A grain kernel is modeled as a triaxial ellipsoid with principal semi-axes lengths of $a = 6.0\text{ mm}$ (major axis), $b = 3.0\text{ mm}$ (intermediate axis), and $c = 2.0\text{ mm}$ (minor axis). The geometric mean diameter ($D_g$) and sphericity ($\phi$) of the kernel are, respectively:
Options:
(A) $2.45\text{ mm}$ and $0.41$
(B) $3.30\text{ mm}$ and $0.55$
(C) $3.67\text{ mm}$ and $0.61$
(D) $4.12\text{ mm}$ and $0.69$
Answer Key:
B
Step-by-Step Solution & Explanation
For a triaxial ellipsoid with principal dimensions $a$, $b$, and $c$:
1. Geometric mean diameter ($D_g$):
$D_g = (a \cdot b \cdot c)^{1/3} = (6.0 \times 3.0 \times 2.0)^{1/3} = (36.0)^{1/3} \approx 3.3019\text{ mm}$
2. Sphericity ($\phi$):
$\phi = \frac{D_g}{a} = \frac{3.3019}{6.0} \approx 0.5503$
Thus, $D_g \approx 3.30\text{ mm}$ and $\phi \approx 0.55$.