Negative Mass Method — Excluded Portion

The idea

When a portion is cut out of a uniform body, the remaining body can be thought of as:

$\text{original body} + \text{negative mass at the location of the removed portion}$

If the original body has mass $m_1$ and CM at $x_1$, and the removed portion has mass $m_2$ and CM at $x_2$:

$x_{cm} = \frac{m_1 x_1 - m_2 x_2}{m_1 - m_2}$

The remaining mass is $m_1 - m_2$ and the CM is shifted away from the removed portion.

Why this works

The original body = remaining body + removed portion.

$m_1 x_1 = (m_1 - m_2) x_{cm,remaining} + m_2 x_2$

Solving for $x_{cm,remaining}$:

$x_{cm,remaining} = \frac{m_1 x_1 - m_2 x_2}{m_1 - m_2}$

Example 1: Circular disc with circular hole

A uniform disc of radius $R$, mass $M$, CM at origin. A circular hole of radius $R/2$ is cut, with centre at $(R/2, 0)$.

Mass of removed disc: $m_2 = M \times \frac{\pi(R/2)^2}{\pi R^2} = \frac{M}{4}$

CM of removed disc: $x_2 = R/2$

$x_{cm} = \frac{M \cdot 0 - \frac{M}{4} \cdot \frac{R}{2}}{M - \frac{M}{4}} = \frac{-MR/8}{\frac{3M}{4}} = \frac{-R/8}{3/4} = -\frac{R}{6}$

The CM shifts to $x = -R/6$ — away from the hole, toward the heavier side. Makes physical sense.

Example 2: Sphere with spherical cavity

A solid sphere of radius $R$, mass $M$, with a spherical cavity of radius $R/2$ scooped out, cavity centre at $(R/2, 0, 0)$.

Mass of removed sphere: $m_2 = M \times \frac{(R/2)^3}{R^3} = \frac{M}{8}$

CM of cavity: $x_2 = R/2$

$x_{cm} = \frac{M \cdot 0 - \frac{M}{8} \cdot \frac{R}{2}}{M - \frac{M}{8}} = \frac{-MR/16}{\frac{7M}{8}} = -\frac{R/16}{7/8} = -\frac{R}{14}$

Example 3: Square with square cutout

A uniform square plate of side $2a$, mass $M$, CM at origin. A smaller square of side $a$ is cut from one corner (say, at position $(a/2, a/2)$).

Mass of removed square: $m_2 = \frac{M}{4}$ (area ratio)

$x_{cm} = \frac{M \cdot 0 - \frac{M}{4} \cdot \frac{a}{2}}{M - \frac{M}{4}} = \frac{-Ma/8}{3M/4} = -\frac{a}{6}$

Similarly $y_{cm} = -\frac{a}{6}$.

The CM always shifts away from the removed portion

This is intuitive: removing mass from one side makes the body heavier on the other side, shifting the CM toward the remaining material.