Lecture 3- 1page

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Vector Mechanics for Engineers: StaticsE i gh t h

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Equilibrium of a Particle

• When the resultant of all forces acting on a particle is zero, the particle isin equilibrium.

• Particle acted upon bytwo forces:

- equal magnitude

- same line of action

- opposite sense

• Particle acted upon by three or more forces:- graphical solution yields a closed

polygon

- algebraic solution

00

0

==

==

∑∑

∑

y x F F

F Rrr

• Newton’s First Law: If the resultant force on a particle is zero, the particle will

remain at rest or will continue at constant speed in a straight line.





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Free-Body Diagrams

Space Diagram: A sketch

showing the physical conditions

of the problem.

Free-Body Diagram: A sketch showing

only the forces on the selected

particle.





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EQUILIBRIUM OF PARTICLE IN 2-D

This is an example of a 2-D or

coplanar force system. If the

whole assembly is in

equilibrium, then particle A is

also in equilibrium.

To determine the tensions in

the cables for a given weight

of the engine, we need to

learn how to draw a free bodydiagram and apply equations

of equilibrium.


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THE WHAT, WHY AND HOW OF A

FREE BODY DIAGRAM (FBD)

Free Body Diagrams are one of the most important things for

you to know how to draw and use.

What ? - It is a drawing that shows

all external forces acting on the particle.

Why ? - It helps you write theequations of equilibrium used to

solve for the unknowns (usually

forces or angles).




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How ?

1. Imagine the particle to be isolated or cut free from itssurroundings.

2. Show all the forces that act on the particle.

Active forces: They want to move the particle.

Reactive forces: They tend to resist the motion.

3. Identify each force and show all known magnitudes

and directions. Show all unknown magnitudes and /

or directions as variables .

FBD at A Note : Engine mass = 250 Kg

A




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Sample Problem 2.4

In a ship-unloading operation, a

3500-lb automobile is supported bya cable. A rope is tied to the cable

and pulled to center the automobile

over its intended position. What is

the tension in the rope?

SOLUTION:

• Construct a free-body diagram for the

particle at the junction of the rope and

cable.

• Apply the conditions for equilibrium by

creating a closed polygon from the

forces applied to the particle.• Apply trigonometric relations to

determine the unknown force

magnitudes.





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Sample Problem 2.4

SOLUTION:

• Construct a free-body diagram for the

particle at A.

• Apply the conditions for equilibrium.

• Solve for the unknown force magnitudes.

°=

°=

° 58sinlb3500

2sin120sin AC AB T T

lb3570= ABT

lb144= AC T





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Sample Problem 2.6

It is desired to determine the drag force

at a given speed on a prototype sailboat

hull. A model is placed in a test

channel and three cables are used to

align its bow on the channel centerline.

For a given speed, the tension is 40 lb

in cable AB and 60 lb in cable AE .

Determine the drag force exerted on

the hull and the tension in cable AC .

SOLUTION:

• Choosing the hull as the free body,

draw a free-body diagram.

• Express the condition for equilibrium

for the hull by writing that the sum of

all forces must be zero.

• Resolve the vector equilibriumequation into two component

equations. Solve for the two unknown

cable tensions.

E E




Vector Mechanics for Engineers: StaticsEi gh t h

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Sample Problem 2.6

SOLUTION:

• Choosing the hull as the free body, draw a

free-body diagram.

°=

==

25.60

75.1ft4

ft7tan

α

α

°=

==

56.20

375.0ft4

ft1.5tan

β

β

• Express the condition for equilibrium

for the hull by writing that the sum of

all forces must be zero.

0=+++= D AE AC AB F T T T Rrrrrr

E E





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Sample Problem 2.6

• Resolve the vector equilibrium equation into

two component equations. Solve for the two

unknown cable tensions.

( ) ( )( ) ( )

( )

( )

( ) jT

iF T

R

iF F

iT jT iT

jT iT T

ji jiT

AC

D AC

D D

AC AC

AC AC AC

AB

r

r

r

rr

rr

rr

rrr

rr

rrr

609363.084.19

3512.073.34

0

lb069363.03512.0

56.20cos56.20sin

lb84.19lb73.3426.60coslb4026.60sinlb40

−++

++−=

=

=

−=+=

°+°=

+−=°+°−=

V M h i f E i S iE E





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Sample Problem 2.6

( )

( ) jT

iF T

R

AC

D AC r

r

r

609363.084.19

3512.073.34

0

−++

++−=

=

This equation is satisfied only if each component

of the resultant is equal to zero

( )( ) 609363.084.1900

3512.073.3400−+==

++−==∑∑

AC y

D AC x

T F F T F

lb66.19

lb9.42

+=

+=

D AC

F

T

V t M h i f E i St tiE E





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EXAMPLE

Write the scalar EofE:

+ → Σ Fx = TB cos 30º – TD = 0

+↑ Σ

Fy = TB sin 30º – 2.452 kN = 0

Solving the second equation gives: TB = 4.90 kN

From the first equation, we get: TD = 4.25 kN

Note : Engine mass = 250 Kg FBD at A

V t M h i f E i St tiE E





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SPRINGS, CABLES, AND PULLEYS

Spring Force = spring constant *

deformation, or

F = k * S

With a

frictionless

pulley, T1

= T2.

V t M h i f E i St tiE

E




Vector Mechanics for Engineers: Staticsi gh t h

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EXAMPLE

Given: Sack A weighs 20 lb.

and geometry is as

shown.

Find: Forces in the cables and

weight of sack B.

Plan:

1. Draw a FBD for Point E.

2. Apply EofE at Point E to solvefor the unknowns (TEG & TEC).

3. Repeat this process at C.

V t M h i f E i St tiE i

E d




Vector Mechanics for Engineers: Staticsi gh t h

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EXAMPLE (continued)

The scalar EofE are:

+ → Σ Fx = TEG sin 30º – TEC cos 45º = 0

+ ↑ Σ Fy = TEG cos 30º – TEC sin 45º – 20 lbs = 0

Solving these two simultaneous equations for the

two unknowns yields:

TEC = 38.6 lb

TEG = 54.6 lb

A FBD at E should look like the one

to the left. Note the assumed

directions for the two cable tensions.

Vector Mechanics for Engineers: StaticsE i

E d




Vector Mechanics for Engineers: Statics gh t h

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EXAMPLE (continued)

+ → Σ Fx = 38.64 cos 45° – (4/5) TCD = 0

+ ↑ Σ Fy

= (3/5) TCD

+ 38.64 sin 45° – WB

= 0

Solving the first equation and then the second yields

TCD

= 34.2 lb and WB

= 47.8 lb .

The scalar EofE are:

Now move on to ring C.

A FBD for C should look

like the one to the left.

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