first force to get the resultant
force, R
Q
P
R = Q + P

14. Triangle Rule Contd.
 Also:
P
Q
R = P + Q
Q + P = P + Q. This is the cummutative law of
vector addition

15. Polygon Rule
 Can be used for the addition of more
than two vectors. Two vectors are
actually summed and added to the
third.

16. Polygon Rule contd.
P
Q
S
P
Q
S
R
R = P + Q + S
(P + Q)

17. Polygon Rule Contd.
 P + Q = (P + Q) ………. Triangle Rule
 i.e. P + Q + S = (P + Q) + S = R
 The method of drawing the vectors is
immaterial . The following method can
be used.

18. Polygon Rule contd.
P
Q
S
P
Q
S
R
R = P + Q + S
(Q + S)

19. Polygon Rule Concluded
 Q + S = (Q + S) ……. Triangle Rule
 P + Q + S = P + (Q + S) = R
 i.e. P + Q + S = (P + Q) + S = P + (Q + S)
 This is the associative Law of Vector
Addition

20. 2.3.3. Vector Subtraction
 P - Q = P + (- Q)
P
Q
P
-Q
P -Q
Q
P
P - Q
Parm. Rule
Triangle Rule

21. 2.4 Resolution of Forces
 It has been shown that the
resultant of forces acting at the
same point (concurrent forces) can
be found.
 In the same way, a given force, F
can be resolved into components.
 There are two major cases.

22. Resolution of Forces: Case 1
 (a)When one of the two components, P is
known: The second component Q is
obtained using the triangle rule. Join the tip
of P to the tip of F. The magnitude and
direction of Q are determined graphically or
by trignometry.
F
P
Q
i.e. F = P + Q

23. Resolution of Forces: Case 2
(b) When the line of action of each component is known: The force, F can be
resolved into two components having lines of action along lines ‘a’ and ‘b’ using the
paralleogram law. From the head of F, extend a line parallel to ‘a’ until it intersects ‘b’.
Likewise, a line parallel to ‘b’ is drawn from the head of F to the point of intersection with
‘a’. The two components P and Q are then drawn such that they extend from the tail of
F to points of intersection.
a
Q F
P b

24. Example
 Determine graphically, the magnitude
and direction of the resultant of the two
forces using (a) Paralleolegram law
and (b) the triangle rule.
900 N
600 N
30o
45o

25. Solution
Solution: A parm. with sides equal to 900 N and 600 N is drawn to scale as shown.
The magnitude and direction of the resultant can be found by drawing to scale.
600 N R
15o
900 N
45o
30o
The triangle rule may also be used. Join the forces in a tip to tail fashion and
measure the magnitude and direction of the resultant.
600 N
R 45o
135o
C
B 30o 900 N
900N
600N
30o
45o

26. Trignometric Solution
Using the cosine law:
R2
= 9002
+ 6002
- 2 x 900 x 600 cos 1350
R = 1390.6 = 1391 N
Using the sine law:
R
B
i e B
The angle of the resul t
sin sin
. . sin
sin
.
tan . .
135
600 600 135
1391
17 8
30 17 8 47 8
1




 

  

ie. R = 139N
47.8o
R
900 N
600N
135o
30o
B

27. Example
 Two structural members B and C are bolted
to bracket A. Knowing that both members
are in tension and that P = 30 kN and Q =
20 kN, determine the magnitude and
direction of the resultant force exerted on the
bracket.
P
25o
50o

28. Solution
Solution: Using Triangle rule:
75o
30 kN
20 kN 105o
 25o
Q
R
R2
= 302
+ 202
- 2 x 30 x 20 cos 1050
- cosine law
R = 40.13 N
Using sine rule:
4013
105
20 20 105
4013
288
288 25 38
401 38
1
. sin
.
.
. .
. . , .
N
Sin Sin
and Sin
Angle R
i e R N
o
o
o
o o o
o
  
  




29. 2.5 RECTANGULAR
COMPONENTS OF FORCE
x
F
j
i
Fx = Fx i
Fy = Fy j
y

30. RECTANGULAR COMPONENTS
OF FORCE CONTD.
 In many problems, it is desirable to resolve
force F into two perpendicular components in
the x and y directions.
 Fx and Fy are called rectangular vector
components.
 In two-dimensions, the cartesian unit vectors
i and j are used to designate the directions of
x and y axes.
 Fx = Fx i and Fy = Fy j
 i.e. F = Fx i + Fy j
 Fx and Fy are scalar components of F

31. RECTANGULAR COMPONENTS
OF FORCE CONTD.
While the scalars, Fx and Fy may be positive or negative, depending on the sense of Fx
and Fy, their absolute values are respectively equal to the magnitudes of the component
forces Fx and Fy,
Scalar components of F have magnitudes:
Fx = F cos and Fy = F sin
F is the magnitude of force F.

