Lecture Notes: Engineering Mechanics - Equilibrium Problem

Introduction

• Lecture on solving equilibrium problems in engineering mechanics.
• Focus on determining the weight of a container supported by three cables attached to a ceiling.

Problem Statements

1. Problem 1: Determine the weight of a container given that the tension in cable AB is 6 kN.
2. Problem 2: Determine the weight of a container given that the tension in cable AD is 4.3 kN.

Key Concepts

• The system is in equilibrium.
• The vector sum of tensions in cables and weight must equal zero.
• Represent tensions in Cartesian vector form using i, j, k components.

Steps to Solving the Problems

General Approach

1. Equilibrium Equation:

   • Sum of tensions (TAB, TAC, TAD) plus weight = 0.
   • Use position vectors to convert tensions into Cartesian form.

2. Tension in Cable AB:

   • Given: TAB = 6 kN
   • Calculate position vector from A to B.
   • Travel directions: +600 mm in y (j), -450 mm in x (i), 0 mm in z (k).
   • Calculate magnitude: sqrt(450² + 600² + 0²) = 750 mm.
   • TAB Cartesian vector:
     • -0.6 TAB i + 0.8 TAB j + 0 k

3. Tension in Cable AC:

   • Calculate position vector from A to C.
   • Travel directions: +600 mm in y (j), -320 mm in z (k), 0 mm in x (i).
   • Calculate magnitude: sqrt(0² + 600² + 320²) = 680 mm.
   • TAC Cartesian vector:
     • 0 i + 0.882 TAC j - 0.471 TAC k

4. Tension in Cable AD:

   • Calculate position vector from A to D.
   • Travel directions: +600 mm in y (j), +500 mm in x (i), +360 mm in z (k).
   • Calculate magnitude: sqrt(500² + 600² + 360²) = 860 mm.
   • TAD Cartesian vector:
     • 0.581 TAD i + 0.698 TAD j + 0.419 TAD k

Solving for Unknowns

Using the Equilibrium Conditions

1. Sum of i Components = 0

   • Equation: -0.6 TAB + 0.581 TAD = 0
   • Given TAB = 6 kN, solve for TAD = 6.20 kN.

2. Sum of j Components = 0

   • Equation: 0.8 TAB + 0.882 TAC + 0.698 TAD - weight = 0
   • Known: TAB, TAD; Unknown: TAC, weight.
   • Use other components to simplify.

3. Sum of k Components = 0

   • Equation: -0.471 TAC + 0.419 TAD = 0
   • Solve for TAC using known TAD: TAC = 5.52 kN.

4. Calculate Weight

   • Substitute known tensions into j components equation.
   • Calculate weight ≈ 14 kN.

Conclusion

• Given tensions, the calculated weight the system can support is approximately 14 kN.
• Problem solving reinforces concepts in vector mechanics.
• Encouraged to solve similar problems and discuss results in comments.

Call to Action

• Subscribe to Engineers Academy for more solutions.
• Solve Problem 2 with given tension in cable AD (4.3 kN) on your own.