Engineering Questions with Answers - Multiple Choice Questions
Consolidation Equation in Polar Coordinates
1 - Question
The transformation from Cartesian to plane coordinates in x-direction is given by ______
a) x=rsinθ
b) x=rcosθ
c) x=rcos2θ
d) x=rsin2θ
View Answer
Answer: b
Explanation: The transformation from Cartesian to plane coordinates in x-direction is given by,
x=rcosθ. Where, x=coordinate in Cartesian form
r=coordinate in polar form.
r2=x2+y2 and θ=tan-1(y/x).
2 - Question
The transformation from Cartesian to plane coordinates in y-direction is given by ______
a) y=rsinθ
b) y=rcosθ
c) y=rcos2θ
d) y=rsin2θ
View Answer
Answer: a
Explanation: The transformation from Cartesian to plane coordinates in y-direction is given by,
y=rsinθ. Where, y=coordinate in Cartesian form
r=coordinate in polar form.
r2=x2+y2 and θ=tan-1(y/x).
3 - Question
The transformation from Cartesian to plane coordinates in x-direction is given by ______
a) z=rsinθ
b) z=rcosθ
c) z=z
d) z=r2sinθcosθ
View Answer
Answer: b
Explanation: In the z-direction, since it does not make any angle with the axis, the z-direction in both the Cartesian as well as the polar coordinates is the same.
∴ z=z.
4 - Question
In polar form the term, ∂r∂x is given by______
a) ∂r∂x=sinθ
b) ∂r∂x=cosθsinθ
c) ∂r∂x=cosθ
d) ∂r∂x=sin2θ
View Answer
Answer: c
Explanation: Since we know, r2=x2+y2 —————-(1)
∴ differentiating (1) with respect to x, we get,
∂r∂x=xr=rcosθr=cosθ.
5 - Question
In polar form the term, ∂r∂y is given by______
a) ∂r∂y=sinθ
b) ∂r∂y=cosθsinθ
c) ∂r∂y=cosθ
d) ∂r∂y=sin2θ
View Answer
Answer: a
Explanation: Since we know, r2=x2+y2 —————-(1)
∴ differentiating (1) with respect to y, we get,
∂r∂y=yr=rsinθr=sinθ.
6 - Question
In polar form the term, ∂θ∂x is given by______
a) ∂θ∂x=sinθr
b) ∂θ∂x=−cosθsinθ
c) ∂θ∂x=−cosθr
d) ∂θ∂x=−sinθr
View Answer
Answer: d
Explanation: Since we know,
θ=tan-1(y/x) —————-(1)
∴ differentiating (1) with respect to x, we get,
∂θ∂x=−sinθr.
7 - Question
In polar form the term, ∂θ∂y is given by______
a) ∂θ∂y=sinθr
b) ∂θ∂y=cosθsinθ
c) ∂θ∂y=cosθr
d) ∂θ∂y=sin2θr
View Answer
Answer: c
Explanation: Since we know,
θ=tan-1(y/x) —————-(1)
∴ differentiating (1) with respect to y, we get,
∂θ∂y=cosθr.
8 - Question
The partial differentiation of excess hydrostatic pressure \overline{u} as a function of r and θ with respect to x is given by _______
a) ∂u¯¯¯∂x=∂u¯¯¯∂rcosθ−1r∂u¯¯¯∂θsinθ
b) ∂u¯¯¯∂x=∂u¯¯¯∂rcosθ−1r∂u¯¯¯∂θcosθ
c) ∂u¯¯¯∂x=∂u¯¯¯∂rsinθ−1r∂u¯¯¯∂θsinθ
d) ∂u¯¯¯∂x=∂u¯¯¯∂rsinθ−1r∂u¯¯¯∂θcosθ
View Answer
Answer: a
Explanation: Partially differentiating the excess hydrostatic pressure \overline{u} with respect to x,
∂u¯¯¯∂x=∂u¯¯¯∂r∂r∂x+∂u¯¯¯∂θ∂θ∂x=∂u¯¯¯∂rcosθ−1r∂u¯¯¯∂θsinθ.
∴ ∂u¯¯¯∂x=∂u¯¯¯∂rcosθ−1r∂u¯¯¯∂θsinθ.
9 - Question
The term ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2 in terms of r and θ is given by _______
a) ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=∂2u¯¯¯∂r2+1r∂u¯¯¯∂r−1r2∂2u¯¯¯∂θ2
b) ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=∂2u¯¯¯∂r2+1r∂u¯¯¯∂r+1r2∂2u¯¯¯∂θ2
c) ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=∂2u¯¯¯∂r2−1r∂u¯¯¯∂r−1r2∂2u¯¯¯∂θ2
d) ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=∂2u¯¯¯∂r2−1r∂u¯¯¯∂r+1c2∂2u¯¯¯∂θ2
View Answer
Answer: b
Explanation: The second order of the partial differentiation of excess hydrostatic pressure u with respect to x is,
∂2u¯¯¯∂x2=(∂∂rcosθ−1rsinθ∂∂θ)(∂u¯¯¯∂rcosθ−1r∂u¯¯¯∂θθsinθ)————−(1)
also, the second order of the partial differentiation of excess hydrostatic pressure \overline{u} with respect to y is,
∂2u¯¯¯∂y2=(∂∂rcosθ−1rsinθ∂∂θ)(∂u¯¯¯∂rcosθ−1r∂u¯¯¯∂θθsinθ) ———-(2)
∴ adding (1) and (2),
∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=∂2u¯¯¯∂r2+1r∂u¯¯¯∂r+1r2∂2u¯¯¯∂θ2.
