Systems

Students are often confused when we say we are speaking of a “system”. It is sometimes difficult to delineate the boundaries of a system; or to determine what should be included versus what are extraneous details that we do not wish to bother with.

System properties are determined by the interactions between objects wtihin the system. If the properties or interactions of the constituent objects within a system are not important in modeling the behavior of the macroscopic system, the system can itself be treated as a single object. Systems may allow interactions between constituent parts of the system and the environment, which may result in the transfer of energy or mass. INdividual objects within a chosen system may behave differently from each other as well as from the system as a whole. The internal structure of a system affects the analysis of that system. As variables external to a system are changes, the system’s substructure may change.

Center of mass

While notions of a system may seem kind of abstract at this point, we feel generally OK looking at a complicated thing like a ship or submarine or airplane or human, and imagining an idealized point mass; something of the same mass as the object we were considering but with all that mass concentrated at a point. The point where such mass is located is called the center of mass. A system can be modeled as a singular object that is located at the system’s center of mass.

For objects or systems with symmetrical mass distributions, the center o fmass is located on lines of symmetry. For a system of discrete masses, the location of the center of mass along a given axis can be calculated using \[\begin{equation} \bar{x}_{cm} = \dfrac{\Sigma m_i x_i}{\Sigma m_i}. \end{equation}\]

If we imagine taking the limit of ever smaller and smaller chunks of an object, we get to calculus. As an example, the linear mass density of a rod or other linear rigid body is the derivative of the rod’s mass with respect to the position of the differential mass element on the rigid body: \[\begin{equation} \lambda = \dfrac{d}{dl} m(l). \end{equation}\]

If a function of mass density is given for a solid, the total mass can be determined by integrating the mass density over the length (one dimension), area (two dimensions), or volume (three dimensions) of the solid. For example, \[\begin{equation} M_{total} = \int \rho(r) dV \end{equation}\]