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Chapter 4: Problem 126

Design a clepsydra (Egyptian water clock) a vessel from which water drains by gravity through a hole in the bottom and which indicates time by the level of the remaining water. Specify the dimensions of the vessel and the size of the drain hole; indicate the amount of water needed to fill the vessel and the interval at which it must be filled. Plot the vessel radius as a function of elevation.

Short Answer

Expert verified
A suitable clepsydra can be designed with a vessel shaped as a truncated cone of height 30 cm, top radius 10 cm and bottom radius 5 cm, filled with a specified amount of water calculated using the volume formula for a frustum of a cone, which needs to be refilled every hour. The function for the radius of the vessel as a function of its elevation is linear.

Step by step solution

01

Understanding the nature of Clepsydra

Clepsydra, or water clock, was used in Egypt for thousands of years. It functioned with the steady dripping of water from a tank to another vessel. The time was then measured based on the water level in the second container.
02

Defining the dimensions of the vessel

For this exercise, let's use a conical vessel, which has a useful property that the water level drops at a uniform rate. The specific dimensions are not given, so let's assume a height \(H\) of 30 cm with the top and bottom radii as \(R_t = 10 \(cm\) and \(R_b = 5 \(cm\) respectively. The shape of the vessel should be a frustrum of a cone (truncated cone).
03

Calculating the drain hole size

The size of the drain hole is calculated from Bernoulli's Equation. If the hole is to be small enough so that the flow rate is laminar throughout, we can use Torricelli's Law: \(v = \sqrt{2gh}\), where \(v\) is the speed of the outflow, \(g\) is acceleration due to gravity and \(h\) is the height of water above the hole. Given a specific time measurement requirement, we can adjust the hole size to achieve it.
04

Indicate the amount of water needed and interval to fill the vessel

The volume \(V\) of a frustum of a cone is given by \(V = \(\frac{1}{3}\) \(\pi h (r1^2 + r1*r2 + r2^2)\). Substituting given values, we can calculate the volume. Suppose we want the water clock to measure the passage of 1 hour. If it takes 1 hour for the tank to empty, then the water should be refilled every hour.
05

Plotting the vessel radius as a function of elevation

Now, we need to define a function for radius of the cone as a function of height. As it's a truncated cone, it can be considered as two similar triangles, from which we get the linear relationship \(r = R_b + h*(R_t - R_b) / H\). Then we plot the radius \(r\) as a function of height \(h\).

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