10 - Question
In case of radial symmetry, ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2 is_________
a) ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=∂2u¯¯¯∂r2+1r∂u¯¯¯∂r
b) ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=∂2u¯¯¯∂r2−1r∂u¯¯¯∂r
c) ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=−∂2u¯¯¯∂r2+1r∂u¯¯¯∂r
d) ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=−∂2u¯¯¯∂r2−1r∂u¯¯¯∂r
View Answer
Answer: a
Explanation: The term ∂2u¯¯¯∂x2+∂2u¯¯¯∂y2 in terms of r and θ is given by,
∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=∂2u¯¯¯∂r2+1r∂u¯¯¯∂r+1r2∂2u¯¯¯∂θ2.
In case of radial symmetry, \overline{u} is independent of θ and hence we get,
∂2u¯¯¯∂x2+∂2u¯¯¯∂y2=∂2u¯¯¯∂r2+1r∂u¯¯¯∂r
11 - Question
In three dimensional consolidation of sand drain, having radial symmetry, the governing consolidation equation is _______
a) ∂u¯¯¯∂t=Cvr(u¯¯¯∂r2+1r∂u¯¯¯∂r)−Cvz∂2u¯¯¯∂z2
b) ∂u¯¯¯∂t=Cvr(u¯¯¯∂r2+1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2
c) ∂u¯¯¯∂t=Cvr(u¯¯¯∂r2−1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2
d) ∂u¯¯¯∂t=Cvz(u¯¯¯∂r2+1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2
View Answer
Answer: b
Explanation: In three dimensional consolidation of sand drain, for the case of radial symmetry,
Cvx=Cvy =Cvr
∴ the governing consolidation equation is,
∂u¯¯¯∂t=Cvr(u¯¯¯∂r2+1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2.
12 - Question
The radial flow part of governing consolidation equation of three dimensional consolidation having radial symmetry is _______
a) ∂u¯¯¯∂t=Cvr(u¯¯¯∂r2+1r∂u¯¯¯∂r)
b) ∂u¯¯¯∂t=Cvr(u¯¯¯∂r2+1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2
c) ∂u¯¯¯∂t=Cvz(u¯¯¯∂r2+1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2
d) ∂u¯¯¯∂t=Cvr(u¯¯¯∂r2+1r∂u¯¯¯∂r)
View Answer
Answer: a
Explanation: In three dimensional consolidation of sand drain, having radial symmetry, the governing consolidation equation is,
∂u¯¯¯∂t=Cvr(u¯¯¯∂r2+1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2, in this the radial flow part of equation is,
∂u¯¯¯∂t=Cvr(u¯¯¯∂r2+1r∂u¯¯¯∂r).
13 - Question
The one dimensional flow part of governing consolidation equation of three dimensional consolidation having radial symmetry is _______
a) ∂u¯¯¯∂t=Cvr∂u¯¯¯∂r2
b) ∂u¯¯¯∂t=Cvr(∂u¯¯¯∂r2+1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2
c) ∂u¯¯¯∂t=Cvz(∂u¯¯¯∂r2+1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2
d) ∂u¯¯¯∂t=Cvr(∂u¯¯¯∂r2+1r∂u¯¯¯∂r)
View Answer
Answer: a
Explanation: In three dimensional consolidation of sand drain, having radial symmetry, the governing consolidation equation is,
∂u¯¯¯∂t=Cvr(∂u¯¯¯∂r2+1r∂u¯¯¯∂r)+Cvz∂2u¯¯¯∂z2, in this the one dimensional flow part of equation is,
∂u¯¯¯∂t=Cvr∂u¯¯¯∂r2.
14 - Question
The equation given by Carillo in 1942 relating the degree of consolidation in one dimensional flow (Uz) and radial flow (Ur) is _______
a) (1-U)=(1-Uz)(1+Ur)
b) (1-U)=(1-Uz)(1-Ur)
c) (1-U)=(1+Uz)(1-Ur)
d) (1-U)=(1+Uz)(1+Ur)
View Answer
Answer: b
Explanation: The equation given by Carillo in 1942 relating the degree of consolidation in one dimensional flow (Uz) and radial flow (Ur) is,
(1-U)=(1-Uz)(1-Ur). This is the combination of the one dimensional flow and radial flow parts from the governing consolidation equation.
15 - Question
The time factor Tv for the vertical flow is given by _______
a) Tv=CvztH2
b) Tv=−CrztH2
c) Tv=CvzH2
d) Tv=CvztH
View Answer
Answer: a
Explanation: The time factor Tv for the vertical flow is given by,
Tv=CvztH2,
Where Cvz is the coefficient of consolidation in z-direction
t=elapsed time
H=average drainage path.