The Center of Mass — Question 1

Question 1

Locating an unseen load

Three compact loads lie on a light horizontal rail. The masses at $x=0$ and $x=3.0\,\mathrm m$ are $2.0$ and $4.0\,\mathrm{kg}$. A third load of unknown positive mass is at $x=6.0\,\mathrm m$. The assembly has $x_{\mathrm{cm}}=4.0\,\mathrm m$.

1. Find the unknown mass.

2. Give the possible range of $x_{\mathrm{cm}}$ as that positive mass varies. Is $x_{\mathrm{cm}}=1.5\,\mathrm m$ possible?

Question 1 - Solution

Locating an unseen load

(a) Derivation and calculation

Use the mass-weighted first moment: $$X=\frac{\sum m_ix_i}{\sum m_i}=\frac{2(0)+4(3)+6m}{6+m}=4.$$ $$24+4m=12+6m\quad\Rightarrow\quad\boxed{m=6.0\ \mathrm{kg}}.$$

(b) Result and interpretation

Rewrite the center as a function of positive finite $m$: $$X(m)=6-\frac{24}{6+m},\qquad\frac{dX}{dm}=\frac{24}{(6+m)^2}>0.$$ $$\lim_{m\to0^+}X=2,\quad\lim_{m\to\infty}X=6 \quad\Rightarrow\quad\boxed{2<X<6\ \mathrm m}.$$ A center at 1.5 m is outside this attainable interval.

Physical check

At the inferred mass, the first moment is 48 kg m and the total mass is 12 kg.