ME 201Engineering Mechanics: Statics

Unit 3.3

3-Dimensional Force Systems

Solving Simultaneous Equations

Using the simultaneous equation solver on

your calculator, solve the following set of

linear equations.

3x + 2y – z = 10

-x + 3y + 2z = 5

x – y – z = -1

Answer:

x = -2, y = 5, z = -6

BXA

BAXAA 11

BAX 1

Equilibrium of a Particle Earlier in Chapter 3 we learned that for

equilibrium:

0 ji yx FF

0F

For 2-D Problems:

0 kji zyx FFF

Or:0 xF

0 yF

0 xF

0 yF

0 zF

For 3-D Problems:

Or:

Steps to Problem Solving1. Draw a Free Body Diagram (FBD), label known and

unknowns Simple sketch of isolated particle

Sketch all forces that act on the particle Active forces

Reactive forces

The sense of an unknown force can be assumed

Label known forces with proper magnitude & direction

2. Apply equations of equilibrium Express each force in Cartesian vector form

Resolve into x, y, and z components (In cases where it is easy to resolve each force into its x, y, and z components Cartesian vectors may be omitted)

Solve 3 equations, 3 unknowns

Example Problem

Given:

W=40 lb

Find:

TAB, TAC, TAD

z

x

y

W

B(-3,-4,8)

C(-3,4,8)

D

A

Example Problem Solution

Solution:

1-FBD

z

x

y

W

B(-3,-4,8) C(-3,4,8)

D

A

z

x

y

FAB FAC

FAD W

222 )8()4()3(

)08()04()03(

kjirF AB

AB

ABABAB F

rF

222 )8()4()3(

)08()04()03(

kjirF AC

AC

ACACAC F

rF

iF ADAD F

kW 40

Example Problem Solution

Solution:

1-FBD

2-Cartesian

Vectors kjiF ABABABAB FFF 848.424.318.

kjiF ACACACAC FFF 848.424.318.

z

x

y

W

B(-3,-4,8) C(-3,4,8)

D

A

z

x

y

FAB FAC

FAD W

:ABF

:ACF

:ADF

:W

kjiF ABABABAB FFF 848.424.318.

kjiF ACACACAC FFF 848.424.318.

iF ADAD F

kW 40

0318.318. ADACAB FFF0 xF

0 yF

0 zF

0424.424. ACAB FF

40848.848. ACAB FF

Solution:

1-FBD

2-Cartesian

Vectors

3-Collect

x, y, & z

components

Example Problem Solution

z

x

y

W

B(-3,-4,8) C(-3,4,8)

D

A

z

x

y

FAB FAC

FAD W

Solution:

1-FBD

2-Cartesian

Vectors

3-Collect

x, y, & z

components

4-Solve

Equations

Example Problem Solution

z

x

y

W

B(-3,-4,8) C(-3,4,8)

D

A

z

x

y

FAB FAC

FAD W

lbFAB 6.23

lbFAD 0.15

0318.318. ADACAB FFF0 xF

0 yF

0 zF

0424.424. ACAB FF

40848.848. ACAB FF

lbFAC 6.23

Review:Solving Simultaneous Equations

Using the simultaneous equation solver on

your calculator, solve the following set of

linear equations. The sum of the solutions

(x + y + z) equals :

A. -3

B. 0

C. 3

D. 21

E. None of the above

3x + 2y – z = 10

-x + 3y + 2z = 5

x – y – z = -1

-2, 5, -6

Videos – Solving 3D ProblemsFundamental Concepts

0 kji zyx FFF

0 xF 0 yF 0 zF

For 3-D Problems:

Or:

Equilibrium of a Particle Earlier in Chapter 3 we learned that for

equilibrium:

0 ji yx FF

0F

For 2-D Problems:

0 kji zyx FFF

Or:0 xF

0 yF

0 xF

0 yF

0 zF

For 3-D Problems:

Or:

Steps to Problem Solving1. Draw a Free Body Diagram (FBD), label known and

unknowns Simple sketch of isolated particle

Sketch all forces that act on the particle Active forces

Reactive forces

The sense of an unknown force can be assumed

Label known forces with proper magnitude & direction

2. Apply equations of equilibrium Express each force in Cartesian vector form

Resolve into x, y, and z components (In cases where it is easy to resolve each force into its x, y, and z components Cartesian vectors may be omitted)

Solve 3 equations, 3 unknowns

Example Problem

Given:

W=40 lb

Find:

TAB, TAC, TAD

z

x

y

W

B(-3,-4,8)

C(-3,4,8)

D

A

Example Problem Solutionz

x

y

W

B(-3,-4,8) C(-3,4,8)

D

A

Example Problem Solution

Solution:

1-FBD

z

x

y

W

B(-3,-4,8) C(-3,4,8)

D

A

z

x

y

FAB FAC

FAD W

222 )8()4()3(

)08()04()03(

kjirF AB

AB

ABABAB F

rF

222 )8()4()3(

)08()04()03(

kjirF AC

AC

ACACAC F

rF

iF ADAD F

kW 40

Example Problem Solution

Solution:

1-FBD

2-Cartesian

Vectors kjiF ABABABAB FFF 848.424.318.

kjiF ACACACAC FFF 848.424.318.

z

x

y

W

B(-3,-4,8) C(-3,4,8)

D

A

z

x

y

FAB FAC

FAD W

:ABF

:ACF

:ADF

:W

kjiF ABABABAB FFF 848.424.318.

kjiF ACACACAC FFF 848.424.318.

iF ADAD F

kW 40

0318.318. ADACAB FFF0 xF

0 yF

0 zF

0424.424. ACAB FF

40848.848. ACAB FF

Solution:

1-FBD

2-Cartesian

Vectors

3-Collect

x, y, & z

components

Example Problem Solution

z

x

y

W

B(-3,-4,8) C(-3,4,8)

D

A

z

x

y

FAB FAC

FAD W

Solution:

1-FBD

2-Cartesian

Vectors

3-Collect

x, y, & z

components

4-Solve

Equations

Example Problem Solution

z

x

y

W

B(-3,-4,8) C(-3,4,8)

D

A

z

x

y

FAB FAC

FAD W

lbFAB 6.23

lbFAD 0.15

0318.318. ADACAB FFF0 xF

0 yF

0 zF

0424.424. ACAB FF

40848.848. ACAB FF

lbFAC 6.23

Example Problems

#3-Find F given max allowable Tension

#2 – Find Tension in each cable#1 – Find maximum mass limited

by max tension in cables

In Class Exercise

Solution

In Class Exercise

Solution

In Class Exercise

Solution