## Sunday, January 3, 2016

### Centre of Mass of Identical Spheres in Contact Problems and Solutions

Three identical spheres are arranged in such a way that each one is in contact with the other two. We would like to identify the center of mass of the system. We shall take some point as the reference point with respect to which, we identify the center of mass position. Let the center of the first sphere is that reference point.

Center of mass is a point which represents the motion of the system when some external force is applied on it. We have all ready explained how to find the  coordinates of center of mass of a system in the previous posts. Here we are going to use that concept to solve this problem.

Let all spheres are having identical radius and their mass is not given in the problem. We can express the mass as the product of volume and density. As the spheres are made up of same materiel, their densities are same and volume is directly proportional to the cube of radii of the sphere.

To identify the center of mass of the system, first we need to identify the center of mass of each sphere and that shall be taken as the reference point of each sphere. We can use simple mathematical understanding as shown in the video lesson presented below, to identify the location of center of mass of each sphere.

The first sphere is at the orizion and the second sphere is on the x- axis. Its center of mass is at a distance double the radius of the sphere as shown. Similarly, we can identify the location of center of mass of third sphere.

By subsisting the values in the formula of center of mass, we can measure the center of mass of the system as shown below.

Center of mass of  T Shaped body

Let us consider a T shaped body having equal dimensions of rods connected. We would like to find the center of mass of that system. As the rod is one dimensional body, mass of the body is directly proportional to its length. We need to fix a reference point from which we can measure the center of mass of the system. Let the junction of the T shape is the reference point. It is the center of horizontal rod and its center of mass lies there itself.

The vertical rod has mass uniformly distributed over the entire length and hence its mass is at its geometrical center of the rod. Thus we can identify the location of center of mass of each rod as shown in the video below. In the place of mass, we can substitute the length of the rod itself. By substituting the values in the mathematical formula of center of mass, we can the center of mass of the system as shown in the below video.

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