Show that the center of mass is well defined, i.e., does not depend on the choice of the origin of reference for radius vectors.

Let's consider that we have a system of particles with masses \(m_1\), \(m_2\), ..., \(m_n\) and position vectors \(r_1\), \(r_2\), ..., \(r_n\) with respect to a reference origin \(O_1\). Let's choose another origin \(O_2\), located at position vector \(R\) with respect to origin \(O_1\). Now, we need to find the position vectors of the particles concerning the new origin \(O_2\).

Let's denote the position vectors of particles concerning origin \(O_2\) by \(r_1'\), \(r_2'\), ..., \(r_n'\). Then, based on vector addition, we obtain the following relationships between position vectors concerning the two origins: \(r_1' = r_1 - R\) \(r_2' = r_2 - R\) \(\cdots\) \(r_n' = r_n - R\)

Now, let's determine the center of mass concerning both origins. The center of mass concerning origin \(O_1\) is given by: \(R_{O_1} = \frac{m_1r_1 + m_2r_2 + \cdots + m_nr_n}{m_1 + m_2 + \cdots + m_n}\) Similarly, the center of mass concerning origin \(O_2\) is given by: \(R_{O_2} = \frac{m_1r_1' + m_2r_2' + \cdots + m_nr_n'}{m_1 + m_2 + \cdots + m_n}\)

Now, let's show that \(R_{O_1}\) and \(R_{O_2}\) are related in such a way that the location of the center of mass remains unchanged. Substitute the position vectors \(r_1'\), \(r_2'\), ..., \(r_n'\) from Step 2 into the expression for \(R_{O_2}\): \(R_{O_2} = \frac{m_1(r_1 - R) + m_2(r_2 - R) + \cdots + m_n(r_n - R)}{m_1 + m_2 + \cdots + m_n}\) \(R_{O_2} = \frac{(m_1r_1 + m_2r_2 + \cdots + m_nr_n) - (m_1 + m_2 + \cdots + m_n)R}{m_1 + m_2 + \cdots + m_n}\) Comparing this expression with the expression for \(R_{O_1}\), we can see that: \(R_{O_2} = R_{O_1} - R\) The expression shows that the position vector of the center of mass concerning the new origin \(O_2\) is the position vector of the center of mass concerning origin \(O_1\) minus the position vector of the new origin \(O_2\) concerning origin \(O_1\). This relationship implies that the center of mass is well-defined and does not depend on the choice of the origin of reference for radius vectors.