ENGR221 Lecture 3
Transcript of ENGR221 Lecture 3
Concurrent Force Systems(II)Concurrent Force Systems(II)
ENGR 221
January 22, 2003
Lecture GoalsLecture Goals
• 2.6 Rectangular Components of a Force
• 2.7 Resultants by Rectangular Components
• 3.2 Free-Body Diagrams
• 3.3 Equilibrium of a particle
Two Dimensional ResultantsTwo Dimensional Resultants The vector components are written as
x yF F i F j
Fx
Fy
F
where,
i - the unit vector in the x direction
j - the unit vector in the y direction
Two Dimensional VectorsTwo Dimensional Vectors Vector Components:In two dimensions, a force can be described using a magnitude |F| and single angle, The components of the vector are Fx and Fy
2 2x y
xy1
yx
F
Fcostan
Fsin
F F
FF
FF
Fx
Fy
F
Two Dimensional Vector - ExampleTwo Dimensional Vector - Example
Given two points(x1,y1) = (4,5) and (x2,y2) = (16, 10), determine the vector from 1 to 2. Determine the components of the vector.
x 2 1
y 2 1
16 4 12
10 5 5
F x x
F y y
Vector can be written as:
x yF
12 5
F i F j
i j
Two Dimensional Vector - ExampleTwo Dimensional Vector - Example
y1
x
1
tan
5tan 0.3948 rad
12
22.62
F
F
2 2x y
2 2
F
12 5 13
F F
The magnitude and angle are
Three Dimensional VectorsThree Dimensional Vectors The vector components are written as
x y zF F i F j F k
where, i - the unit vector in the x direction j - the unit vector in the y direction k - the unit vector in the z direction
Three Dimensional VectorsThree Dimensional Vectors Vector Components:In two dimensions, a force can be described using a magnitude |F| and three angles,x, y, and z The components of the vector are Fx, Fy, and Fz.
2 2 2x y zF F F F
Three Dimensional Three Dimensional VectorsVectors
Vector Components:The three angles,x, y, and z are defined as:
xx
yy
zz
cosF
cosF
cosF
F
F
F
Three Dimensional Three Dimensional VectorsVectors
Vector Components:These vector cosines are
x x
y y
z z
F cos
F cos
F cos
F
F
F
x y zF F i F j F k
Three Dimensional Three Dimensional VectorsVectors
Vector Components:Substitute into the vector
x y z
x y z
F F cos F cos F cos
F cos cos cos
i j k
i j k
The magnitude is |F| and unit vector is
x y zcos cos cosi j k
Three Dimensional Three Dimensional VectorsVectors
Vector Components:The unit vector can be written as:
The magnitude is |F| and unit vector is 2 2 2x y z
2 2 2x y z
1
cos cos cos 1
x y z
x y z
cos cos cosi j k
i j k
Three Dimensional Vector - ExampleThree Dimensional Vector - Example
Given two points(x1,y1,z1) = (4,5,6) and (x2,y2 ,z2) = (16, 10,12), determine the vector from 1 to 2. Determine the components of the vector.
x 2 1
y 2 1
z 2 1
16 4 12
10 5 5
12 6 6
F x x
F y y
F z z
Vector can be written as:
x y zF
12 5 6
F i F j F k
i j k
Three Dimensional Vector - ExampleThree Dimensional Vector - Example
2 2 2x y z
2 2 2
F
12 5 6 14.3178
F F F
The magnitude is
Three Dimensional Vector - ExampleThree Dimensional Vector - Example
The directional cosines are
xx x
yy y
zz z
12cos 0.8381 0.577 rad or 33.06
F 14.3178
5cos 0.3492 1.214 rad or 69.56
F 14.3178
6cos 0.4191 1.138 rad or 65.22
F 14.3178
F
F
F
Three Dimensional Vector - ExampleThree Dimensional Vector - Example
The vector can be written as
x y zF F cos cos cos
14.3178 0.8381 0.3492 0.4191
i j k
i j k
Class - ProblemClass - Problem
Determine the magnitude and directional cosines of the vector.
A 700 820 900i j k
Why?Why?
Why do we need to a procedure to find the directional cosine or unit vector of the vector?
Force - ExampleForce - Example
A tower guy wire is anchored by means of a bolt at A. The tension in the wire is 2500 N. Determine (a) the components Fx, Fy, and Fz of the force acting on the bolt,(b) the angles x, y, and z defining the direction of the force.
Force - ExampleForce - Example
The first step is to determine the coordinate system for the vector AB. If we place the forces acting on bolt A (tension). The force acts along the direction of the wire, so we length of the vector to fined the unit vector
x
y
z
40 m
80 m
30 m
d
d
d
Force - ExampleForce - ExampleThe magnitude of d is
The unit vector is
2 2 2d 40 m 80 m 30 m
94.34 m
40 m 80 m 30 md 94.34 m
94.34 m 94.34 m 94.34 m
94.34 m 0.424 0.848 0.318
0.424 0.848 0.318
i j k
i j k
i j k
Force - ExampleForce - ExampleThe magnitude of the force is 2500 N so that the force vector is
So that Fx = -1060 N, Fy = 2120 N and Fz = 795 N.
