In this first post (of what is likely to be a long series) I will delve into center of mass.
Center of Mass
Wikipedia defines center of mass as follows:
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point at any given time where the weighted relative position of the distributed mass sums to zero.
It goes onto say:
This is the point to which a force may be applied to cause a linear acceleration without an angular acceleration. Calculations in mechanics are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of Newton's laws of motion.
So the need to understand the center of mass in physics simulations is clear.
In practice, we would be dealing with symmetrical shapes, like a circle or a rectangle. Hopefully it will be intuitively obvious that the center of mas of circle is the center of the circle, or for a rectangle, where the center of mass is where the diagonals intersect. This is the result of the symmetry principle, which states that:
If a region R is symmetric about a line l, then the centroid of R lies on l.This will get us a very long way in our simple 2D physics simulations. The rest of this post - which is about how the center of mass of a shape / lamina can be derived - can be skipped if you are not fond of maths. However, it may provide useful background when we come to discuss moment of inertia.
With that, let's delve into this subject very slow, first by looking at the center of mass in a one-dimensional context.
Centre of Mass of a 1D system of particles
Consider a long thin rod of negligible mass ersting on a fulcrum. Now suppose we place objects having masses m1 and m2 at distances x1 and x2 from the fulcrum, rather like a seesaw.
Intuitively, it should be clear that the rod will balance if and only if m1 * x1 = m2 * x2. The center of mass (shown as x=0) is the balanceing point of the system, or the point where the fulcrum should be placed to make the system balance, ie the center of mass of this system.
The numerator is called the first moment of the system with repspect to the origin (if the context is clear, first moment will be referred simpy as moment of the system).
The denominator is the total mass of the system.
In general, if n masses m1, m2,...,mn are palced at x1, x2,..,xn, then the center of mass of the system is given by:
Center of mass in 2D
We can extend the above to 2D context, or put another way, locate the mass of a system of point masses in a plane.
Again, we need to find the moment of the masses.
In 2D, the moment of a mass around the x-axis which is free to rotate is given by the product of the mass of the element and the distance (displacement) from the axis, ie y.
Similarly, the moment of a mass around the y-axis is the product of its mass and its distance from the axis, ie x.
So, if we let m1 be a point mass located at point (x1, y1) in the plane and m2 be a point mass located at (x2, y2), like below, then then the moment Mx of mass m1 with respect to the x-axis is given by Mx = m1 * y1. Similarly, the moment My of mass m1 with respect to the y-axis is given by My = m1 * x1.
If we have m1, m2,.., mn masses located in the xy-opakne at points (x1, y1), (x2, y2),..., (xn, yn), and let M be the total mass of the system. Then the moments Mx, and My of the system with repsect to the x- and y-axes, respectively are given by:
The x-coordinate of the centre of mass is the sum of the moments about the y-axis, divided by the total mass of the system
The y-coordinate of the centre of mass is the moment of the whole system around the x-axis divided by the total mass of the system.
Center of Mass of Thin Plate
So far we have looked at systems of point masses on a line and in a plane. However, what we want is to find the center of mass of a "shape" in 2D, or systems in which the mass of the system is distrbuetd continuously acrsos a thin sheet of material, where the sheet is thin enough that it an be treaetd as if it is 2D - such sheet is called laimina. SSo how do we calculate the center of mas of a limina?
Laminas are often represented by a two-dimensional region in a plane. The geometric center of such a region is called its centroid. Assuming that the density of the lamina is constant, the center of mass of the lamina depends only on the shape of the corresponding region. As with the system of point mases, we need to find the total mass of the lamina, as well as the moments of the lamina with respect to the x- ad y-axes.
To do this, we imagine diving the region into elemental mases each with mass dm, with the center of each mass specified relative to the reference coordinate system axes. The first moment of the "region" with respect to the x-axis is given by:
and
The coordinate of the center of mass (x, y) are:
and
Centroid
Quite often you will see the term centroid used interchangeably with center of mass.
Wikipedia explains as follows:
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any object in n-dimensional Euclidean space.
In geometry, one often assumes uniform mass density, in which case the barycenter or center of mass coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin.
The key being that if density is constant through the region, the terms mean the same thing. The calculation of the centroid becomes simpler, as follows:
As mentioned earlier, for symmetrical regions identifying the centroid is simple and intuitively obvious. But for non-symmetric shapes, you can either look it up (eg you can find a list of centroids here) or derive it yourself. We will try a few ourselves in the following posts.
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