Equilibrium A particle (mass, but a size that can be neglected) is in equilibrium provided it is at rest if originally at rest or has constant velocity if originally in motion A particle (mass, but a size that can be neglected) is in equilibrium provided it is at rest if originally at rest or has constant velocity if originally in motion Typically the term “static equilibrium” refers to an object at rest Typically the term “static equilibrium” refers to an object at rest To maintain equilibrium, the resultant force acting on a particle must be equal to zero, ∑ F = 0 (the ∑ F is the vector sum of all forces acting on the particle) To maintain equilibrium, the resultant force acting on a particle must be equal to zero, ∑ F = 0 (the ∑ F is the vector sum of all forces acting on the particle) The equilibrium equation, ∑ F = 0, can be used to determine unknown forces acting on an object in equilibrium The equilibrium equation, ∑ F = 0, can be used to determine unknown forces acting on an object in equilibrium

Free-body diagrams A sketch showing the particle isolated or “free” from its surroundings where all the forces acting on the particle are shown A sketch showing the particle isolated or “free” from its surroundings where all the forces acting on the particle are shown Drawing a free-body diagram involves the following steps Drawing a free-body diagram involves the following steps –Draw a sketch of the particle isolated from its surroundings –Draw vectors representing all of the forces acting on the isolated particle Active forces – tend to set the particle in motion Active forces – tend to set the particle in motion Reactive forces – result from constraints or supports and tend to prevent motion Reactive forces – result from constraints or supports and tend to prevent motion –Label the known forces with their magnitudes/directions and use letters to represent the unknown forces –Choose a coordinate system

Springs For a linear elastic spring, the length of the spring will change in direct proportion to the force acting on it For a linear elastic spring, the length of the spring will change in direct proportion to the force acting on it The “elasticity” of a spring is characterized by its spring constant or stiffness, k The “elasticity” of a spring is characterized by its spring constant or stiffness, k The magnitude of the force exerted on a linearly elastic spring as it is deformed a distance “s” from its unloaded position is: The magnitude of the force exerted on a linearly elastic spring as it is deformed a distance “s” from its unloaded position is: F = ks = k(ℓ – ℓ o ) k = spring constant or stiffness s = (ℓ – ℓ o ) = difference in spring’s deformed length from its undeformed length

Cables/cords and pulleys The force in a cable/cord is in tension and the line of action of that force is collinear with the cable/rope The force in a cable/cord is in tension and the line of action of that force is collinear with the cable/rope The tension in a cable/cord is the same on both sides of a pulley The tension in a cable/cord is the same on both sides of a pulley

Coplanar force systems System of forces is 2-D or coplanar System of forces is 2-D or coplanar Orienting a coordinate system such that forces lie in the x-y plane, ∑ F = (∑ F x ) i + (∑ F y ) j = 0 Orienting a coordinate system such that forces lie in the x-y plane, ∑ F = (∑ F x ) i + (∑ F y ) j = 0 Since a vector is zero if and only if its components are zero, ∑ F x = 0, ∑ F y = 0 Since a vector is zero if and only if its components are zero, ∑ F x = 0, ∑ F y = 0 EXAMPLES (pg 98 – 104) EXAMPLES (pg 98 – 104)

Three-dimensional force systems System of forces is 3-D System of forces is 3-D Equilibrium, ∑ F = (∑ F x ) i + (∑ F y ) j + (∑ F z ) k = 0 Equilibrium, ∑ F = (∑ F x ) i + (∑ F y ) j + (∑ F z ) k = 0 Since a vector is zero if and only if its components are zero, ∑ F x = 0, ∑ F y = 0, ∑ F z = 0 Since a vector is zero if and only if its components are zero, ∑ F x = 0, ∑ F y = 0, ∑ F z = 0 EXAMPLES (pg 112 – 116) EXAMPLES (pg 112 – 116)