Hourglass curve equation

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In summary, to find the equation of the curve of an hourglass, you can use a combination of 8 functions for a 3D representation. These functions involve the variables y and z, and use the height and diameter of the hourglass as constants. It also involves scaling, choosing the type of sand, and experimenting to determine the correct amount of sand. The equation can be written as f(y,z) = n(y3 + z3) + a for positive y, f(y,z) = n(-y3 - z3) - a for negative y, and f(y,z) = n(y3 + z3) - a for negative y.
  • #1
ssj5harsh
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I need to find the equation of the curve of an hourglass. Known: diameter, height. Time taken for the sand to be completely emptied is 1 minute. I would like it if someone would tell me where or how to start. I can't think of anything.

Thanks in advance.
 
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  • #2
The curve of an hourglass? An hourglass can be many different shapes, starting with a cone. The information you give as "known" has nothing to do with the shape.
 
  • #3
i was looking for the same thing today and came across this question

what i came up with was a combination of 8 functions for a 3d representation, although i am not certain if they are correct. also of note is that this is for it laying on its side. anyway, here goes:

x = y3 + z3 + 1, For x, 0 to h
-x = y3 + z3 + 1, For x, -h to 0
-x = -y3 + z3 + 1, For x, -h to 0 (?)
x = -y3 + z3 + 1, For x, 0 to h (?)
x = y3 + z3 - 1, For x, -h to 0
-x = y3 + z3 - 1, For x, 0 to h
-x = -y3 + z3 - 1, For x, 0 to h (?)
x = -y3 + z3 - 1, For x, -h to 0 (?)

where 2h = height of hourglass

from here it (should) should just be a matter of scaling, choosing of sand, and experimental trial and error to come up with the correct amount of the particularly chosen sand.

again, am not sure if this is correct. . . .

Best Regards,
 
Last edited:
  • #4
whoops, let's try this again. . . .

f(y,z) = n(y3 + z3) + a; For y = 0, and positive y; For x, 0 to h
f(y,z) = n(-y3 - z3) - a; For negative y; For x, -h to 0
f(y,z) = n(y3 - z3) - a; For negative y; For x, -h to 0
f(y,z) = n(-y3 + z3) + a; For y = 0, and positive y; For x, 0 to h
f(y,z) = n(y3 + z3) - a; For negative y; For x, 0 to h
f(y,z) = n(-y3 - z3) + a; For y = 0, and positive y; For x, -h to 0
f(y,z) = n(y3 - z3) + a; For y = 0, and positive y; For x, -h to 0
f(y,z) = n(-y3 + z3) - a; For negative y; For x, 0 to h

where h = 1/2 hourglass height,
a = the cubed root of the hourglass annulous radius,
and
n = a height to width coefficient
 
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  • #5


I would first start by defining the variables in this equation. The diameter and height of the hourglass can be represented by the variables d and h, respectively. The time taken for the sand to be completely emptied can be represented by the variable t.

Next, I would consider the shape of an hourglass. It is essentially two cones connected at their bases. Therefore, we can use the formula for the volume of a cone: V = 1/3 * π * r^2 * h, where r is the radius of the base. Since the hourglass has equal diameters at both ends, we can use d/2 as the radius for each cone. Thus, the total volume of the hourglass can be represented as: V = 1/3 * π * (d/2)^2 * h + 1/3 * π * (d/2)^2 * h = 2/3 * π * (d/2)^2 * h.

Now, we need to consider the rate at which the sand is flowing through the hourglass. Since we know that it takes 1 minute for the sand to be completely emptied, we can say that the volume of sand flowing through the hourglass per unit time is constant. This can be represented by the equation: V/t = constant.

Combining these two equations, we can come up with the final equation for the hourglass curve: V/t = 2/3 * π * (d/2)^2 * h.

To find the specific equation for a given hourglass, we would need to know the specific values for d and h. We can then plug those values into the equation and solve for the constant, giving us the complete equation for the hourglass curve.

I hope this helps you in your search for the equation of the hourglass curve. Good luck!
 

1. What is the "Hourglass Curve Equation" and what is it used for?

The Hourglass Curve Equation is a mathematical formula that describes the shape of an hourglass, where the width narrows at the center and widens at the top and bottom. This equation is primarily used in physics and engineering to model the behavior of hourglass-shaped objects under different forces and stresses.

2. How is the Hourglass Curve Equation derived?

The Hourglass Curve Equation is derived from the principle of conservation of energy, where the sum of potential and kinetic energy at any point must remain constant. By applying this principle to an hourglass shape, the equation can be derived and used to analyze the behavior of hourglass-shaped objects.

3. Can the Hourglass Curve Equation be applied to other shapes besides an hourglass?

While the Hourglass Curve Equation is specifically developed for hourglass-shaped objects, it can also be applied to other shapes that share similar characteristics, such as a tapered cone or a funnel. However, it may not accurately model the behavior of other shapes that differ significantly from an hourglass.

4. What are some real-life applications of the Hourglass Curve Equation?

The Hourglass Curve Equation has various real-life applications, including designing hourglass-shaped structures like bridges and dams, analyzing the behavior of hourglass-shaped containers or vessels under pressure, and even modeling the shape and movement of hourglass-shaped organisms, such as certain types of fish.

5. Is the Hourglass Curve Equation a perfect representation of an hourglass shape?

No, the Hourglass Curve Equation is a simplified mathematical model that may not perfectly represent the shape and behavior of a real hourglass. Factors such as material properties, friction, and external forces can impact the accuracy of the equation in real-world scenarios. However, it is a useful tool for understanding and predicting the behavior of hourglass-shaped objects in certain situations.

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