Calculate Centroid of Hemispherical Dome w/ Top Removed

[jiboom]

7)
Pile foundations are the part of a structure used to carry and transfer the load of the structure to the bearing ground located at some depth below ground surface.
A concrete pile for a domestic dwelling has a radius of 1000 mm at its base, and 2000 mm at the top. It is 2000 mm deep and the sides slope uniformly.
Find the volume of concrete needed to form this pile.
A plinth is formed in the shape of a hemispherical dome, radius 4 m, with its top 1 m removed. Using integration, find the position of the plinth’s centroid along its axis of symmetry.

i think i just need volume of a frustrum for first part,with base radius 2000 and top radius 1000 and height 2000??
s0
2pi[4+2+1]/3= 14pi/3


how do i go about the second part? what do i need to ingrate for this shape?

 

[haruspex]

Suppose you set the origin at the sphere's centre and take the x-axis to be the axis of symmetry of the plinth. Can you apply the 'washer' method to find the volume? Finding the moment about the y-axis is very similar. Do you know a formula for the x coordinate of the mass centre?

 

[jiboom]

Suppose you set the origin at the sphere's centre and take the x-axis to be the axis of symmetry of the plinth. Can you apply the 'washer' method to find the volume? Finding the moment about the y-axis is very similar. Do you know a formula for the x coordinate of the mass centre?

not heard of the washer method. i know the formula for x bar for a hemisphere but the question wants it from integration

 

[haruspex]

The washer method just slices a volume of revolution into discs perpendicular to the axis. Write down the expression for the area of one such disc, thickness dx, and integrate. Thst gives the volume. Repeat the process with an extra factor x to get the moment of the disc element and integrate again.

 

[Nickie]

For the second part, you will need to use the formula for calculating the centroid of a hemisphere with the top removed. The formula is x = (3r/8)(1 + sin(θ)), where r is the radius of the hemisphere and θ is the angle of the removed portion. In this case, r = 4 m and θ = π/2 (since the top is removed, the angle is 90 degrees or π/2 radians). So, plugging in the values, we get x = (3*4/8)(1 + sin(π/2)) = 1.5 m. This means that the centroid of the plinth is located 1.5 m above the base, along the axis of symmetry.