The Center of Mass — Question 8

Question 8

A hollow shell plus an offset bead

A uniform spherical shell of mass $9.0\,\mathrm{kg}$ has center $O$ and radius $0.50\,\mathrm m$. A compact $1.0\,\mathrm{kg}$ bead is attached at a point on its surface. The assembly center is measured at $(0.030,0.040,0)\,\mathrm m$ relative to $O$.

1. Find the bead position and check that it lies on the shell.

2. Could the same measured center arise from a bead at $(0,0,0.50)\,\mathrm m$?

Question 8 - Solution

A hollow shell plus an offset bead

(a) Derivation and calculation

Spherical symmetry makes the shell’s first moment vanish about $O$: $$(9+1)\vec R=9\vec0+1\vec r_b.$$ $$\boxed{\vec r_b=10(0.030,0.040,0)=(0.30,0.40,0)\ \mathrm m},\qquad |\vec r_b|=\sqrt{0.30^2+0.40^2}=0.50\ \mathrm m.$$

(b) Result and interpretation

The alternative bead location would instead produce $$\boxed{\vec R_{\mathrm{alt}}=\frac{1}{10}(0,0,0.50)=(0,0,0.050)\ \mathrm m}.$$ Its magnitude matches the measured center magnitude, but its direction does not. The full vector measurement excludes it.

Physical check

The combined center lies one tenth of the way from the shell center to the bead.