Balancing A Lamina With Point Masses

What It Means to Balance a Lamina

A lamina is a thin planar object whose mass is distributed across a region. If its mass is not centered at the origin, its first moments about the coordinate axes are generally nonzero. One way to balance it about the origin is to add a point mass whose first moments cancel those of the lamina.

The key is to keep track of which coordinate contributes to which first moment. A point at \((a,b)\) contributes \(mb\) to the first moment about the \(x\)-axis, and \(ma\) to the first moment about the \(y\)-axis. Both combined first moments must be zero for the combined center of mass to be at the origin.

First Moments and the Centroid

For a lamina occupying a planar region \(D\) with density \(\rho(x,y)\), its mass is \(M=\iint_D \rho(x,y)\,dA\). Its first moment about the \(x\)-axis weights each small mass by its vertical coordinate; its first moment about the \(y\)-axis weights each small mass by its horizontal coordinate. With this convention,

When \(M>0\), the centroid coordinates are defined by \(\bar{x}=Q_y/M\) and \(\bar{y}=Q_x/M\). Equivalently, the moments are \(Q_x=M\bar{y}\) and \(Q_y=M\bar{x}\). This reversal of subscripts is worth noticing: the moment about the \(x\)-axis uses \(y\), the distance from that axis.

A point mass \(m\) at \((a,b)\) contributes \(mb\) to \(Q_x\) and \(ma\) to \(Q_y\). First moments add, so after the point mass is added, the combined first moments are \(Q_x+mb\) and \(Q_y+ma\). The combined mass is \(M+m\), which is positive when \(M>0\) and \(m\ge 0\). Therefore, the combined centroid is the origin exactly when both combined first moments are zero.

Why the Equations Work

The result follows directly from the definitions, and it explains why both equations are needed. The centroid of a collection of masses has coordinates equal to the total first moment about the \(y\)-axis divided by the total mass, and the total first moment about the \(x\)-axis divided by the total mass. After adding the point mass, those coordinates are therefore

Since \(M+m>0\), the first fraction is zero exactly when \(M\bar{x}+ma=0\), and the second is zero exactly when \(M\bar{y}+mb=0\). Thus the combined centroid is at the origin if and only if both balancing equations hold. Cancelling just one moment would put the centroid on one axis, not necessarily at the origin.

Finding the Location When the Mass Is Known

If \(m>0\) is given and the location is unrestricted, solve the two balancing equations for \(a\) and \(b\). The point must be located opposite the lamina’s centroid, with its distance from the origin scaled by the ratio of the lamina mass to the added mass.

Finding the Mass at a Prescribed Location

Sometimes the location \((a,b)\) is fixed and the mass is unknown. In that case, both equations must yield the same value of \(m\), and that value must be positive for an ordinary added point mass. When \(a\) and \(b\) are both nonzero, the candidate values are \(-M\bar{x}/a\) and \(-M\bar{y}/b\); they must agree. If either coordinate is zero, use the original balancing equations rather than dividing by that coordinate.

When No Single Point Mass Works

A prescribed point does not always allow a balance. The required cancellation must happen in both coordinate directions with the same mass. In vector form, the condition is \(m(a,b)=-M(\bar{x},\bar{y})\). For \(m>0\), the added point’s position vector must point in the direction opposite the lamina’s centroid vector, and its distance must have the appropriate scale. If those two vectors are not proportional in the required way, one scalar mass cannot satisfy both equations.

A Common Sign and Coordinate Mistake

A frequent error is to interchange which coordinate contributes to each axis moment. For the lamina in the first example, the original moments are \(Q_x=-12\text{ kg}\cdot\text{m}\) and \(Q_y=24\text{ kg}\cdot\text{m}\). A mistaken calculation might assign \(ma\) to the \(x\)-axis and \(mb\) to the \(y\)-axis. It would then produce the incorrect location \((2\text{ m},-4\text{ m})\) for a \(6\text{ kg}\) mass.

Check that proposed location with the correct definitions: its combined \(x\)-axis moment is \(-12+(6)(-4)=-36\text{ kg}\cdot\text{m}\), and its combined \(y\)-axis moment is \(24+(6)(2)=36\text{ kg}\cdot\text{m}\). Neither is zero. The correction is to remember that \(Q_x\) uses the \(y\)-coordinate and \(Q_y\) uses the \(x\)-coordinate; applying the correct equations gives \((-4\text{ m},2\text{ m})\).

Practical Checks Before Accepting a Balance

After calculating a location or mass, substitute it into both combined-moment equations. A solution that makes only one equation zero is not a balance about the origin. Also check that the calculated mass is positive when the problem asks for an added physical point mass. A negative result indicates that the prescribed location and lamina do not permit the requested balance by adding positive mass.

These calculations use first moments, not moments of inertia. First moments determine the centroid and whether the combined center of mass is at the origin. They do not by themselves describe how mass is distributed relative to a rotational axis.