Chapter 3 Lecture Notes: Equilibrium in 2D and 3D

Chapter 3: Equilibrium - Sum of Forces Equals Zero

Overview

  • This chapter extends the knowledge of resultant forces from Chapter 2.

  • Focuses on understanding and applying Newton's First Law of Motion.

Objectives

  • Be able to draw a free body diagram.

  • Apply equations of equilibrium to solve 2D and 3D problems.

Free Body Diagrams

  • Start with a real picture of the situation.

  • Draw all external forces acting on the object.

  • Label all the external forces.

  • Examples:

    • A crane holding up a weight: Tension in the cable and the weight of the object are the external forces.

    • Forces at point A: Tension in cables AB, AC, and AD.

    • A boat moored by cables: Tension in each cable with given angles.

Equations of Equilibrium

  • Sum of forces equals zero for static equilibrium (no movement).

  • ∑F=0\sum F = 0

Example Problem 1: Cable System with a 40 kg Weight

  • Assume particle A is in equilibrium, meaning ∑F=0\sum F = 0.

  • Draw a free body diagram at point A.

  • Identify all external forces acting on the particle.

Forces at Point A

  • Tension in cable AD ($\F_d$).

  • Tension in cable AB ($\F_b$).

  • External load of 40 kg acting straight down.

  • Draw the forces on x and y axes, including the angle of 30 degrees.

Force Types

  • Active forces: Forces that want to move the particle.

  • Reactive forces: Forces that tend to resist movement.

Converting Mass to Weight

  • Convert kilograms (mass) to Newtons (force) by multiplying by the gravitational constant (9.81 m/s²).

  • 40 kg×9.81ms2=392.4 N40 \text{ kg} \times 9.81 \frac{\text{m}}{\text{s}^2} = 392.4 \text{ N}

  • Forces must be in Newtons or pounds.

Equations of Equilibrium in Cartesian Form

  • ∑F=F<em>b+F</em>c+Fd=0\sum F = F<em>b + F</em>c + F_d = 0

  • Scalar form: Sum of forces in x and y directions equals zero.

    • ∑Fx=0\sum F_x = 0

    • ∑Fy=0\sum F_y = 0

X-Component Equation

  • F<em>bcos⁡(30)−F</em>d=0F<em>b \cos(30) - F</em>d = 0

Y-Component Equation

  • Fbsin⁡(30)−392.4=0F_b \sin(30) - 392.4 = 0

Solving the Equations

  • Solve for $\F_b$ in the y-component equation:

    • Fb=392.4sin⁡(30)=784.8 NF_b = \frac{392.4}{\sin(30)} = 784.8 \text{ N}

  • Solve for $\Fd$ using the value of $\Fb$:

    • Fd=784.8cos⁡(30)=679.9 NF_d = 784.8 \cos(30) = 679.9 \text{ N}

Example Problem 2

  • Given angles or side lengths, determine forces at point A in equilibrium.

  • Sum of forces equals zero.

  • ∑F=F<em>b+F</em>c+Fd=0\sum F = F<em>b + F</em>c + F_d = 0

Equations

  • X-component: −F<em>bcos⁡(30)+F</em>c(45)=0-F<em>b \cos(30) + F</em>c \left(\frac{4}{5}\right) = 0

  • Y-component: F<em>bsin⁡(30)+F</em>c(35)−550=0F<em>b \sin(30) + F</em>c \left(\frac{3}{5}\right) - 550 = 0

Solutions:

  • $\F_b = 478$ pounds

  • $\F_c = 518$ pounds

3D Equilibrium

Objectives

  • Draw a free body diagram in 3D.

  • Write and solve scalar equations for equilibrium in 3D.

Setup

  • Sum forces at point A, considering tensions in different cables.

  • Free body diagram at point A.

  • ∑F=0\sum F = 0

Cartesian Setup

  • Using I, j, and k components.

  • Sum forces in each axis (x, y, and z) separately.

Example Problem: Cords Supporting a 600 N Load

  • Determine tensions in chords AB, AC, and AD.

  • Sum forces at point A.

  • Label forces as $\Fc$, $\Fb$, and $\F_d$.

Axis

  • x-axis coming into the computer.

  • y-axis along the horizontal.

  • z-axis straight up and down.

Force Components

  • $\Fb = Fb \sin(30)i + F_b \cos(30)j$

  • $\Fc = -Fc i$

Position Vector

  • The unit vector is $\frac{r{AD}}{|r{AD}|}$

  • rAD=12+(−2)2+22=9=3r_{AD} = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{9} = 3

  • F<em>D=F</em>D(13i−23j+23k)F<em>D = F</em>D \left( \frac{1}{3}i - \frac{2}{3}j + \frac{2}{3}k \right)

Important Note

  • Ensure correct position vector calculations.

Equations in Unit Vector Form

  • Each force is written in unit vector form (I, j, and k components).

  • Easier to solve by combining all I, j, and k components together.

Sum of Forces Equations

  • ∑Fx=0\sum F_x = 0

  • ∑Fy=0\sum F_y = 0

  • ∑Fz=0\sum F_z = 0

Solving for Forces

  • Solve for $\F_d$ using the z-component equation.

  • Solve for other forces by substitution.

Key Points

  • Equation of equilibrium based on the free body diagram.

  • Write position and unit vectors using 3D coordinates.

  • Sum of forces equals zero.

This concludes the lecture on 2D and 3D equilibrium.