Question

The minimum possible void ratio of uniformly graded spherical grains is _______

(a) 0.20

(b) 0.25

(c) 0.30

(d) 0.35

I got this question in an international level competition.

My question is taken from Packing of Uniform Spheres topic in chapter Preliminary Definitions and Relationships of Geotechnical Engineering I

Answer 1

The correct answer is (d) 0.35

To explain: Consider a unit cube of soil having spherical particles of diameter d.

Volume of the solids Vs= (π/6)=0.5236

The total volume V=(1-cosα)*(√(1+2cosα))

Minimum void ratio is possible in densest state with α=60°

∴  V=(1-cos60°)*(√(1+2cos60°))

V=0.7071

Volume of the voids Vv=V-Vs=0.7071-0.5236

Vv=0.1835

Voids ratio e= Vv/ Vs=0.1835/0.5236

∴ e=0.35.