The Center of Mass — Question 5

Question 5

Reassembling grouped masses

A $6.0\,\mathrm{kg}$ assembly has center $(1,2)\,\mathrm m$ and a $4.0\,\mathrm{kg}$ assembly has center $(6,-3)\,\mathrm m$. Their internal shapes are unspecified.

1. Find the center of mass of both assemblies together.

2. The first assembly is reshaped without moving its own center. Can the overall center change? Prove your answer.

Question 5 - Solution

Reassembling grouped masses

(a) Derivation and calculation

The group centers contain all required first-moment information: $$M=6+4=10\ \mathrm{kg},\qquad M\vec R=6(1,2)+4(6,-3)=(30,0)\ \mathrm{kg\,m}.$$ $$\boxed{\vec R=(3,0)\ \mathrm m}.$$

(b) Result and interpretation

For any grouping of particles, $$\sum_i m_i\vec r_i=\sum_g\sum_{i\in g}m_i\vec r_i =\sum_g M_g\vec R_g.$$ Reshaping a group with fixed mass and fixed group center leaves this sum unchanged. Therefore $\Delta\vec R=\boxed{\vec0}$.

Physical check

The combined center divides the line between the two group centers in the inverse ratio of their masses.