STATIC EQUILIBRIUM OF A PARTICLE (3-D)

Learning Objectives

1). To draw a free body diagram (FBD) of an object that is

modeled as a particle.

2). To evaluate the forces required for static equilibrium of an

object that is modeled as a particle.

3). To do an engineering estimate of these quantities.

Force Classifications

External Forces: applied forces which are typically known or

prescribed (e.g., forces due to cables, springs, gravity, etc.).

Reaction Forces: constraining forces at supports, intended to

prevent motion (usually nonexistent unless system is externally

loaded).

Free Body Diagram (FBD)

Free Body Diagram (FBD): a graphical sketch of the system

showing a coordinate system, all

external/reaction forces and moments,

and key geometric dimensions.

Benefits:

1). Provides a coordinate system to establish a solution

methodology.

2). Provides a graphical display of all forces/moments acting on

the rigid body.

3). Provides a record of geometric dimensions needed for

establishing moments of the forces.

Newton’s First Law

Given no net force, a body at rest will remain at rest and a body

moving at a constant velocity will continue to do so along a

straight path ( )R F 0 .

Static Equilibrium

Vector Equation:

Component Equations:

N

R i x y z

i 1

F F F i F j F k 0

x y z0 0 0F F F

Problem Solving

1). Draw complete FBD.

2). Choose an xyz reference frame.

3). Evaluate the geometrical parameters.

4). Write equations of static equilibrium.

5). Count number of scalar equations and number of unknowns.

6). Solve equations of static equilibrium.

Note:

Fdrag = ½ ρ CD SA ν2

Where ρ = density of air

CD = Drag Coefficient

SA = Frontal Surface Area

ν = velocity

3-D Particle Equilibrium

Example 2

Given: A 500 lb crate is suspended in static equilibrium using rigid strut AD and cables AB and AC.

Find:

a) Write expressions for cables ABT and ACT and strut DAF

in terms of their unknown magnitudes and their known unit vectors.

b) Write the equations of static equilibrium and determine the magnitudes of the tensions in cables AB and AC and the axial load in strut AD.

c) Is the axial load in strut AD in tension or compression?

3-D Particle Equilibrium

Group Quiz

Group #: ____________ Group Members: 1) ______________________________

(Present Only)

Date: ______________ Period: _________ 2) ______________________________

3) ______________________________

4) ______________________________

Given: A weather balloon is tethered by the lines AB, AC, and AD. The mass of the balloon and the

gas it contains is 80 kg. and the (upward) buoyancy force on the balloon is 1000 N.

Find: a) Draw a free body diagram of particle A.

b) Write force vectors ABF , ACF , ADF , and bF in terms of

their unknown tensions. c) Estimate the tensions in lines AB, AC, and AD. d) Set up the equilibrium equations and (if time permits)

solve for the unknown tension in each line.

Solution: