The center of mass (or center of gravity) is the point that represents the overall motion of a system. For a rigid body or system of particles, it is the point where, if all mass were concentrated there, the system would move identically to the real system. This concept elegantly simplifies the motion of complex objects.
Mathematical definition of center of mass
For a system of \( n \) particles with masses \( m_1, m_2, \ldots, m_n \) at positions \( \mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_n \), the position of the center of mass is defined as:
\[ \mathbf{r}_{\text{cm}} = \frac{\sum_i m_i \mathbf{r}_i}{\sum_i m_i} = \frac{\sum_i m_i \mathbf{r}_i}{M} \]
where \( M = \sum_i m_i \) is the total mass of the system.
Cartesian coordinates
Breaking down into x, y, z components:
\[ x_{\text{cm}} = \frac{\sum_i m_i x_i}{M}, \quad y_{\text{cm}} = \frac{\sum_i m_i y_i}{M}, \quad z_{\text{cm}} = \frac{\sum_i m_i z_i}{M} \]
Continuous objects
For a continuous body (not a discrete system of particles), the discrete sums are replaced by integrals:
\[ x_{\text{cm}} = \frac{\int x \, dm}{M} = \frac{\int x \, \rho(\mathbf{r}) \, dV}{M} \]
where \( \rho(\mathbf{r}) \) is the mass density and the integral is taken over the entire volume of the body.
Example 1: Three particles on a line
Three particles lie on the \( x \)-axis:
- \( m_1 = 2 \) kg at \( x_1 = 0 \) m
- \( m_2 = 3 \) kg at \( x_2 = 4 \) m
- \( m_3 = 5 \) kg at \( x_3 = 6 \) m
Solution:
Total mass: \( M = 2 + 3 + 5 = 10 \) kg
Center of mass position:
\[ x_{\text{cm}} = \frac{2(0) + 3(4) + 5(6)}{10} = \frac{0 + 12 + 30}{10} = \frac{42}{10} = 4.2 \text{ m} \]
The center of mass is at \( x = 4.2 \) m. Note that it is not simply midway between the first two particles, but shifted toward the heavier masses (especially \( m_3 \)).
Example 2: Two objects on a plane
Two objects on the \( xy \)-plane:
- Object A: mass \( m_A = 4 \) kg at \( \mathbf{r}_A = (2, 3) \) m
- Object B: mass \( m_B = 6 \) kg at \( \mathbf{r}_B = (5, 1) \) m
Solution:
Total mass: \( M = 4 + 6 = 10 \) kg
Center of mass coordinates:
\[ x_{\text{cm}} = \frac{4(2) + 6(5)}{10} = \frac{8 + 30}{10} = 3.8 \text{ m} \]
\[ y_{\text{cm}} = \frac{4(3) + 6(1)}{10} = \frac{12 + 6}{10} = 1.8 \text{ m} \]
The center of mass is at the point \( (3.8, 1.8) \) m.
Significance of center of mass
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Represents translational motion: If an external force acts on the system, the center of mass moves as if all the mass were concentrated there.
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Decouples motion: Complex object motion can be separated into motion of the center of mass (translational) and motion about the center of mass (rotational).
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Conservation laws: The center of mass is crucial in conservation of momentum (Chapter 9) and energy.
Important note: Symmetry
For symmetric bodies with uniform density, the center of mass coincides with the geometric center. Examples:
- Sphere: at geometric center
- Cylinder: on axis, at midpoint
- Rectangular plate: at intersection of diagonals
But for asymmetric shapes or non-uniform density, integration is necessary.
What you should be able to do
- Define center of mass for discrete and continuous systems
- Calculate \( x_{\text{cm}}, y_{\text{cm}}, z_{\text{cm}} \) from weighted sums
- Recognize symmetry and apply it to reduce calculation
- Set up and evaluate integrals for complex bodies
Preview of §9.2
Now that we have defined center of mass, the next step is motion. How does Newton's second law apply to the center of mass?
📚 See also: Halliday Vol 1, Ch 9, §9.1.
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