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

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

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

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, rx is given by______
a) rx=sinθ
b) rx=cosθsinθ
c) rx=cosθ
d) rx=sin2θ

Explanation: Since we know, r2=x2+y2 —————-(1)
∴ differentiating (1) with respect to x, we get,
rx=xr=rcosθr=cosθ.

5 - Question

In polar form the term, ry is given by______
a) ry=sinθ
b) ry=cosθsinθ
c) ry=cosθ
d) ry=sin2θ

Explanation: Since we know, r2=x2+y2 —————-(1)
∴ differentiating (1) with respect to y, we get,
ry=yr=rsinθr=sinθ.

6 - Question

In polar form the term, θx is given by______
a) θx=sinθr
b) θx=cosθsinθ
c) θx=cosθr
d) θx=sinθr

Explanation: Since we know,
θ=tan-1(y/x) —————-(1)
∴ differentiating (1) with respect to x, we get,
θx=sinθr.

7 - Question

In polar form the term, θy is given by______
a) θy=sinθr
b) θy=cosθsinθ
c) θy=cosθr
d) θy=sin2θr

Explanation: Since we know,
θ=tan-1(y/x) —————-(1)
∴ differentiating (1) with respect to y, we get,
θy=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¯¯¯rcosθ1ru¯¯¯θsinθ
b) u¯¯¯x=u¯¯¯rcosθ1ru¯¯¯θcosθ
c) u¯¯¯x=u¯¯¯rsinθ1ru¯¯¯θsinθ
d) u¯¯¯x=u¯¯¯rsinθ1ru¯¯¯θcosθ

Explanation: Partially differentiating the excess hydrostatic pressure \overline{u} with respect to x,
u¯¯¯x=u¯¯¯rrx+u¯¯¯θθx=u¯¯¯rcosθ1ru¯¯¯θsinθ.
∴ u¯¯¯x=u¯¯¯rcosθ1ru¯¯¯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+1ru¯¯¯r1r22u¯¯¯θ2
b) 2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r2+1ru¯¯¯r+1r22u¯¯¯θ2
c) 2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r21ru¯¯¯r1r22u¯¯¯θ2
d) 2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r21ru¯¯¯r+1c22u¯¯¯θ2

Explanation: The second order of the partial differentiation of excess hydrostatic pressure u with respect to x is,
2u¯¯¯x2=(rcosθ1rsinθθ)(u¯¯¯rcosθ1ru¯¯¯θθsinθ)(1)
also, the second order of the partial differentiation of excess hydrostatic pressure \overline{u} with respect to y is,
2u¯¯¯y2=(rcosθ1rsinθθ)(u¯¯¯rcosθ1ru¯¯¯θθsinθ) ———-(2)
2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r2+1ru¯¯¯r+1r22u¯¯¯θ2.

10 - Question

In case of radial symmetry, 2u¯¯¯x2+2u¯¯¯y2 is_________
a) 2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r2+1ru¯¯¯r
b) 2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r21ru¯¯¯r
c) 2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r2+1ru¯¯¯r
d) 2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r21ru¯¯¯r

Explanation: The term 2u¯¯¯x2+2u¯¯¯y2 in terms of r and θ is given by,
2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r2+1ru¯¯¯r+1r22u¯¯¯θ2.
In case of radial symmetry, \overline{u} is independent of θ and hence we get,
2u¯¯¯x2+2u¯¯¯y2=2u¯¯¯r2+1ru¯¯¯r

11 - Question

In three dimensional consolidation of sand drain, having radial symmetry, the governing consolidation equation is _______
a) u¯¯¯t=Cvr(u¯¯¯r2+1ru¯¯¯r)Cvz2u¯¯¯z2
b) u¯¯¯t=Cvr(u¯¯¯r2+1ru¯¯¯r)+Cvz2u¯¯¯z2
c) u¯¯¯t=Cvr(u¯¯¯r21ru¯¯¯r)+Cvz2u¯¯¯z2
d) u¯¯¯t=Cvz(u¯¯¯r2+1ru¯¯¯r)+Cvz2u¯¯¯z2

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+1ru¯¯¯r)+Cvz2u¯¯¯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+1ru¯¯¯r)
b) u¯¯¯t=Cvr(u¯¯¯r2+1ru¯¯¯r)+Cvz2u¯¯¯z2
c) u¯¯¯t=Cvz(u¯¯¯r2+1ru¯¯¯r)+Cvz2u¯¯¯z2
d) u¯¯¯t=Cvr(u¯¯¯r2+1ru¯¯¯r)

Explanation: In three dimensional consolidation of sand drain, having radial symmetry, the governing consolidation equation is,
u¯¯¯t=Cvr(u¯¯¯r2+1ru¯¯¯r)+Cvz2u¯¯¯z2, in this the radial flow part of equation is,
u¯¯¯t=Cvr(u¯¯¯r2+1ru¯¯¯r).

13 - Question

The one dimensional flow part of governing consolidation equation of three dimensional consolidation having radial symmetry is _______
a) u¯¯¯t=Cvru¯¯¯r2
b) u¯¯¯t=Cvr(u¯¯¯r2+1ru¯¯¯r)+Cvz2u¯¯¯z2
c) u¯¯¯t=Cvz(u¯¯¯r2+1ru¯¯¯r)+Cvz2u¯¯¯z2
d) u¯¯¯t=Cvr(u¯¯¯r2+1ru¯¯¯r)

Explanation: In three dimensional consolidation of sand drain, having radial symmetry, the governing consolidation equation is,
u¯¯¯t=Cvr(u¯¯¯r2+1ru¯¯¯r)+Cvz2u¯¯¯z2, in this the one dimensional flow part of equation is,
u¯¯¯t=Cvru¯¯¯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)

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