The anti-matter method

How do you find the centre of mass of an object with a hole in it? The trick is elegant: use the anti-matter method, treating the hole as an object of negative mass.

The idea is that an object with a hole is equivalent to the full object plus a “plug of anti-matter” that exactly fills the hole with negative mass. Adding the two, the plug’s matter cancels the real matter in the hole’s region, and what remains is the pierced object. Since the CM is linear in the masses, you can apply the usual formula by putting in a contribution with a minus sign.

Principle — Anti-matter method

Object with a hole = whole object ($+m_1$, at $\vec{s}_1$) $+$ hole ($-m_2$, at $\vec{s}_2$): $\ev{\vec{s}_\text{CM} = \frac{m_1\,\vec{s}_1 - m_2\,\vec{s}_2}{m_1 - m_2}}$

The denominator is $m_1 - m_2$, i.e. the effective mass left once the hole is removed. The numerator subtracts the contribution of the missing region. The result: the CM shifts to the side opposite the hole, because it is as if mass had been removed from one side.

Anti-matter method: the disc with a hole = whole disc $+$ hole of negative mass. The CM shifts to the side opposite the hole.

The worked exercises Circle with a hole and Disc with a tangent hole apply the method to concrete cases.

Links

Topics: Centre of mass Concepts: Centre of mass Skills: Calculating the centre of mass

Related exercises: Worked exercise — The cat on the trolley · Rocket exploding mid-trajectory · Disc with a tangent hole