The power of integral calculus is once again exploited in this entry, this time to determine the centers of mass of one- and two-dimensional objects. Before getting to that, however, some preliminary “discourse” will be engaged in to set the stage.
Everyone (of my age at least) can relate to the scenario involving two children playing on a see-saw. If the children have equal mass and are sitting an equal distance from the fulcrum, they can achieve perfect balance; the fulcrum in this scene is located at the center of mass. If, however, two differing masses are involved (all other things remaining equal), the side containing the greater mass will rotate downward on the fulcrum. This brings us to a very important term for this and other concepts involving rotational motion. This term is called the MOMENT of a force and is defined below:
The moment of a force is a measure of its tendency to rotate about an axis or a point. The moment can be influenced by two quantities: the object’s mass and its distance from the axis or point of rotation; this distance can be referred to as the moment arm.
Moment=Force x Distance , with units measured in ft-lb, kgf-m, etc.
In our example, when two children of the same mass are positioned an equal distance on opposing sides of the fulcrum, a state of equilibrium is achieved. If a bird suddenly joins in the fun and lands on the head of one of the participants, the mass on that end is increased and begins to rotate downward; this increase in mass has created a moment. A moment can also be created if one participant increases his/her position relative to the fulcrum. The moment arm on that side has now been lengthened, thereby creating a moment.
When finding the center of mass in one dimension, the same principles apply; this is a very straight forward procedure if the object has uniform mass density over its entire length. Complications arise, however, if the object’s mass density is not uniform throughout. To address this issue, the object is analyzed as a collection of very small points, each having its own mass and unique position (moment arm) within the main object; each of these will be referred to as “point-mass”. As in the playground scenario described earlier, the further each point-mass is situated from the axes (or point) of rotation, the greater its contribution will be to the moment and, thereby, its influence on the location of the center of mass. The calculation for center of mass is built upon the concept of weighted averages; while the most frequently occurring outcomes have a significant say with respect to the overall average, the extreme outliers can also have a measurable impact.
Before weighted averages can be referenced in this context, the notation and the underlying concept that will be utilized throughout must be introduced. This is initiated below in the Mean Value Theorem of integral calculus.
Mean Value Theorem of Integral Calculus
NOTE: In the calculations that follow, x- and y-components of the moment appear. Since centers of mass occur at a point of equilibrium, force due to gravity is ignored and omitted from the units chosen to represent those quantities. I wanted clarification on this item here since moments are once again called upon when dealing with torque. In that application, force is included in the units of measurement when describing moments of inertia.
A reference was made earlier to “point-masses” and their relative position within the object containing them. A direct connection between this and weighted averages exists and is presented below.
Weighted Average
The end result in the first two examples in the following image are common sense and serve as a “trial run” on the theory developed above; all three can be related directly to our playground scenario described earlier.
One Dimensional Center of Mass
While limiting ourselves here to one-dimension would be silly, attempting to illustrate centers of mass in three dimensions on a 2-D surface could be considered reckless. For this reason, two-dimensions will be the extent to which this topic is pursued here.
Two Dimensions: x-component
Two Dimensions: y-component
In the examples that follow, centers of mass are determined using the theory developed above. Interactive links also provide the opportunity to change one or more parameters in these examples to observe variations in the various integrals involved.
Constant Function
Click on the link provided here to interact with centers of mass on one dimension.
Linear Function
One Image from the exploration that follows……
Click on the link provided here to explore centers of mass defined by linear functions.
Quadratic Function
Click on the link provided here to explore centers of mass defined by quadratic functions.
The examples that follow have mass density increasing exponentially along the x-axis. With exponentials in the mix, the need for a new method of integration (by parts) emerges; the power of WolframAlpha is also introduced to do the “heavy lifting”.
Constant Function: Exponential Increase in Mass Density
The x-component of the center of mass in the example above can be calculated manually using integration by parts; this procedure is included here.
Integration by Parts
Click on the link provided here to explore centers of mass resulting from an exponential increase in mass density.
While integration by parts can be exploited to evaluate all integrals of this form, the process can become a time-consuming one. The following example is one result obtained from the exploration directly above; it is included here with the intent of introducing students to the power of WolframAlpha. The x-component of the center of mass is shown in a screenshot of the WolframAlpha app available on any device. Interested students have the option of verifying this and other results manually using integration by parts.
Quadratic Function: Exponential Increase in Mass Density
WolframAlpha App
I’ve included a link here to the web-based version of the app shown above. To verify the y-component of the center of mass in the final example shown above, click on WolframAlpha.
Thanks for reading.
References
Courant, Richard., John, Fritz (1999). Introduction to Calculus and Analytics: Classics of Mathematics. New York, NY: Springer-Verlag Berlin Heidelberg.
Larson, R., Hostetler, R. P., & Edwards, B. H. (1995). Calculus of a Single Variable: Early Transcendental Functions. Lexington, MA: D.C. Heath.
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