True/false. Check true or false, as appropriate:14
(a) When a fluid is subjected to a steady shear stress, it will reach a state of equilibrium in which no further motion occurs.
(b) Pressure and shear stress are two examples of a force per unit area.
(c) In fluid mechanics, the basic conservation laws are those of volume, energy, and momentum.
(d) Absolute pressures and temperatures must be employed when using the ideal gas law.
(e) The density of an ideal gas depends only on its absolute temperature and its molecular weight.
(f) Closely, the density of water is 1,000 kg/m3, and the gravitational acceleration is 9.81 m/s2.
(g) To convert pressure from gauge to absolute, add approximately 1.01 Pa.
(h) To convert from psia to psig, add 14.7, approximately.
(i) The absolute atmospheric pressure in the classroom is roughly one bar.
(j) If ρ is density in g/cm3 and μ is viscosity in g/cm s, then the kinematic viscosity ν = μ/ρ is in stokes.
(k) For a given liquid, surface tension and surface energy per unit area have identical numerical values and identical units.
(l) A force is equivalent to a rate of transfer of momentum.
(m) Work is equivalent to a rate of dissipation of power per unit time.
(n) It is possible to have gauge pressures that are as low as −20.0 psig.
(o) The density of air in the classroom is roughly 0.08 kg/m3.
(p) Pressure in a static fluid varies in the vertically upward direction z according to dp/dz = −ρgc.
(q) At any point, the rate of change of pressure with elevation is dp/dz = −ρg, for both incompressible and compressible fluids.
(r) A vertical pipe full of water, 34 ft high and open at the top, will generate a pressure of about one atmosphere (gauge) at its base.
(s) The horizontal force on one side of a vertical circular disk of radius R immersed in a liquid of density ρ, with its center a distance R below the free surface, is πR3ρg.
(t) For a vertical rectangle or dam of width W and depth D, with its top edge submerged in a liquid of density ρ, as in Fig. 1.15, the total horizontal thrust of the liquid can also be expressed as D 0 ρghW dh, where h is the coordinate measured downward from the free surface.
(u) The horizontal pressure force on a rectangular dam with its top edge in the free surface is Fx. If the dam were made twice as deep, but still with the same width, the total force would be 2Fx.
(v) A solid object completely immersed in oil will experience the same upward buoyant force as when it is immersed in water.
(w) Archimedes’ law will not be true if the object immersed is hollow (such as an empty box with a tight lid, for example).
(x) The rate of pressure change due to centrifugal action is given by ∂p/∂r = ρr2ω, in which ω is the angular velocity of rotation.
(y) To convert radians per second into rpm, divide by 120π.
(z) The shape of the free surface of a liquid in a rotating container is a hyperbola.
(A) The hydrostatic force exerted on one face of a square plate of side L that is held vertically in a liquid with one edge in the free surface is F. If the plate is lowered vertically by a distance L, the force on one face will be 3F.
Shear stresses for air and water—E. Consider the situation in Fig. 1.8, with h = 0.1 cm and V = 1.0 cm/s. The pressure is atmospheric throughout.
(a) If the fluid is air at 20 ◦C, evaluate the shear stress τa (dynes/cm2). Does τ vary across the gap? Explain.
(b) Evaluate τw if the fluid is water at 20 ◦C. What is the ratio τw/τa?
(c) If the temperature is raised to 80 ◦C, does τa increase or decrease? What about τw?
Centroid of triangle—E. A triangular plate held vertically in a liquid has one edge (of length B) coincident with the surface of the liquid; the altitude of the plate is H. Derive an expression for the depth of the centroid. What is the horizontal force exerted by the liquid, whose density is ρ, on one side of the plate?
Newspaper statements about the erg—E. In the New York Times for January 18, 1994, the following statement appeared: “An erg is the metric unit scientists use to measure energy. One erg is the amount of energy it takes to move a mass of one gram one centimeter in one second.” (This statement related to the earthquake of the previous day, measuring 6.6 on the Richter scale, in the Northridge area of the San Fernando Valley, 20 miles north of downtown Los Angeles.)
Also in the same newspaper, there was a letter of rebuttal on January 30 that stated in part: “This is not correct. The energy required to move a mass through a distance does not depend on how long it takes to accomplish the movement. Thus the definition should not include a unit of time.”
A later letter from another reader, on February 10, made appropriate comments about the original article and the first letter. What do you think was said in the second letter?
Energy to place satellite in orbit—M . “NASA launched a $195 million astronomy satellite at the weekend to probe the enigmatic workings of neutron stars, black holes, and the hearts of galaxies at the edge of the universe . . . The long-awaited mission began at 8:48 a.m. last Saturday when the satellite’s Delta–2 rocket blasted off from the Cape Canaveral Air Station.”13
This “X-ray Timing Explorer satellite” was reported as having a mass of 6,700 lbm and being placed 78 minutes after lift-off into a 360-mile-high circular orbit (measured above the earth’s surface).
How much energy (J) went directly to the satellite to place it in orbit? What was the corresponding average power (kW)? The force of attraction between a mass m and the mass Me of the earth is GmMe/r2, where r is the distance of the mass from the center of the earth and G is the universal gravitational constant. The value of G is not needed in order to solve the problem, as long as you remember that the radius of the earth is 6.37 × 106 m and that g = 9.81 m/s2 at its surface.
Oil and water in rotating container—E. A cylindrical container partly filled with immiscible layers of water and oil is placed on a rotating turntable. Develop the necessary equations and prove that the shapes of the oil/air and water/oil interfaces are identical.
Rotatin mercury mirror—M . Physicist Ermanno Borra, of Laval University in Quebec, has made a 40-in. diameter telescopic mirror from a pool of mercury that rotates at one revolution every six seconds.12 (Air bearings eliminate vibration, and a thin layer of oil prevents surface ripples.) By what value Δz would the surface at the center be depressed relative to the perimeter, and what is the focal length (m) of the mirror? The mirror cost Borra $7,500. He estimated that a similar 30-meter mirror could be built for $7.5 million. If the focal length were unchanged, what would be the new value of Δz for the larger mirror? Hint: the equation for a parabola of focal length f is r2 = 4fz.
Force on V-shaped dam—M . A vertical dam has the shape of a V that is 3 m high and 2 m wide at the top, which is just level with the surface of the water upstream of the dam. Use two different methods to determine the total force (N) exerted by the water on the dam.
Grand Coulee dam—E. The Grand Coulee dam, which first operated in 1941, is 550 ft high and 3,000 ft wide. What is the pressure at the base of the dam, and what is the total horizontal force F lbf exerted on it by the water upstream?
Pressure variations in air—M . Refer to Example 1.5 concerning the pressure variations in a gas, and assume that you are dealing with air at 40 ◦F. Suppose further that you are using just the linear part of the expansion (up to the term in z) to calculate the absolute pressure at an elevation z above ground level. How large can z be, in miles, with the knowledge that the error amounts to no more han 1% of the exact value?
Oil and gas well pressures—M . A pressure gauge at the top of an oil well 18,000 ft deep registers 2,000 psig. The bottom 4,000-ft portion of the well is filled with oil (s = 0.70). The remainder of the well is filled with natural gas (T = 60 ◦F, compressibility factor Z = 0.80, and s = 0.65, meaning that the molecular weight is 0.65 times that of air). Calculate the pressure (psig) at (a) the oil/gas interface, and (b) the bottom of the well.