Given a system of three particles with masses $m_1$, $m_2$, and $m_3$ located at positions $x_1$, $x_2$, and $x_3$ along the x-axis, which of the following expressions correctly gives the x-coordinate $x_{\mathrm{cm}}$ of the center of mass of the system?

• A. $x_{\mathrm{cm}}=\frac{m_1{x}_1+m_2{x}_2+m_3{x}_3}{m_1+m_2+m_3}$
• B. $x_{\mathrm{cm}}=\frac{x_1+x_2+x_3}{m_1+m_2+m_3}$
• C. $x_{\mathrm{cm}}=\frac{m_1+m_2+m_3}{x_1+x_2+x_3}$
• D. $x_{\mathrm{cm}}=\frac{m_1{x}_1+m_2{x}_2+m_3{x}_3}{3}$

Answer: (A)

Verified step by step guidance

1. Recall that the center of mass (COM) of a system of particles is the weighted average of their positions, where the weights are their respective masses.
2. Write down the general formula for the x-coordinate of the center of mass for multiple particles: \(x_{cm} = \frac{\sum m_i x_i}{\sum m_i}\), where \(m_i\) and \(x_i\) are the mass and position of the \(i^{th}\) particle.
3. For three particles with masses \(m_1\), \(m_2\), and \(m_3\) located at positions \(x_1\), \(x_2\), and \(x_3\) respectively, substitute these into the formula to get: \(x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3}\).
4. Understand that this formula gives the balance point of the system along the x-axis, taking into account both the positions and masses of the particles.
5. Compare this expression with the given options to identify the correct formula representing the x-coordinate of the center of mass.