§12.2 — Center of Mass and Center of Gravity

If you push a pencil at exactly the right spot, it stays balanced. Push at the wrong spot and it tips over. What makes the difference? The center of mass.

Definition of Center of Mass

For a system of particles (or a continuous object), the center of mass is the special point where all the mass effectively "lives."

For Discrete Particles

If \( n \) particles with masses \( m_1, m_2, \ldots, m_n \) are at positions \( \vec r_1, \vec r_2, \ldots, \vec r_n \):

\[ \vec r_{cm} = \frac{m_1 \vec r_1 + m_2 \vec r_2 + \cdots + m_n \vec r_n}{m_1 + m_2 + \cdots + m_n} \]

Or compactly:

\[ \vec r_{cm} = \frac{\sum_i m_i \vec r_i}{\sum_i m_i} = \frac{\sum_i m_i \vec r_i}{M} \]

where \( M = \sum_i m_i \) is the total mass.

For Continuous Objects

For a continuous object:

\[ \vec r_{cm} = \frac{\int \vec r \, dm}{M} \]

Simple Example: Two Masses

Two masses on the x-axis: \( m_1 = 2 \) kg at \( x_1 = 0 \) and \( m_2 = 3 \) kg at \( x_2 = 4 \) m.

\[ x_{cm} = \frac{2 \times 0 + 3 \times 4}{2 + 3} = \frac{12}{5} = 2.4 \text{ m} \]

The center of mass is at 2.4 m (closer to the heavier mass).

Center of Gravity

The center of gravity is the point where gravitational force effectively acts.

In a uniform gravitational field:

\[ \vec r_{cg} = \frac{\sum_i m_i g \vec r_i}{\sum_i m_i g} = \frac{\sum_i m_i \vec r_i}{\sum_i m_i} = \vec r_{cm} \]

Therefore: in a uniform gravitational field, center of gravity = center of mass.

Mathematical note: if \( g \) is constant (always true near Earth's surface), center of gravity and center of mass coincide.

Three Types of Equilibrium

If you suspend an object from different points, what happens?

1. Stable Equilibrium

Suspend the object from a point above its center of mass.

• If the object tilts slightly, gravity's torque restores it
• The object oscillates back to center

Example: a simple pendulum

2. Unstable Equilibrium

Suspend the object from a point below its center of mass (if possible!).

• If it tilts slightly, gravity's torque tips it further
• The object falls

Example: a pencil balanced on its tip (practically impossible)

3. Neutral Equilibrium

Suspend the object from its own center of mass (if possible).

• No torque exists
• The object is in equilibrium at any orientation

Example: a sphere floating in water

Stability of Objects on a Surface

For an object resting on a surface:

• The object is stable if its center of mass lies within the base of support
• The object becomes unstable if the center of mass moves outside the base

Example: Tilting a Cylinder

A uniform cylinder on a tilted surface.

• Center of mass is at mid-height
• The cylinder remains in equilibrium as long as the vertical line through the center of mass falls within the base
• If tilted too far, the vertical line exits the base; the cylinder rolls

Calculating Center of Mass for Regular Shapes

For simple (symmetric) shapes, the center of mass is at the geometric center.

Shape	Center of Mass
Uniform rod	middle of rod
Disk / sphere	geometric center
Triangle	intersection of medians (1/3 from base)
Solid hemisphere	\( \frac{3R}{8} \) from base

Role of Center of Mass in Equilibrium

For a Rigid Body

If all external forces pass through a single axis at the center of mass:

• First condition: \( \sum \vec F = 0 \) ✓
• Second condition: \( \sum \vec \tau_{cm} = 0 \) ✓ (all forces pass through axis)

In General

For equilibrium to exist, both conditions must hold: forces balance AND torques balance. The center of mass is the natural point for computing torques in equilibrium problems.

What You Should Know

• Center of mass is the effective "balance point": \( \vec r_{cm} = \frac{\sum m_i \vec r_i}{M} \)
• In uniform gravity: center of gravity = center of mass
• Three equilibrium types: stable (above), unstable (below), neutral (at itself)
• Objects balance if center of mass lies within the base of support

Preview of §12.3

Now we know where the center of mass is. Next, let's see how to apply this to rigid bodies in equilibrium — forces at different points, different torques.

📚 See also: Halliday Vol 1, Ch 12, §12.2 — Center of mass and gravity. 🔗 Reference: §8 (Center of mass in motion), §12.1 (Equilibrium conditions).

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