Spherical Hourglass

Let the above picture represent a spherical hourglass, where the bulbs are sections of spheres. Assume that the two bulbs have equal size. In order to calculate the volume of a spherical hourglass, double the volume of a single spherical bulb.

A cross section of a sphere can be describe by the equation x^2 + y^2 = r^2. Surprised? A sphere is a three-dimensional circular object.

By the diagram above, we calculate the volume of a spherical bulb giving radius r and height h. Using the Method of Discs with the discs rotating around the y-axis (x=0):

Top constraint: y = h
Bottom constraint: y = 0
Left constraint: x = 0
Right constraint: x = √(r^2 - y^2)

r(y) = √(r^2 - y^2)

And the volume of one of the spherical bulbs is:

h
∫ (r(y))^2 dy * π
0

h
∫ r^2 - y^2 dy * π
0

h
[ r^2 * y - y^3/3 ] * π
0

(r^2 * h - h^3/3) * π

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. To get the volume of the spherical hourglass, double the volume of a spherical bulb:

V = 2 * π * (r^2 * h - h^3/3)

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Note that if h = r, the bulbs are two half-spheres and that hourglass' volume:

V = 2 * π * (r^2 * r - r^3/3) = 2 * π * 2/3 * r^3 = 4/3 * π * r^3

Turns out to be the volume of a sphere.


Interesting how the mathematics checks out. With that I wish you a great day/night!

Eddie


This blog is property of Edward Shore. 2014

Parabolic Hourglass

Let the above picture represent a "parabolic hourglass", where the two bulbs are outlined by a parabolic curve. Assume that each of the bulb is equal in size.

To calculate the volume of the parabolic hourglass, first calculate the volume of one of the two bulbs. The total volume of the parabolic hourglass is twice the volume of a bulb.

Examining one of the bulbs, suppose the edge of the bulb can be described by the equation y = x^2 - b. See the diagram below, where we impose a cross section of the bulb on the Cartesian plane.

Note that:

(1) The small art of the bulb is placed on the x-axis, and

(2) The origin (point (0,0)) is placed in the center of the base.

To calculate the volume, we are going to use the Disc Integration Method with the discs rotating around the y-axis. The general formula for this method with y-axis
(x = 0) as the axis of rotation is:

d
∫ (r(y))^2 dy * π
c

We have the following constraints:

Upper: y = h
Lower: y = 0
Left: x = 0
Right: x = √(y + b)

And r(y) = √(y + b) - 0 = √(y + b)

With c = 0 and d = h, the volume of one bulb is:

h
∫ (√(y + b))^2 dy * π
0

h
∫ y + b dy * π
0

h
[ y^2/2 + b*y ] * π
0

π * (h^2/2 + b*h) (I)

If we want to determine the volume of the bulb in terms of a (outer radius), use the equation y = x^2 + b and the point (a, h) to determine that:

h = a^2 - b

b = a^2 - h (II)

Substitute equation (II) into (I) to get:

π * (h^2/2 + a^2 * h - h^2)

π * (a^2 * h - h^2/2) (III)

Remember that this the volume of one bulb. The parabolic hourglass consists of two equally sized bulbs.

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Therefor the volume of the parabolic hourglass is:

V = 2 * π * (h^2/2 + b * h) = 2 * π * (a^2 * h - h^2/2)

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Eddie


This blog is property of Edward Shore. 2014