The Center of Mass — Question 10

Question 10

Bounds from incomplete mass maps

A $2.0\,\mathrm{kg}$ component is known only to lie entirely within $0\le x\le1.0\,\mathrm m$. A $3.0\,\mathrm{kg}$ component lies within $4.0\le x\le6.0\,\mathrm m$. Their densities are nonnegative; no uniformity is assumed.

1. Find sharp bounds on the combined center.

2. Can one report the midpoint of those bounds as an exact center? Explain how the extremes can be approached.

Question 10 - Solution

Bounds from incomplete mass maps

(a) Derivation and calculation

Nonnegative density places each component center within its support: $$0\le x_1\le1,\qquad4\le x_2\le6,\qquad X=\frac{2x_1+3x_2}{5}.$$ Both weights are positive, hence $$\boxed{X_{\min}=\frac{2(0)+3(4)}5=2.4\ \mathrm m},\qquad \boxed{X_{\max}=\frac{2(1)+3(6)}5=4.0\ \mathrm m}.$$

(b) Result and interpretation

The midpoint of the interval is $$X_{\mathrm{mid}}=\frac{2.4+4.0}{2}=3.2\ \mathrm m,$$ which is a possible value but is not determined by the support bounds. Concentrating mass near the left or right endpoints approaches the corresponding bound; ideal point masses attain it.

Physical check

No assumption of uniformity is needed for the bounds, but additional density information is needed for an exact center.