Equilibrium and Elasticity

1. Equilibrium and Elasticity Equilibrium Center of gravity Static Equilibrium Indeterminate Structures Elasticity Tension, Compression and Shearing pps by C Gliniewicz

2. For an object to be stable, it must be in equilibrium. Equilibrium is a state in which two requirements must be met. First, the linear momentum of its center of mass must be constant. Second, the angular momentum about its center of mass must be constant. Note that there are four possible combinations: p=0 and L=0, p=0 and L≠0, p ≠0 and L=0, and p ≠0 and L ≠0. We will deal with only the first possibility, that the linear momentum is zero and the angular momentum is zero. The first condition of equilibrium is that the sum of all the forces is zero. The second condition of equilibrium is the sum of all torque is zero. Since we are only considering situations where the momenta equal zero, the situation is considered to be in static equilibrium. pps by C Gliniewicz

3. Center of gravity is different from the center of mass. Recall that the center of mass is the point where all the mass can be considered to be concentrated. Gravity is produced by a large object which may have its mass distributed in a non-uniform manner. Therefore its attraction to another object may be non-uniform. The center of gravity is the point where the gravitational force effectively acts on the body due to the non-uniform distribution of mass. Only if the force of gravity is the same for all points in the body is the center of gravity the same as the center of mass. In the case of an object on the earth, the force of gravity varies little from place to place, horizontally and vertically. Since this difference is very small, the difference between the center of mass and center of gravity is very small and can be considered to be the same. To determine the forces existing is situations where the momenta are zero require summing the forces to zero and picking a suitable point to sum the torques to zero to find the unknown quantities. pps by C Gliniewicz

4. On occasion, we run into problems which require more equations that we have available. For instance, the forces on each leg of a table with four legs. There are not enough equations to solve for all the unknowns. These are indeterminate structures. One can find answers in the example of the table by placing a scale under each leg. The problem arises because in real life, objects are not perfectly rigid. They bend slightly causing differences in the forces. Our equations require perfectly rigid objects which never deform. A perfectly rigid table would wobble if the feet were not all touching the floor. In the real world tables deform slightly. The floor deforms slightly. Everything deforms as the forces on it change. Atoms and molecules making up solids are bound together in a three dimensional lattice. The bonds between them act like springs. This means that all solids, to some extent, are elastic. An elastic material is one which can be deformed slightly and then return to the original shape. Objects can be deformed by tension, compression or shearing. All deformations are caused by a stress and produce a strain. pps by C Gliniewicz

5. The modulus is the ratio of the stress to the strain. For tensile and compression, the ratio is called Young’s modulus and is symbolized by E or Y. The shear modulus is calculated the same way. Shear occurs when the force is parallel to the surface area. Hydraulic stress is strain on a volume due to the three dimensional forces. The bulk modulus is the ratio of the hydraulic stress the volume of the object. The pressure is the bulk modulus multiplied by the strain. Yield strength is the value of the stress when the object is permanently deformed. The ultimate strength is the value of the stress when the object breaks. pps by C Gliniewicz