Section 4.1: Center of Mass
The center of mass is the weighted average position of all the mass in a system. It simplifies the analysis of motion for extended bodies and systems of particles.
Center of Mass (Discrete Particles):
\( \vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i} \)
- \( m_i \): Mass of the i-th particle
- \( \vec{r}_i \): Position vector of the i-th particle
Center of Mass (Continuous Body):
\( \vec{R}_{cm} = \frac{1}{M} \int \vec{r} \, dm \)
- \( M \): Total mass of the body
- \( dm \): Infinitesimal mass element at position \( \vec{r} \)
Example 1
Two particles, m₁ = 3 kg at x₁ = 2 m and m₂ = 5 kg at x₂ = 6 m, are on a line. Find the center of mass.
\( x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = \frac{3*2 + 5*6}{3+5} = \frac{36}{8} = 4.5 \, \text{m} \)
Practice Problems
- Three particles of masses 2 kg, 4 kg, 6 kg are at positions 1 m, 3 m, 5 m along a line. Find the center of mass.
- A uniform rod of length 4 m has mass 10 kg. Determine its center of mass.
- Two particles with masses 3 kg and 7 kg are at coordinates (0,0) and (4,0). Find the center of mass.
- A uniform square plate of side 2 m has mass 8 kg. Locate its center of mass.
- Four particles of equal mass at vertices of a rectangle: find the center of mass.