Centroid of trapezoidal

The centroid of the trapezoidal with its one face perpendicular is given be

X=-1/3 (-2b^2-2ab+a^2)/(a+b)

Y=1/3 h (b+2a)/(a+b)

Where a, b, h and origin is as shown in the figure below.

Similarly, the centroid of any trapezoidal is given by

x= 1/3(3ab+3bc+b^2+6ac+3a^2+2c^2)/(b+2a+c)

y=1/3 h (b+3a+c)/(b+2a+c)

Where a, b ,c, h and origin is as shown in the figure below.

The result was obtained using MathLab 7.0

MEASUREMENT OF LAND IN NEPAL

In Nepal we use different units for measurement of land. In hilly regions we use Ropani-Ana-Paisa-Dam system while in Southern parts i.e. at Terai region we use Bhigha-Kattha-Dhur system.

The conversion between these units and also to the SI unit is essential as we come across this situation often.

The conversion factor for these units is as follows

Ropani-Ana-Paisa-Dam system

• 1 Ropani =16Ana
• 1 Ana =4 Paisa
• 1Paisa =4 Dam

Bhigha-Kattha-Dhur system

• 1 Bhigha= 20 Kattha
• 1 Kattha= 20 Dhur

For inter conversion between Ropani and Bhigha

• 1Bhigha=13 Ropani

Similarly in standard units we may use

• 1 Ropani =74feet X 74 feet
• 1 m=3.218 feet

An Excel sheet has been developed for the general use which can be downloaded in this page.

Derivation of Combined Angle

Abstract: Derivation of combined angle may become useful when you want to know the magnitude and direction of force due to bend in both XY and XZ plane. For eg in design of bend in pipeline.

Let us consider a straight line which deflects at O along OA as shown in the figure. The deflection angles are also shown in the figure.

Let R be the position vector of A. OB is projection of OA on XY plane and AB is projection of OA on a plane parallel to YZ plane. Similarly, OC is projection of line OB in X axis and BC is projection of OB on axis parallel to Y axis.

From figure it can inferred that

OB=R cosα

AB=R sinα

OC=OB cosβ = R cosα cosβ

BC= OB sinβ =R cosα sinβ

Therefore, the position vector of A is OA which is given by

R=|R| (cosα cosβ i + cosα sinβ j + sinα k )

And the unit vector along R is given by

r= (cosα cosβ i + cosα sinβ j + sinα k )

Now the combined deflection angle is given by

r . i =|r| |i| cosδ

Or, r . i =1*1 cosδ

Or, cosδ = cosα cosβ+0+0

Or, δ =cos^-1(cosα cosβ)

The angle δ lies in the plane OAC. This plane is shown in the figure in hatch line.

Forces

If F be the magnitude of force due to the deflection δ (e.g. deflection in pipeline) then the vertical and horizontal component of this force is given by,

F_x1 =F r . i =F cosα cosβ

F_x2 =F r . j =F cosα sinβ

F_x =F

=F cosα

F_y =F r . j

=F sinα

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