# ME-B41 Lab 1: Hydrostatics. Experimental Procedures

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1 ME-B41 Lab 1: Hydrostatics In this lab you will do four brief experiments related to the following topics: manometry, buoyancy, forces on submerged planes, and hydraulics (a hydraulic jack). Each experiment should take less than 15 minutes and the write-up should take 1 hour or less for each. Manometry: Experimental Procedures WHAT YOU ARE GOING TO DO: You will find the density of the fluid in one manometer by measuring the pressure applied to that manometer using a second manometer with a fluid of known density. WHAT YOU SHOULD LEARN: You should learn how a manometer works and how the elevation in a manometer depends on the density of the fluid in the manometer. DETAILS: In this experiment there are two U-tube manometers. One leg of each manometer is connected by a flexible tube to a hand pump. The purple fluid in one manometer has a known density of ρ=1 gm/cm 3. The flexible tube connecting the manometers is filled with air. The red fluid in the second manometer has an unknown density. You are to find the density of the red fluid. Procedure: p1. Sketch the set-up on the back of the attached data sheet. (This is for your reference when you write the lab report). p2. Apply a vacuum to the tubes between the manometers using the hand pump. Be careful: a short gentle stroke on the pump is very effective. Do not overflow the manometer. p3. Measure the difference in height between the two legs of each U-tube (both purple and red) and record on the data sheet. Release the vacuum using the red lever on the hand pump nozzle. p4. Repeat steps p2 and p3 three more times, each time with a different applied vacuum, to obtain a total of four readings. Calculations: c1. Use the equation for hydrostatic pressure of a constant density fluid to write an expression for the difference between air pressure in the tube between manometers and atmospheric pressure using the difference in height between the two legs of U-tube for the purple fluid; repeat for red fluid. c2. Set the equations in c1 equal. c3. Use equation in c2 to find density of the red fluid for the first applied vacuum; repeat for all readings. c4. Calculate the average density of the red fluid found from the four measurements. Write-Up: Include the following in your write-up: 1. Derivation of equations in steps c1 and c2; Sample of calculation (step c3). 2. Table of red fluid density for each applied pressure. Also include the average density in the table. 3. Discuss briefly (a few sentences per topic) Why can the effect of the air in the tube between the two manometers be neglected? Sources of error. 4. Manometry Data Sheet

2 MANOMETRY DATA SHEET Run #1 Run #2 Run #3 Run #4 h Purple liquid (ρ = g/cm 3 ) in. in. in. in. h Red liquid in. in. in. in. h signifies height difference Note any apparant sources of error in your measurements:

3 Buoyancy: WHAT YOU ARE GOING TO DO: You will find the density (mass/volume) of two different solid bodies in two different ways: 1) measure the volume of the body directly; 2) measure the volume using buoyancy relations. WHAT YOU SHOULD LEARN: You should learn how to get an accurate measure of a solid's volume using buoyancy. DETAILS: In this experiment, you are to find the density two different cylindrical bodies--one solid metal and one hollow plastic. Procedure: p1. You may ignore the weight of the cord, rod, hook, etc. in this experiment. p2. Sketch the metal body (body a) and measure its dimensions to the nearest 0.1 cm. p3. Screw the small hook all of the way into the rod at the bottom of the scale until the hook bottoms out. Zero the scale using the thumbwheel below the display. Weigh the body by hanging it from the scale, so that it is not submerged in the water. Be careful not to drop the weight in the water or torque the shaft connected to the scale. Record the weight. p4. Use the cord between the body and the scale so that the body is fully immersed in the water. Record the weight of the body when it is fully immersed (convert to N). This is equivalent to the tension force in the cord. p5. Sketch the experimental setup on the back of your data sheet. (This is for your reference when you write the lab report). p6. Repeat steps p2 through p5 for the plastic hollow body (body b), except here you must use the a rigid rod between the body and the scale to keep the body fully immersed in the water in step p4. The rigid rod is needed because the body floats, and you need to record the net upward force of the body on the scale when the body is submerged. To do this, screw the threaded rod into the plastic hollow body. Remove the hook from the rod attached to the scale and slide the narrow end of the rod attached to the plastic body into the rod attached to the scale. (Use caution so that the rod does not bend.) Calculations: c1. Using the dimensions that you measured in 1 above, calculate the volume of the body a in cm 3. c2. Calculate the density of the body using the weight of the body and the volume from step c1, in gm/cm 3. c3. Draw a free body diagram of the body when it is fully submerged in the fluid. Include the weight force, buoyancy force, and tension for in the string to the scale. c4. Using the principles of buoyancy and a force balance on the body, calculate the volume of the body in cm 3. c5. Calculate the density of the body using the volume from step c4, in gm/cm 3. c6. Repeat calculations for body b. Write-Up: Include the following in your write-up: 1. Calculations of volume and density (c1 and c2) based on dimensions for each body. 2. Free Body Diagram of each body and calculations of volume and density (c3, c4, and c5) for each body. 3. Discuss briefly (a few sentences per topic) Which method for calculating density is more accurate? Why? Sources of error. 4. Buoyancy Data Sheet Note: 1 lbf = N = 444,820 gm-cm/s 2

