Chapter 9

• An object with forces acting on it, but that is not moving, is said to be in equilibrium.
• The first condition for equilibrium is that the forces along each coordinate axis add to zero.
• Materials can be under compression, tension, or shear stress.
• If the force is too great, the material will exceed its elastic limit; if the force continues to increase the material will fracture.
• The second condition of equilibrium is that there be no torque around any axis; the choice of axis is arbitrary.
• Choose an object at a time, and make a free-body diagram 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.
• If a force in your solution comes out negative (as F A will here), it just means that it’s in the opposite direction from the one you chose.
• If there is a cable or cord in the problem, it can support forces only along its length.
• These same principles can be used to understand forces within the body.
• The angle at which this man’s back is bent places an enormous force on the disks at the base of his spine, as the lever arm for F M is so small.
• 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.
• 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.
• 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.
• The ultimate strengths of materials under tensile stress, compressional stress, and shear stress have been measured.
• The stones or bricks in a round arch are mainly under compression, which tends to strengthen the structure.
• Compressional stress is the opposite of tensional stress.
• The change in length of a stretched object depends on the applied force, its length and cross-sectional area, and the material from which it is made.
• The material factor in this change in length is called Young’s modulus, which has been measured for many materials.
• In tensile stress, forces tend to stretch the object.
• When designing a structure, it is a good idea to keep anticipated stresses less than 1/3 to 1/10 of the ultimate strength.
• Beyond the elastic limit, the material is permanently deformed, and it breaks at the breaking point.
• Pointed arches can be built that require considerably less horizontal force.
• In order for an object to be in equilibrium, there must be no net force on it along any coordinate, and there must be no net torque around any axis.
• A dome is similar to an arch, but spans a two-dimensional space.
• The Young’s modulus is the stress divided by the strain.
• A horizontal beam will be under both tensile and compressive stress due to its own weight.
• The Romans developed the semicircular arch about 2000 years ago, allowing wider spans to be built than could be done with stone or brick slabs.
• Unfortunately, the horizontal forces required for a semicircular arch can become quite large, which is why many Gothic cathedrals have “flying buttresses” to keep them from collapsing.
• An object at rest is in equilibrium; the study of such objects is called statics.
• If the stress is too great, the object will fracture.
• Shear stress tends to deform an object.
• An object in static equilibrium can be either in stable, unstable, or neutral equilibrium.