Center of gravity
The force of gravity on each little bit of an object is gmi where g is the local gravitational ‘constant’ and mi is the mass of the bit. For objects that are small compared to the radius of the earth (a reasonable assumption for all but a few special engineering calculations) the gravity constant is indeed constant from one point on the object to another (see box A.1 on page A.1 for a discussion of the meaning and history of g.)
The near-earth gravity forces acting on a system are equivalent to a single force, mg, acting at the system’s center of mass. For the purposes of calculating the net force and moment from near-earth (constant g) gravity forces, a system can be replaced by a point mass at the center of gravity.
The words ‘center of mass’ and ‘center of gravity’ both describe the same point in space. Although the result we have just found seems plain enough, here are two things to ponder about gravity when viewed as an inverse square law (and thus not constant like we have assumed) that may make the result above seem less obvious.
• The net gravity force on a sphere is indeed equivalent to the force of a point mass at the center of the sphere. It took the genius Isaac Newton 3 years to deduce this result and the reasoning involved is too advanced for this book.
• The net gravity force on systems that are not spheres is generally not equivalent to a force acting at the center of mass (this is important for the understanding of tides as well as the orientational stability of satellites).
A recipe for finding the center of mass of a complex system.
You find the center of mass of a complex system by knowing the masses and mass centers of its components. You find each of these centers of mass by
• Treating it as a point mass, or
• Treating it as a symmetric body and locating the center of mass in the middle, or
• Using integration, or
• Using the result of an experiment (which we will discuss in statics), or
• Treating the component as a complex system itself and applying this very recipe.
The recipe is just an application of the basic definition of center of mass (eqn. ??) but with our accumulated wisdom that the locations and masses in that sum can be the centers of mass and total masses of complex subsystems.
One way to arrange one’s data is in a table or spreadsheet, like below.
The first four columns are the basic data. They are the x, y, and z coordinates of the subsystem center of mass locations (relative to some clear reference point), and the masses of the subsystems, one row for each of the N subsystems.
One next calculates three new columns (5,6, and 7) which come from each coordinate multiplied by its mass. For example the entry in the 6th row and 7th column is the z component of the 6th subsystem’s center of mass multiplied by the mass of the 6th subsystem.
Then one sums columns 4 through 7. The sum of column 4 is the total mass, the sums of columns 5 through 7 are the total mass-weighted positions. Finally the result, the system center of mass coordinates, are found by dividing columns 5-7 of row N+1 by column 4 of row N+1.
Of course, there are multiple ways of systematically representing the data. The spreadsheet-like calculation above is just one way to organize the calculation.
Summary of center of mass
All discussions in mechanics make frequent reference to the concept of center of mass because
For systems with distributed mass, the expressions for gravitational moment, linear momentum, angular momentum, and energy are all simplified by using the center of mass.
Simple center of mass calculations also can serve as a check of a more complicated analysis. For example, after a computer simulation of a system with many moving parts is complete, one way of checking the calculation is to see if the whole system’s center of mass moves as would be expected by applying the net external force to the
system. These formulas tell the whole story if you know how to use them:
Comments: The advantage of finding the expression for in terms of r and as in eqn. (??) is that you can easily find the center of mass of any size circular cut-out located at any distance d on the x-axis. This is useful in design where you like to select the size or location of the cut-out to have the center of mass at a particular location.