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ENGI 1313 Mechanics I

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ENGI 1313 Mechanics I

Lecture 12:3D Particle Equilibrium

- to introduce the concept of the free-body diagram for a particle.
- to show how to solve particle equilibrium problems using the equations of equilibrium

- to examine and apply Chapter 3 objectives in 3D space

- Revised Problem Set
- Problem 6 with spring deleted
- Problem added on Dot Product

- A plate with a mass of 150 kg is supported by three cables and is in equilibrium. Find the tension force in each cable.

- What is known?
- Plate mass
- Coordinate geometry

- What is needed?
- Convert mass to weight (force)
- Determine cable forces
- Cartesian force vectors
- Magnitude, direction and sense

- Cartesian force vectors

- Draw FBD at A
- Define Cartesian Force Vectors

FA = W= 150 kg (9.806 m/s2) = 1471 N

A

FAC

FAB

FAD

- Combine Like Terms
- x, y and z directions

FA = W= 150 kg (9.806 m/s2) = 1471 N

A

FAC

FAB

FAD

- Multiply Fx by 1.5 and add to Fy

FA = W= 150 kg (9.806 m/s2) = 1471 N

A

FAC

FAB

FAD

- Multiply Fx by 3 and add to Fz

FA = W= 150 kg (9.806 m/s2) = 1471 N

A

FAC

FAB

FAD

- Assumptions
- In this course for analysis of all pulley systems
- Weightless
- Zero friction
- Tension cables

- In this course for analysis of all pulley systems
- Fixed Pulley
- Class 1
- Fixed axle

- Used to change direction of the pull force
- Mechanical advantage of 1

- Class 1

- Moveable Pulley
- Class 2
- Floating axle

- Used to multiply forces
- Mechanical advantage of 2

- Class 2

- Compound Pulley
- Combination of fixed and moveablepulley system

D

C

- The "scale" consists of a known weight W which is suspended at A from a cord of total length L. Determine the weight w at B if A is at a distance y for equilibrium. Neglect the sizes and weights of the pulleys.

D

C

- Examine Pulley System
- Known weight, W1
- Find weight w2at B for equilibrium position y

W1

W1

W1

w2 = ?

FBD = W1

FBC = W1

B

w2

D

C

- Draw FDB at Point B

W1

W1

W1

w2 = ?

How to determine ?

d/2

h

(L-y)/2

D

C

- Determine
- Total cable length, L
- Triangle geometry
- Neglect pulley size and weight

W1

w2

- The joint of a space frame is subjected to four member forces. Member OA lies in the x–y plane and member OB lies in the y–z plane. Determine the forces acting in each of the members required for equilibrium of the joint.

- Draw 3D FBD at Point O
- Define position and unitvectors for F1

z

F1

O

F3

45

y

40

F2

x

F4

- Draw 3D FBD at Point O
- Define position and unitvectors for F2

z

F1

O

F3

45

y

40

F2

x

F4

- Draw 3D FBD at Point O
- Define position and unitvectors for F3 and F4

z

F1

O

F3

45

y

40

F2

x

F4

- Unit and Force Vectors

z

F1

O

F3

45

y

40

F2

x

F4

- Fx Equilibrium

z

F1

O

F3

45

y

40

F2

x

F4

- Fz Equilibrium

z

F1

O

F3

45

y

40

F2

x

F4

- Fy Equilibrium

z

F1

O

F3

45

y

40

F2

x

F4

Hibbeler (2007)

- http://en.wikipedia.org
- Hibbeler (2007)
- http://wps.prenhall.com/esm_hibbeler_engmech_1