The major, intermediate, and minor triaxial dimensions of an agricultural grain kernel are $a = 6.0\text{ mm}$, $b = 4.0\text{ mm}$, and $c = 3.0\text{ mm}$, respectively. The sphericity of the grain kernel is ________ (round off to 3 decimal places).
Answer Key:
0.693
Step-by-Step Solution & Explanation
1. Geometric mean diameter $D_g$:
$D_g = (a \cdot b \cdot c)^{1/3} = (6.0 \times 4.0 \times 3.0)^{1/3} = (72.0)^{1/3} \approx 4.160168\text{ mm}$
2. Sphericity $\phi$ is defined as the ratio of geometric mean diameter to longest diameter $a$:
$\phi = \frac{D_g}{a} = \frac{4.160168}{6.0} = 0.693361 \approx 0.693$