# Soil Mechanics Questions and Answers – Packing of Uniform Spheres

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This set of Soil Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “Packing of Uniform Spheres”.

1. The loosest stable arrangement of equal sized spheres is obtained when the sphere centres form a ______ space lattice.
a) triangular
b) rhombohedral
c) rectangular
d) hexagonal

Explanation: When the sphere centres form a rectangular space lattice, then they form a packing known as cubic packing, wherein each sphere is in contact with six surrounding neighbouring spheres.

2. The porosity of cubic packing is ______
a) 42.66%
b) 47.64%
c) 25.95%
d) 30.21%

Explanation: Consider a unit cube of soil having spherical particles of diameter d.
Volume of each spherical particle = (π/6)d3
Total volume of container = 1*1*1=1
No. of solids in the container = (1/d)*(1/d)*(1/d)=(1/d3)
Volume of the solids Vs = (π/6)*d3*(1/d3)=(π/6)
Volume of the voids Vv = 1-(π/6)
Voids ratio e = (1-(π/6))/(π/6)
Porosity n = e/(1+e) = 0.9099/(+0.9099) = 0.4764
In percentage, n = 47.64%.

3. The angle of orientation of the spheres in the maximum possible voids ratio is ______
a) 60°
b) 90°
c) 30°
d) 75°

Explanation: The soil will have maximum possible voids when the soil grains are arranged in a cubical array of spheres. In a cubical array, the angle of orientation α = 90°.
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4. The porosity of intermediate packing factor is given by ______
a) n = 1-[1/(6*(1-cosα)*( √(1-2cosα) )]
b) n = (1-cosα)*( √(1+2cosα) )
c) n = 1-[π/(6*(1-cosα)*(√(1-2cosα))]
d) n = 6*(1-cosα)*(√(1-2cosα))

Explanation: The formula was given by Slichter. In an intermediate packing, the centres of any 8 spheres, originally arranged in a cubic packing form the corners of the rhombohedron, with an acute face angle α.

5. The volume of the solids in a unit cell is given by ______
a) π/6
b) π/4
c) π/6
d) 2π/3

Explanation: Consider a unit cell,
Volume of each spherical particle = (π/6)*d3
Total volume of container=1*1*1=1
No. of solids in the container=(1/d)*(1/d)*(1/d)=(1/d3)
Volume of the solids Vs=(π/6)*d3*(1/d3)=(π/6).

6. The volume of solids in a unit cell is constant.
a) True
b) False

Explanation: The volume of solids in a unit cell is (π/6) as it does not change with the acute face angle α. Whereas, the total volume of the unit cell varies with acute face angle α.

7. In the densest state of packing, each sphere is in contact with ______ neighbouring spheres.
a) 6
b) 12
c) 8
d) 18

Explanation: In the densest state of packing, sphere centres form a rhombohedral array with face angle α=60°. Thus, each sphere is in contact with 12 neighbouring spheres.

8. The total volume of rhombohedron with face angle 60° is _______
a) 0.5
b) 0.7071
c) 0.9099
d) 1

Explanation: The total volume of rhombohedron is given by,
V = (1-cosα)*(√(1+2cosα))
V = (1-cos60°)*(√(1+2 cos60°))
V = 0.7071.

9. The minimum possible void ratio of uniformly graded spherical grains is _______
a) 0.20
b) 0.25
c) 0.30
d) 0.35

Explanation: 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.

10. The volume if voids of a rhombohedral array of spherical grains with α=75° is _______
a) 0.4266
b) 0.7439
c) 0.3895
d) 0.3595

Explanation: Given,
α=75°
The total volume V=(1-cosα)*(√(1+2cosα))
∴ V=(1-cos75°)*(√(1+2cos75°))
V=0.9131
Volume of the solids Vs=(π/6)=0.5236
Volume of the voids Vv=V-Vs=0.9131-0.5236
Vv=0.3895.

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