4 BUOYANCY DATA SHEET d a = cm (diameter of body a) h a = cm (height of body a) W a air = lbf = N (weight of body a in air) W a liquid = lbf = N (weight of body a in liquid) d b = cm (diameter of body b) h b = cm (height of body b) W b air = lbf = N (weight of body b in air) W b liquid = lbf = N (weight of body b in liquid) Note any apparant sources of error in your measurements:

6 Force on a Submerged Plane Data Sheet t = in (door thickness) h = in (door height) w = in (door width) d = in (vertical distance from point of rotation to liquid/air interface) l = in (vertical distance from point of rotation to attachment site of resisting cable) T = lbf (tension in resisting cable) Note any apparant sources of error in your measurements:

7 Practical Application--Hydraulic Jack: WHAT YOU ARE GOING TO DO: You will take apart a hydraulic jack and determine the work done by the jack in lifting a weight. WHAT YOU SHOULD LEARN: You should learn how a hydraulic jack is used to obtain leverage. DETAILS: The workings of common instruments are not always intuitive. For instance, a hydraulic jack lifts very heavy objects without requiring a lot of human effort to operate it. This is because a jack incorporates leverage, as do similar tools such as a block and tackle, a winch, and a prybar. In leveraging effort, the application of a small force through a long distance has the same effect as a greater force applied through a shorter distance. There are three identical hydraulic jacks on display for this lab: 1) jack with no fluid so that you can take it apart to see how it works; 2) jack with parts of it cut away to expose its inner workings; 3) jack that works and is used to make measurements. Below is a schematic of the pump. In the schematic, valve body contains one check valve which opens on pump upstroke and one which opens on pump downstroke. Pump Piston W Ram Piston Air Vent Pump Volume Valve Body Ram Volume Reservoir Release Valve Note the following: * Assume all three jacks in this lab are identical in every respect, so that you can make measurements on one jack and apply them to another jack. * Remember that energy is conserved (it is not created or destroyed, although it can be transformed from one type to another). * Work = W = F ds, where F is the vector force and s is the vector displacement. Note that for a constant force parallel to the displacement we can use the magnitudes and W = F ds = Fs. * Assume that the fluid used in the jack is incompressible and isothermal (remains at the same temperature) throughout the exercise. In other words, assume that all of your effort applied to the jack handle goes into raising the weight and doesn't go into raising the temperature of the fluid via compression or friction. * The jack's "leverage" is determined by the relative sizes of the ram (large) cylinder diameter and the pump (small) cylinder diameter. These will determine the distance each piston travels in the ram and pump cylinders.

8 Procedure: p1. Take apart the fluid-less model and measure: The distance from the mark on the handle to the pivot point. (Measure the distance parallel to the pump handle.) The distance from the pump piston connection point to the pivot point. The inside diameters of both the ram cylinder and the pump cylinder (not the piston diameter). Pump Handle Mark Pump Piston Connection Pivot Point Pump Piston Pump Piston and Handle Assembly p2. Figure out how the hydraulic jack works. Note the function of the valves. p3. Lift the weight by pumping the handle several times. Make sure that the weight is high enough so that it does not interfere with a complete stroke of the handle. Record the weight including the weight of the platform and bar (11.1 lbf). p4. Measure the distance traveled by the pump piston in one stroke of the handle by standing a ruler up next to the pump and noting how far the wire marker moves during a pump stroke. p5. Measure the distance traveled by the ram piston (ram plunger) in three pumps of the handle by standing a ruler up next to the pump and noting how far the wire marker moves during the pump strokes. Calculations: c1. Sketch a free body diagram of the ram piston. c2. Calculate the pressure in the hydraulic fluid in the chamber between the pump piston and the ram piston using the free body diagram of the ram piston. c3. Sketch a free body diagram of the pump piston and handle assembly (including the pivot point and the pump piston connection). c4. Calculate the minimum force that needs to be applied to the mark on the handle for the given weight based on the free body diagram in c3 by summing the moments about the pivot point. c5. Calculate the work done by the pump piston in one stroke of the handle. c6. Calculate the work done by the ram piston in one stroke of the handle. (You measured the travel for three strokes of the handle so you need to divide this measurement by 3 before calculating the work). Write-Up: Include the following in your write-up: 1. Calculation of the pressure in the hydraulic fluid (c1 and c2). 2. Calculation of the minimum force applied at the mark on the handle to lift the weight (c3 and c4). 3. Calculation of the work done by the pump piston and the ram piston (c5 and c6). 4. Discuss briefly (a few sentences per topic) Describe how the hydraulic jack works. What is the function of each of the three valves? Why is the work done by both pistons the same? In an analysis of the hydraulic jack we neglected friction due to o-rings, hinges, etc. Why can we do this?

