CE 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, OR 2. 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 × a Applying the condition for equilibrium  F = m × a then, m × a = 0 since m ≠ o then, a = 0 this 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 N Ball 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  s F = acting force k = spring constant or stiffness s = 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 – L 0 L0L0 L s (-ve) -F +F L0L0 s (+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