F
2500 N 0.424 0.848 0.318
1060 N 2120 N 795 N
F
i j k
i j k
Force - ExampleForce - Example
The angles are
x x
y y
z z
cos 0.424 2.009 rad or 115.1
cos 0.848 0.559 rad or 32.0
cos 0.318 1.247 rad or 71.5
Resultant ForcesResultant Forces
The components of vectors are used to find the resultants acting on object. Using the unit vectors, the components of forces are
x x y y z z R F R F R F 2 2 2x y z
yx zx y z
R
cos cos cosR R R
R R R
RR R
Class - ProblemClass - Problem
Cable AB is in tension, 450 lbs, determine the components of the force exerted on the plate at A.
Equilibrium of a Particle in SpaceEquilibrium of a Particle in Space
The components of the forces in equilibrium
x y z0 =0 0F F F
Example-EquilibriumExample-Equilibrium
A 200 kg cylinder is hung by means of two cable AB and AC, which are attached at the top of a vertical wall holds the cylinder in the position shown. Determine the magnitude of P and the tension in each of the cables.
Example-EquilibriumExample-Equilibrium
Define the coordinate system of the points and find the force vectors.
2 200 kg 9.81 m/s
1962 N
W mg j j
j
P P i
Example-EquilibriumExample-Equilibrium
The vector AB is
x
y
z
1.2 m
12 m 2 m 10 m
8 m
d
d
d
2 2 2
1.2 m 10 m 8 m
1.2 m 10 m 8 m 12.86 m
AB i j k
AB
��������������
Example-EquilibriumExample-Equilibrium
The unit vector in the AB direction is
The tension vector TAB is
AB
1.2 m 10 m 8 m
12.86 m 12.86 m 12.86 m
0.0933 0.778 0.622
ABi j k
AB
i j k
��������������
AB AB AB
AB AB AB0.0933 0.778 0.622
T T
T i T j T k
��������������
Example-EquilibriumExample-Equilibrium
The vector AC is
x
y
z
1.2 m
12 m 2 m 10 m
10 m
d
d
d
2 2 2
1.2 m 10 m 10 m
1.2 m 10 m 10 m 14.19 m
AC i j k
AC
��������������
Example-EquilibriumExample-Equilibrium
The unit vector in the AB direction is
The tension vector TAB is
AC
1.2 m 10 m 10 m
14.19 m 14.19 m 14.19 m
0.0846 0.705 0.705
ACi j k
AC
i j k
��������������
AC AC AC
AC AC AC0.0846 0.705 0.705
T T
T i T j T k
��������������
Example-EquilibriumExample-Equilibrium
Apply the equilibrium condition.
Combine the vectors AB AC0 0F T T W P ��������������������������������������������������������
AB AC
AB AC
AB AC
0
0.0933 0.0846
0.778 0.705 1962 N
0.622 0.705
F
T T P i
T T j
T T k
Example-EquilibriumExample-Equilibrium
Break the components of the forces:
Use (3) to get a relationship for TAB and TAC
AB ACx
AB ACy
AB ACz
0 0.0933 0.0846 1
0 0.778 0.705 1962 N 2
0 0.622 0.705 3
F T T P
F T T
F T T
AB AC AC
0.7051.133
0.622T T T
Example-EquilibriumExample-Equilibrium
Plug into (2) solve for magnitude of TAC and TAB
Use the values to solve for P
AC AC
AC
AB
0.778 1.133 0.705 1962 N
1236 N
1401 N
T T
T
T
AB AC0.0933 0.0846
0.0933 1401 N 0.0846 1236 N
235 N
P T T
Class -ProblemClass -ProblemA weight W is supported by three cables. Determine the value of W, knowing that the tension in the cable DC is 975 lb.
Class-ProblemClass-Problem
The tension force in AC is 28 kN, determine the the required values of tension in AB and AD so that the resultant force of the three forces applied at A is vertical. Determine the resultant force.
Free Body DiagramsFree Body Diagrams
The first step in solving a problem is drawing a free-body diagram (FBD). This step is the most crucial and important solving any problem. It defines weight of the body, the known external forces, and unknown external forces. It defines the constraints and the directions of the forces. If the FBD is drawn correctly the solving of the problem is trivial.
Free Body DiagramsFree Body Diagrams
Construction of a free body diagram.
Step 1: D
Step 2:
Step 3:
Decide which body or combination of bodies are to be shown on the free-body diagram.
Prepare drawing or sketch of the outline of the isolated or free body.
Carefully trace around the boundary of the free-body and identify all the forces exerted by contacting or attracting bodies that were removed during isolation process.
Free Body DiagramsFree Body Diagrams
Construction of a free body diagram(cont.)
Step 4: CChoose the set of coordinate axes to be used in solving the problem and indicate their directions on the free-body diagram. Place any dimensions required for solution of the problem on the diagram.
FBD - ExamplesFBD - ExamplesWhat is the free-body diagram of the weight?
FBD - ExamplesFBD - ExamplesThe diagram has the given angles and the know magnitudes.
TDA
TDB
TDC
W
FBD - ExamplesFBD - ExamplesWhat is the free-body diagram of the tower for the resultant force acting vertically?
FBD - ExamplesFBD - ExamplesThe resulting free-body diagram can solve the problem by attaching points.
TAB
TAD
TAC
R
Homework (Due 1/29/03)Homework (Due 1/29/03)
Problems:
2-55, 2-61, 2-63, 2-83, 3-3, 3-4