9 5. Hydraulic Jack Data Sheet

10 HYDRAULIC JACK DATA SHEET rh= in. rp= in. dr= in. dp= in. W= lbf. dist pump = in. dist ram = in. distance from mark on handle to pivot pt. distance from pump piston connection to pivot pt. diameter of ram cylinder (not the piston) diameter of pump cylinder (not the piston) weight including weight platform and bar distance traveled by pump(1 stroke of pump) distance traveled by ram(3 strokes of pump) Note any apparant sources of error in your measurements:

11 Appendix: Fluid Statics When a fluid has no motion at all, the pressure gradient in the fluid balances the body forces in the fluid. This can be expressed mathematically as p = ρ f (1), where is the vector gradient operator, p is the pressure, ρ is the fluid density, and f is the vector body force. The most common body force is gravity. If z is positive upward from the surface of the earth, then the body force f is the acceleration of gravity, g, in the negative z direction. The body force is zero in the x and y direction. Equation (1) becomes dp dz = - ρ g (2). The minus sign on the right hand side results from the acceleration of gravity in the negative z direction. If the density and acceleration of gravity are independent of z (which they usually are in a laboratory situation), equation (2) can be integrated to p2-p1= - ρ g (z2-z1) (3). Manometry: From equation (3) it is apparent that the height of a static fluid column can be used to measure a pressure difference, if the density of the fluid is known. This is the basis of a device known as a manometer. In a manometer, the difference in height of fluid in the legs of a U-tube indicate the difference in pressure applied to each of the two legs. Consider the example shown below. p = p 2 atm p 1 g z 1 z 2 z=0 The left leg of the manometer is exposed to an unknown pressure p1. The pressure in the right leg is atmospheric, so p2=patm. The coordinates z1 and z2 are measured from an arbitrary reference plane. Substituting into equation (3) results in p1= ρ g (z2 - z1) +patm Multiple fluids in a manometer are handled similarly by careful and consistent application of equation (3). Buoyancy:

12 Two principles, developed by Archimedes in about 300 B.C., describe buoyancy forces on submerged or partially submerged bodies. These two principles are: 1) The buoyant force acting on a body, Fb, equals the weight of the fluid displaced by a body. This can be written as Fb = ρ g Vdisplaced (4), where Vdisplaced is the volume of fluid displaced by the body. 2) A floating body displaces its own weight in fluid. This is written as W = ρ g Vdisplaced (5), where W is the weight of the body. Forces on submerged plane surfaces: The fluid static pressure will exert a resultant force on a submerged surface that is given by F = p da A (6), where A is the "wetted" area over which the pressure acts. Consider the submerged plane surface shown below. Free Surface p = p =0 atm h θ point O y=0 da g h c Resultant force F y c y p Plane with surface area A y Here the coordinate y is parallel to the plane at angle θ with respect to the fluid surface. The depth to an incremental area, da, is h. Assuming that the reference plane is at the surface, we substitute z1=0, p1=patm=0, and z2=-h=(y sin θ) into equation (3) to obtain the pressure, p, acting on da at any arbitrary z2 on the surface as p = ρ g y sin θ (7).

13 Substituting (7) into (6) yields F = ρ g sin θ y da A (8). But the integral is simply the centroid of the surface times the area of the surface, yc A. The quantity in brackets is the depth of the centroid of the surface, hc, times the area of the surface, since hca = (ycent sin θ) A. Now (8) can be rewritten as F = ρ g hc A (9). Thus, all that is needed to calculate the force on the submerged plane surface is the area of the surface that is in contact with the fluid, A, and the distance from the fluid surface to the centroid of the surface, hc. Now the question that remains is where does the resultant force, F, act on the submerged surface? This can be answered by considering the moment balance about point O, where yp is the distance parallel to the surface to the center of pressure where F acts. The moment balance is y p F = y df A (10). Noting that df = p da and using (7) and (9), equation (10) can be rewritten as Simplifying results in y p (ρ g h c A) = y (ρ g y sin θ) da A (11). y p h c A = sin θ y 2 da A (12). But the integral is simply the moment of inertia about the horizontal axis, Ix, where x is perpendicular to the plane of the paper through point O. Noting that Ix = Ixc + yc 2 A, where Ixc is the moment of inertia about the centroidal horizontal axis, (12) can be rewritten as yp hc A = sin θ Ixc + sin θ yc 2 A (13). But hc = sin θ yc so we can finally write y p - y c = I xc y c A (14). Equation (14) gives the position of the position where a single force, F, equivalent to the distributed hydrostatic force would be applied.

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