A legendary Dutch boy saved Holland by plugging a hole in a...

A legendary Dutch boy saved Holland by plugging a hole in a dike with his finger, which is $1.20 \mathrm{cm}$ in diameter. If the hole was $2.00 \mathrm{m}$ below the surface of the North Sea (density $\left.1030 \mathrm{kg} / \mathrm{m}^{3}\right),(\mathrm{a})$ what was the force on his finger? (b) If he pulled his finger out of the hole, how long would it take the released water to fill 1 acre of land to a depth of $1 \mathrm{ft}$, assuming the hole remained constant in sixe? (A typical U.S. family of four uses 1 acre-foot of water, $1234 \mathrm{m}^{3},$ in 1 year.

Key Concepts

Hydrostatic Pressure

This concept refers to the pressure exerted by a fluid at a certain depth due to the weight of the overlying fluid. It is calculated using the formula P = ?gh, where ? is the density of the fluid, g is the acceleration due to gravity, and h is the depth below the surface. Understanding hydrostatic pressure is essential for analyzing forces within fluids.

Force Calculation from Pressure

The force exerted on a surface by a fluid is found by multiplying the pressure by the area over which the pressure acts (F = P × A). This principle allows one to determine the total force acting on an object submerged in a fluid or in contact with it, which is a fundamental aspect of fluid mechanics.

Flow Rate from an Orifice (Torricelli’s Law)

Torricelli's law gives the speed of a fluid flowing out of an orifice under the influence of gravity, expressed as v = ?(2gh) for an ideal fluid. This speed, when multiplied by the cross?sectional area of the orifice, yields the volumetric flow rate (Q = A?(2gh)). This law is crucial for analyzing fluid discharge through openings or leaks.

Unit Conversion and Volume Calculations

Accurate unit conversion is essential when solving physics problems that involve different measurement systems, such as converting diameters to areas or volumes (e.g., converting acre-foot volumes to cubic meters). Mastery of unit conversion and dimensional analysis ensures correct application of formulas and consistent results across different units.

Transcript

00:01 Hi, so in this problem, we have a volume of water with a height of two meters.

00:08 So you write h equals two meters, and on the bottom of this volume, we have a hole, the diameter of 1 .2 centimeters.

00:19 And this water is actually seawater, so the density is going to equal row s, which is 1 ,30 kilograms per meters cubed.

00:32 In part a, we're trying to figure out if there was a plug stopping the flow of water at the bottom of this hole.

00:43 What is going to be the force acting on that plug? well, we can first figure out the pressure, which is just equal to the density of water times gravity, gravitation and acceleration times depth.

00:59 And we know that the force is going to equal to the pressure times the air.

01:04 Area acting upon it, and the area is just equal to pi r squared.

01:15 And in this case, the diameter is given, so we have pi d over two squared, which is just equal to 1 over 4 pi d squared.

01:28 So once we substitute everything back in, we get 1 over 4 row s, g, h, pi, d squared.

01:36 So for row s, we have 130 kilograms per meters cubed, and times 9 .8 meters per second squared times the height, which is equal to 2 meters, or the depth, times pi, and our diameter is equal to 1 .2 centimeters, so that is just equal to 0 .012 meters squared.

02:13 And once you do the computation, you should get, 2 .2 .2 .2 .2 .2.

02:16 To 8 newtons.

02:19 The second part of our question, we're asked to figure out when the plug is gone, how long would it take to fill up a volume of water of one acre foot? this is equal to 1 ,234 meters cubed.

02:37 To start this out, we can use the bernoulli's equation.

02:41 So let's quickly write that down.

02:50 It's p2 plus row gy2 plus 1⁄2.

02:55 Row b2 squared.

02:58 Now let's look at this equation...

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