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Vector Mechanics for Engineers: Statics

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Equilibrium of a particle

Chapter 3

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Contents

• Equilibrium of a Particle

• Free-Body Diagrams

• Sample Problem 2.4

• Sample Problem 2.6

• Examples

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Introduction

• The objective for the current chapter is to investigate the effects of forces

on particles:

- Introduce the concept of free-body diagram for a particle.

- relations between forces acting on a particle that is in a

state of equilibrium.

• The focus on particles does not imply a restriction to miniscule bodies.

Rather, the study is restricted to analysis in which the size and shape of

the bodies is not significant so that all forces may be assumed to be

applied at a single point.

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Equilibrium of a Particle• When the resultant of all forces acting on a particle is zero, the particle is

in equilibrium.

• Particle acted upon by

two 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

yx FF

FR

• 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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Example

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Coplanar Force System

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Problem 2.44 (also see Sample Problems 2.4-2.6 in the text)

Knowing that = 55, determine the tension in bar AC and in rope BC.

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Problem 2.51

Two forces P and Q are applied as shown to an aircraft

connection. Knowing that the connection is in equilibrium and

that P = 400 lb and Q = 520 lb, determine the magnitudes of the

forces exerted on the rods A and B.

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Springs

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Pulleys

Pulleys

• Ideal pulleys simply change the direction of a force.

• The tension on each side of an ideal pulley is the same.

• The tension is the same everywhere in a given rope or cable if ideal pulleys are used.

• In a later chapter non-ideal pulleys are introduced (belt friction and bearing friction).

50 lb

Vertical

force

Horizontal

force

50 lbExample

Determine the tension T required to support the 100 lb block shown below.

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Pulleys

Example: (Problem 6-68 in Statics, 9th Ed. by Hibbeler) Determine the force P needed to

support the 100-lb weight. Each pulley has a weight of 10 lb. Also, what are the cord

reactions at A and B?

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Problem 2.70

A 350-lb load is supported by the rope-and-pulley arrangement shown.

Knowing that = 35, determine:

a) The angle

b) The magnitude of the force P which should be exerted on the free end of

the rope to maintain equilibrium. (Hint: The tension is the same on either

side of an ideal pulley).

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Examples

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Examples

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Equilibrium in Space

Equilibrium of a particle in space

If an object in is equilibrium and if the problem is represented in three dimensions,

then the relationship F = 0 can be expressed as:

Fx = 0

Fy = 0

Fz = 0

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Example

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Example

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Example

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Example

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Problem 2.103

Three cables are used to tether a balloon as shown. Determine

the vertical force P exerted by the balloon at A knowing that the

tension in cable AB is 60 lb.