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EQUILIBRIUM AND ELASTICITY - Coggle Diagram
EQUILIBRIUM AND ELASTICITY
The Conditions for Equilibrium
An object with forces acting on it, but with
zero net force, is said to be in equilibrium.
The first condition for equilibrium:
The second condition of equilibrium is that there be no torque around any axis; the choice of axis is arbitrary.
Solving Statics Problems
Choose one object at a time, and make a free- body diagram by showing all the forces on it and where they act.
Choose a coordinate system and resolve forces into components.
Write equilibrium equations for the forces.
Choose any axis perpendicular to the plane of the forces and write the torque equilibrium equation. A clever choice here can simplify the problem enormously.
Solve.
Stability and Balance
If the forces on an object are such that they tend to return it to its equilibrium position, it is said to be in stable equilibrium.
If, however, the forces tend to move it away from its equilibrium point, it is said to be in unstable equilibrium.
An object in stable equilibrium may become unstable if it is tipped so that its center of gravity is outside the pivot point. Of course, it will be stable again once it lands!
People carrying heavy loads automatically adjust their posture so their center of mass is over their feet. This can lead to injury if the contortion is too great.
Elasticity; Stress and Strain
Hooke’s law: the change in length is proportional to the applied force.
This proportionality holds until the force reaches the proportional limit. Beyond that, the object will still return to its original shape up to the elastic limit. Beyond the elastic limit, the material is permanently deformed, and it breaks at the breaking point.
The change in length of a stretched object depends not only on the applied force, but also on its length, cross-sectional area and the material from which it is made.
The material factor, E, is called the elastic modulus or Young’s modulus, and it has been measured for many materials.
The three types of stress for rigid objects:
The shear strain, where G is the shear modulus:
If an object is subjected to inward forces on all sides, its volume changes depending on its bulk modulus. This is the only deformation that applies to fluids.
Fracture
If the stress on an object is too great, the object will fracture. The ultimate strengths of materials under tensile stress, compressional stress, and shear stress have been measured.
When designing a structure, it is a good idea to keep anticipated stresses less than 1/3 to 1/10 of the ultimate strength.
A horizontal beam will be under both tensile and compressive stress due to its own weight. Therefore, it must be made of a material that is strong under both compression and tension.
Arches and Domes
The stones or bricks in a round arch are mainly under compression, which tends to strengthen the structure.
Unfortunately, the horizontal forces required for a semicircular arch can become quite large. The pointed arch was an improvement, but still needed external supports, or “flying buttresses.”
The pointed arches require considerably less horizontal force than a round arch.
Trusses and Bridges
One way to span a wide space is to use a truss—a framework of
rods or struts joined at their ends into triangles.
Each truss member is under either tension or compression; if the
mass is small, these forces act along the strut.
On a real bridge, the load will not, in general, be centered. The maximum load rating for a bridge must take this into account.
For larger bridges, trusses are too heavy. Suspension bridges are one solution; the roadway is suspended from towers by closely spaced vertical wires.