32. Example
 Determine the resultant of the three
forces below.
25o
45o
350 N
800 N
600 N
60o
y
x

33. Solution
 F x = 350 cos 25o
+ 800 cos 70o
- 600 cos 60o
= 317.2 + 273.6 - 300 = 290.8 N
 F y = 350 sin 25o
+ 800 sin 70o
+ 600 sin 60o
= 147.9 + 751 + 519.6 = 1419.3 N
i.e. F = 290.8 N i + 1419.3 N j
Resultant, F
F N
  
 

2908 1419 3 1449
1419 3
2908
78 4
2 2
1 0
. .
tan
.
.
.

F = 1449 N 78.4o
25o
45o
350 N
800 N
600 N
60o
y

34. Example
 A hoist trolley is subjected to the three
forces shown. Knowing that = 40o ,
determine (a) the magnitude of force, P for
which the resultant of the three forces is
vertical (b) the corresponding magnitude of
the resultant.

1000 N
P
2000 N
 

35. Solution
1000 N
P
2000 N
40o
40o
(a) The resultant being vertical means that the
horizontal component is zero.
 F x = 1000 sin 40o
+ P - 2000 cos 40o
= 0
P = 2000 cos 40o
- 1000 sin 40o
=
1532.1 - 642.8 = 889.3 = 889 kN
(b)  Fy = - 2000 sin 40o
- 1000 cos 40o
=
- 1285.6 - 766 = - 2052 N = 2052 N

36. 2.6. EQUILIBRIUM OF A PARTICLE
A particle is said to be at equilibrium when the resultant of all the forces acting on it is
zero. It two forces are involved on a body in equilibrium, then the forces are equal and
opposite.
.. 150 N 150 N
If there are three forces, when resolving, the triangle of forces will close, if they are in
equilibrium.
F2 F1 F2
F3
F1
F3

37. EQUILIBRIUM OF A PARTICLE
CONTD.
If there are more than three forces, the polygon of forces will be closed if the particle is
in equilibrium.
F3
F2 F2
F3 F1 F4
F1
F4
The closed polygon provides a graphical expression of the equilibrium of forces.
Mathematically: For equilibrium:
R =  F = 0
i.e.  ( Fx i + Fy j) = 0 or  (Fx) i +  (Fy) j

38. EQUILIBRIUM OF A PARTICLE
CONCLUDED
 For equilibrium:
  Fx = 0 and
  F y = 0.
 Note: Considering Newton’s first law
of motion, equilibrium can mean that
the particle is either at rest or moving in
a straight line at constant speed.

39. FREE BODY DIAGRAMS:
 Space diagram represents the sketch
of the physical problem. The free body
diagram selects the significant particle
or points and draws the force system
on that particle or point.
 Steps:
 1. Imagine the particle to be isolated or
cut free from its surroundings. Draw or
sketch its outlined shape.

40. Free Body Diagrams Contd.
 2. Indicate on this sketch all the forces
that act on the particle.
 These include active forces - tend to
set the particle in motion e.g. from
cables and weights and reactive forces
caused by constraints or supports that
prevent motion.

41. Free Body Diagrams Contd.
 3. Label known forces with their
magnitudes and directions. use letters
to represent magnitudes and directions
of unknown forces.
 Assume direction of force which may
be corrected later.

