ce 201 - statics lecture 7. equilibrium of a particle condition for the equilibrium of a particle a...

CE 201 - Statics

Lecture 7

EQUILIBRIUM OF A PARTICLE

CONDITION FOR THE EQUILIBRIUM OF A PARTICLE

A particle is in EQUILIBRIUM if:1.it is at rest, OR2.it is moving with constant velocity

The term "EQUILIBRIUM" is often used to describe a particle at rest. For a particle to be in EQUILIBRIUM, Newton's First Law of Motion must be satisfied.

Newton's First Law of Motion

"IF THE RESULTANT FORCE ACTING ON A PARTICLE IS ZERO, THEN THE PARTICLE IS IN EQUILIBRIUM".

F = 0

Newton's Second Law of Motion

F = m × aApplying the condition for equilibrium

F = m × athen,

m × a = 0since m ≠ o then, a = 0this means that the particle acceleration is equal to zero, therefore the particle is moving at constant velocity.

THE FREE-BODY DIAGRAM

To apply the equilibrium equation ( F = 0), all known and unknown forces must be included in the equation.

The equilibrium equation will best be applied when a free-body diagram of the particle is drawn.

What is a free-body diagram?

Example

Suppose that we have a ball supported on a surface with force F is applied.

How can we draw a free-body diagram of the ball?

1.Isolate the ball from all surroundings

2.Indicate all known and unknown forces acting on the ball.

F = 50 N

30

F = 50 N

30

W

R

In this case, we have the following forces acting on the particle:F = 50 NBall weight, W = ?Surface reaction, R = ?

F = 50 N

30

F = 50 N

30

W

R

Example

Draw a free-body diagram of the following system:

Tips•Label known forces: should be labeled with their magnitude and direction•Label unknown forces: should be labeled using letters•Assume +ve magnitude of unknown forces. If magnitude of an unknown force was obtained –ve, then the direction of the force is opposite to the direction assumed.•Apply equilibrium equation

F

F

W

R1

R2

Connections

Two types of connections will be discussed:

1. Springs

2. Cables and Pulleys

Springs

If a spring is subjected to a force, the length of the spring will change in direct proportion to the force acting on it (if it is a linear elastic spring).

Example

In this case, the following equation can be used:

F = k sF = acting forcek = spring constant or stiffnesss = deformed distance measured from its unloaded position (elongated or compressed)

If ( s ) is +ve, then ( F ) pulls on the spring, while if ( s ) is –ve, then ( F ) pushes on the spring.

s = L – L0

L0

Ls (-ve)

-F

+F

L0s (+ve)

L

Cables and Pulleys

• All cables are assumed to have negligible weight and can not be stretched

• Cables can support only tension or pulling forces

T T