Fluid force problems

Transcription

1 Roberto s Notes on Integral Calculus Chapter 5: Basic applications of integration Section 1 Fluid force problems What ou need to know alread: How to use the four step process to set up an integral. What ou can learn here: How to appl this process to the computation of hdrostatic force. You ma remember from our phsics courses that if a force F is applied uniforml on a surface of area A, then the pressure P acting on the surface is given b: F P F PA A Moreover, when the force in question is the one applied b a fluid on an object immersed in it at a fixed depth h, then the corresponding pressure is given b the formula: P gh densit gravit depth Wh did ou use p for densit? That is not a letter p, but the Greek letter ( rho ) commonl used to identif densit. I am tring to avoid the letter d, as it is used to identif the differential of an integral, or the derivative of a function. If everthing is constant, these two formulae can be combined to compute the total force exerted b the liquid on the object. But what if one of the quantities involved is not constant? In particular, how do we compute the force exerted b a liquid on an object that: extends verticall, so that the pressure is not constant? has a horizontal width that changes verticall, so that the area at different depth are different? Then we ll have to use the good-old slice-approximate-add-limit process, which will help us generate an integral. Strateg to set up an integral describing the fluid force acting on an object 1. Draw a clear picture representing the object in the fluid (polgonal region in shaded rectangle in the picture).. Identif the acceleration of gravit g and the densit of the liquid.. Identif a vertical reference frame and use it consistentl (up arrow). w h Integral Calculus Chapter 5: Basic applications of integration Section 1: Fluid force problems Page 1

2 4. Identif the top and bottom values, a and b, where the object is located. 5. Divide the object into thin horizontal slices (horizontal slice of width w). 6. Construct expressions representing the depth h and width w of a generic slice, as functions of the variable in the chosen reference frame. 7. Combine these quantities into the integral: b F g h w( ) d a B using the existing reference frame we notice that: the densit is 945 and the gravit is g 9.81 The depth of a generic slice in position is h 4.5 The width of each slice is w x Therefore, the required force is given b the integral: Example: F d 6 Example: Assume that a large tank is designed with vertical ends in the shape of the x region between the curves and 4.5, measured in metres, as shown here. Assume also that the tank is filled with oil with a densit of 945 kg/m X Y A plate is submerged in water as shown in the picture, with all units in meters. To compute the fluid force acting on either side of it, we can measure distances from the vertex of the triangle. In this wa: The depth is metres more than the position: h The width can be obtained b using similar triangles: As usual, the thickness of each slice is densit of the fluid is 1000 kg / m. w w 6, gravit is 9.81 Therefore, the force acting on the slice is given b: m / sec and the Integral Calculus Chapter 5: Basic applications of integration Section 1: Fluid force problems Page w

3 and the total force is given b: F dgha F 4905 d N 6 0 Summar The fluid force acting on an object immersed in it at variable depth, or with variable width can be computed b constructing an integral through the four-step process. Common errors to avoid Take our time to analze and understand each required element of the integral. Set up the frame of reference at the beginning and use it consistentl for all elements of the integral. Remember to alwas use horizontal slices and to identif the surface of the fluid Learning questions for Section I 5-1 Review questions: 1. Describe how to use the four-set process to set up an integral representing the force exerted b a fluid on a surface. Integral Calculus Chapter 5: Basic applications of integration Section 1: Fluid force problems Page

4 Memor questions: 1. Which formula describes the pressure exerted b a fluid at a given depth?. When setting up the integral representing the force exerted b a fluid on a trapezoidal surface, what shape will we assume for the individual slices?. When setting up the integral representing the force exerted b a fluid on a surface, in which direction do we slice the surface that is being analzed? Theor questions: 1. When setting up the integral representing the force exerted b a fluid on a surface, what does dx represent?. When setting up the integral representing the force exerted b a fluid on a surface, which two quantities are usuall considered constant?. Wh is an integral needed to compute a hdrostatic force? 4. What happens to the hdrostatic force acting on the plate if it is immersed at the same depth in a deeper pool? 5. What happens to the hdrostatic force acting on the plate if it is immersed at the lower depth in a pool of the same depth? 6. Which phsics principle is a consequence of the fact that a solid object immersed in a fluid is subject to a lower force on the top than it is at the bottom? Application questions: 1. A clindrical tank is placed on the ground, so that the circular bases are vertical. Such bases have a diameter of metres and the tank is full of water. One of the circular ends is built so that the section consisting of its bottom 5 cm is easil removable b detaching a set of bolts. Set up an integral describing the hdrostatic force acting on the removable portion when the tank is full. No need to compute the integral.. A clindrical tank is resting horizontall on the ground (the circular bases are vertical) and is filled with a special liquid that has a densit of 94 kg per cubic metre. If the radius of the clinder s base is metres, set up an integral describing the force exerted b the liquid on each base of the tank. No need to compute the integral. A fish tank is 1 metre long, 45 cm wide and filled to a depth of 0 cm. Determine the hdrostatic force on one of the shorter walls b assuming that it uses fresh water. 4. A dam has the shape of an inverted isosceles trapezoid, 50 m on top, 0 on bottom, 0 in height, with water level 4 metres below top. Find the total fluid force acting on it. 5. A fish tank is 60 cm tall, 1 meter wide and 50 cm deep, and is filled up to 5 cm from the top with low salinit water at a densit of 1015 kg/m. Determine the hdrostatic force on each of the four vertical sides. Integral Calculus Chapter 5: Basic applications of integration Section 1: Fluid force problems Page 4

5 6. The bottom portion of the wall of an aquarium tank is in the shape of a semicircle of radius metres. Its top is 5 metres below water surface. What is the total force exerted b the water on the window? 7. A monitoring window must be built into the wall of a tank designed to contain a solution with a densit of 10 grams per litre. The shape of the window is best described as the region bounded b the parabola x and the line (all units in metres), with its top located 8 metres below the surface of the solution. Set up the integral that describes the force that will be exerted b the solution on the window. Identif the phsical or geometrical meaning of each piece of the integral, but no need to compute the integral. 8. An aquarium window has the shape of the parabola x, with the water surface at = (all the units in metres). Which integral describes the force exerted b the seawater of the aquarium on this window? Identif the geometric or phsical meaning of each piece of the integral, but no need to compute it. 9. An underwater window has the shape of the region that is above the x-axis, below cos x, and contains a piece of the -axis. The surface of the water is 1 meter above the top of the window (all the units in metres). Set up an integral that represents the force exerted b the water on this window. No need to evaluate this integral. 10. The viewing window of an aquarium has the shape of an inverted Norman window, with the bottom semicircle having a radius of 1m and the top rectangular portion having a height of m. The top of the window is 1m below the surface. Which integral represents the force exerted b the water on this window? No need to compute the integral. 11. A plugged eaves trough is 5 metres long and has a cross section in the shape of a right trapezoid, 9 cm deep, 8 cm wide at the top and 5 at the bottom. If it is filled with water to a depth of 6 cm, what is the force that the water is exerting on each of the two ends of the trough? 1. A pop can has a diameter of 8 cm and a height of 10 cm. If it is full of a tpe of pop that has a densit of 98 kg/m, what force is acting on the side of the can? 1. A punch bowl is made in the shape of a box with two ends slanted at 0 degrees from the vertical. The bowl is 0 cm wide, 5 cm high and is totall full. What force is exerted on each slanted end? What questions do ou have for our instructor? Integral Calculus Chapter 5: Basic applications of integration Section 1: Fluid force problems Page 5

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