42. Example
 The crate below has a weight of 50 kg. Draw
a free body diagram of the crate, the cord BD
and the ring at B.
CRATE
B ring C
A
D
45o

43. Solution
(a) Crate
FD ( force of cord acting on crate)
50 kg (wt. of crate)
(b) Cord BD
FB (force of ring acting on cord)
FD (force of crate acting on cord)
CRATE
C
45o
B
A
D

44. Solution Contd.
(c) Ring
FA (Force of cord BA acting along ring)
FC (force of cord BC acting on ring)
FB (force of cord BD acting on ring)

45. Example

46. Solution Contd.
F
F
F
BC
AC
o
o AC
 
sin
cos
. .............( )
75
75
373 1
 Fy = 0 i.e. FBC sin 75o
- FAC cos 75o
- 1962 = 0
F
F
F
BC
AC
AC


 
1962 0 26
0 966
20312 0 27 2
.
.
. . ......( )
From Equations (1) and (2), 3.73 FAC = 2031.2 + 0.27 FAC
FAC = 587 N
From (1), FBC = 3.73 x 587 = 2190 N

47. RECTANGULAR COMPONENTS
OF FORCE (REVISITED)
x
j
i
Fx = Fx i
Fy = Fy j
y
F = Fx + Fy
F = |Fx| . i + |Fy| . j
|F|2 = |Fx|2 + |Fy|2
F | | | | | |
F Fx Fy
 
2 2

48. 2.8 Forces in Space
 Rectangular Components
Fy
Fx
Fz
j
i
k
F 

49. Rectangular Components of a Force
in Space
F = Fx + Fy + Fz
F = |Fx| . i + |Fy| . j + |Fz| . k
|F|2 = |Fx|2 + |Fy|2 + |Fz|2
| | | | | | | |
F Fx Fy Fz
  
2 2 2
| | | | cos | | | | cos | | | |cos
, cos
,
Fx F Fy F Fz F
Cos Cos and Cos are called direction ines of
angles and
x y z
x y z
x y z
  
  
  
  

50. Forces in Space Contd.
i.e. F = F ( cos x i + cos y j + cos z k) = F 
F can therefore be expressed as the product of scalar, F
and the unit vector  where:  = cos x i + cos y j + cos z k.
 is a unit vector of magnitude 1 and of the same direction as F.
 is a unit vector along the line of action of F.

51. Forces in Space Contd.
Also:
x = cos x, y = cos y and z = cos z - Scalar vectors
i.e. magnitudes.
x
2
+ y
2
+ z
2
= 1 = 2
i.e. cos2
x, + cos2
y + cos2
z = 1
Note: If components, Fx, Fy, and Fz of a Force, F are known,
the magnitude of F, F = Fx
2
+ Fy
2
+ Fz
2
Direction cosines are: cos x = Fx/F , cos y = Fy/F and cos2
z = Fz/F

53. Force Defined by Magnitude and two Points
on its Line of Action Contd.
Unit vector,  along the line of action of F = MN/MN
MN is the distance, d from M to N.
 = MN/MN = 1/d ( dx i + dy j + dz k )
Recall that: F = F 
F = F  = F/d ( dx i + dy j + dz k )
F
Fd
d
F
Fd
d
F
Fd
d
d x x d y y d z z
d d d d
d
d
d
d
d
d
x
x
y
y
z
z
x y z
x y z
x
x
y
y
z
z
  
     
  
  
, ,
, ,
cos , cos , cos
2 1 2 1 2 1
2 2 2
  

54. 2.8.3 Addition of Concurrent Forces
in Space
The resultant, R of two or more forces in space is obtained by
summing their rectangular components i.e.
R =  F
i.e. Rx i + Ry j + Rz k =  ( Fx i + Fy j + Fz k )
= ( Fx) i + ( Fy)j + ( Fz )k
R x =  Fx, Ry =  Fy , Rz =  Fz
R = Rx
2
+ Ry
2
+ Rz
2
cos x = Rx/R cos y = Ry/R cos z = Rz/R

56. Solution
Solution:
Position vector of BH = 0.6 m i + 1.2 m j - 1.2 m k
Magnitude, BH = 0 6 12 12 18
2 2 2
. . . .
   m


BH
BH BH BH BH
BH
x y z
BH
BH
m i m j m k
T T T
BH
BH
N
m
m i m j m k
T N i N j N k
F N F N F N
   
    
  
   




  
| | .
( . . . )
| |. | |
| | .
. . .
( ) (500 ) (500 )
, ,
1
18
0 6 12 12
750
18
0 6 12 12
250
250 500 500

57. 2.9 EQUILIBRIUM OF A
PARTICLE IN SPACE
 For equilibrium:
 Fx = 0, Fy = 0 and Fz =
0.
 The equations may be used to
solve problems dealing with the
equilibrium of a particle involving
no more than three unknowns.