Equilibrium is the state of an object where: Equilibrium the linear momentum,, of the center of mass is constant Feb. 19, 2018 the angular momentum,, about the its center of mass, or any other point, is constant Our concern this chapter will be with static equilibrium which means the object is not moving
The Requirements of Equilibrium With respect to translational motion and our definition of equilibrium we have and constant With respect to rotational motion and our definition of equilibrium we also have and constant
Center of Gravity The gravitational force on a rigid body is the vector sum of the gravitational forces acting on all the individual elements of the body. The gravitational force on a body effectively acts at a single point called, called the center of gravity (cog) of the body. If is the same for all elements of the body, then the body's center of gravity (cog) is coincident with the body's center of mass (com).
Indeterminate Structures For the problems in this chapter we have only three independent equations, usually two balance of forces equations and one balance of torques equation. Therefore, if a problem has more than three unknowns, we cannot solve it. These are called indeterminate. The problem is that we have assumed that the bodies to which we apply the equations of static equilibrium are perfectly rigid (meaning they do not deform when forces are applied to it). There are no such bodies To solve indeterminate equilibrium problems we must supplement our equilibrium equations with some knowledge of elasticity, the branch of physics and engineering that describes how bodies how real bodies deform when forces are applied to them.
Elasticity In solid metallic objects, the atoms settle into equilibrium positions in a three dimensional lattice. The atoms are held in position by interatomic forces. These interatomic forces act like small springs, making the lattice extremely rigid All "rigid" bodies are somewhat elastic, meaning we can change their dimensions slightly by pulling, pushing, twisting, or compressing it. In all types of deformations, the stress (force per unit area) produces a strain. Stress and strain are proportional to each other, with the proportionality constant called the modulus of elasticity
Elasticity (cont'd) In testing tensile strength, the stress in slowly increased until the object breaks. For a large range of applied stress, the stress strain graph is linear. For this, the object will return back to its original shape when the stress is removed. The stress strain equation only applies in this initial part of the graph. If the stress is increased beyond the yield strength, S y, the object becomes permanently deformed, and if it still continues the object eventually breaks at a stress called the ultimate strength.
Elasticity (cont'd) Tension & Compression The stress is the force (F) per unit area (A), F/A, with the force being applied perpendicular to the area. The strain is the dimensionless quantity ΔL/L. For tension and compression the modulus is called the Young's modulus. Our stress strain equation then becomes Although Young's modulus for an object may be almost the same for compression and tension, the ultimate strength may be different. Concrete is very strong in compression but very weak in tension. Shearing The stress is still a force per unit area, but the force is in the plane of the area rather than perpendicular to it. The strain is the dimensionless quantity Δx/L. The modulus is called the shear modulus Shearing stress is critical in the buckling of shafts that rotate under load and in bone fractures caused by bending. Hydraulic Stress The stress is the fluid pressure (which is measure as a force per unit area). The strain is the change in volume, ΔV/V. The modulus is the bulk modulus In general, solids are less compressible than liquids in which the atoms or molecules are less tightly coupled to their neighbors.
Example Ans: (a) 1.71kN down (b) 2.51kN up An 82.0kg diver stands at the edge of a light 5.00m diving board, which is supported by two narrow pillars 1.60m apart. Find the magnitude and direction of the force exerted on the diving board (a) by pillar A (b) by pillar B
Example Ans: 34.0 cm A nonuniform 80.0g meterstick balances when the support is placed at the 51.0cm mark. At what location should a 5.00g tack be placed so that the stick will balance at the 50.0 cm mark?
Example Ans: 5.1m As shown in the figure, a 10.0m long bar is attached by a frictionless hinge to a wall and held horizontal by a light rope that makes an angle θ = 49 o with the bar. The bar is uniform and weight 66.5N. What distance x from the hinge should a 10.kg mass be suspended for the tension in the rope to be 110N? x