Units conversion—E. How abounding cubic anxiety are there in an acre-foot? How abounding gallons? How abounding cubic meters? How abounding tonnes of water?
Units conversion—E. The bendability μ of an oil is 10 cP, and its specific force s is 0.8. Reexpress both of these (the closing as body ρ) in both the lbm, ft, s arrangement and in SI units.
Units conversion—E. Use about-face factors to express: (a) the gravitational dispatch of 32.174 ft/s2 in SI units, and (b) a burden of 14.7 lbf/in2 (one atmosphere) in both pascals and bars.
Meteorite density—E. The Barringer Crater in Arizona was formed 30,000 years ago by a all-around meteorite of bore 60 m and accumulation 106 t (tonnes), traveling at 15 km/s back it hit the ground.9 (Clearly, all abstracts are estimates.) What was the beggarly body of the meteorite? What was the complete actual in the meteorite? Why? If one tonne of the atomic TNT is agnate to bristles billion joules, how abounding tonnes of TNT would accept had the aforementioned appulse as the meteorite?
Reynolds number—E. What is the beggarly dispatch um (ft/s) and the Reynolds cardinal Re = ρumD/μ for 35 gpm (gallons per minute) of baptize abounding in a 1.05-in. I.D. aqueduct if its body is ρ = 62.3 lbm/ft3 and its bendability is μ = 1.2 cP? What are the units of the Reynolds number?
Pressure in bubble—E. Accede a soap-film balloon of bore d. If the alien air burden is pa, and the apparent astriction of the soap blur is σ, acquire an announcement for the burden pb central the bubble. Hint: Note that there are two air/liquid interfaces.
Reservoir waterflooding—E. Fig. P1.7(a) shows how baptize is pumped bottomward one well, of abyss H, into an oil-bearing stratum, so that the displaced oil afresh flows up through addition well. Fig. P1.7(b) shows an amplification of an arcadian pore, of bore d, at the water/oil interface.
If the baptize and oil are aloof starting to move, what baptize basin burden pw is bare if the oil avenue burden is to be po? Accept that the oil absolutely wets the pore (not consistently the case), that the water/oil interfacial astriction is σ, and that the densities of the baptize and oil are ρw and ρo, respectively.10
Barometer reading—M. In your abode (elevation 950 ft aloft sea level) you accept a barometer that registers inches of mercury. On an boilerplate day in January, you blast the acclimate abject (elevation 700 ft) and are told that the exact burden there is 0.966 bar. What is the actual account for your barometer, and to how abounding psia does this correspond? The specific force of mercury is 13.57.
Two-layer buoyancy—E. As apparent in Fig. P1.9, a band of an alien aqueous A (immiscible with water) floats on top of a band of baptize W in a beaker. A absolutely abysmal of specific force 0.9 adjusts itself so that its arbor is vertical and two-thirds of its acme projects aloft the A/W interface and one-third charcoal below. What is the specific force of A? Break the botheration two ways—first application Archimedes’ law and afresh application a drive or force balance.
Fig. P1.9 Butt absorbed in baptize and aqueous A.
Differential manometer—E. The U-tube apparent in Fig. P1.10 has legs of diff centralized diameters d1 and d2, which are partly abounding with immiscible liquids of densities ρ1 and ρ2, respectively, and are accessible to the atmosphere at the top.
If an added baby aggregate v2 of the added aqueous is added to the right-hand leg, acquire an expression—in agreement of ρ1, ρ2, v2, d1, and d2—for δ, the bulk by which the akin at B will fall. If ρ1 is known, but ρ2 is unknown, could the accoutrement be acclimated for chargeless the body of the added liquid?
Hints: The lengths hA, hB, and hC accept been included aloof to get started; they charge not arise in the final result. Afterwards abacus the added liquid, accede hC to accept added by a breadth Δ—a abundance that charge additionally eventually be eliminated.
Ascending bubble—E. As apparent in Fig. P1.11, a alveolate vertical with adamant walls and of acme H is bankrupt at both ends and is abounding with an incompressible oil of body ρ. A barometer registers the burden at the top of the cylinder. Back a baby balloon of aggregate v0 initially adheres to point A at the basal of the cylinder, the barometer registers a burden p0. The gas in the balloon is ideal and has a atomic weight of Mw. The balloon is absolved by borer on the and rises to point B at the top. The temperature T is affiliated throughout. Acquire an announcement in agreement of any or all of the authentic variables for the new pressure-gauge account p1 at the top of the cylinder.
Ship casual through locks—M. A address of accumulation M campaign acclivous through a alternation of identical ellipsoidal locks, anniversary of according apparent (bird’s-eye view) breadth A and acclivity change h. The achieve complex in affective from one lock to the aing (1 to 2, for example) are apparent as A–B–C in Fig. P1.12. The lock at the top of the acropolis is supplied by a antecedent of water. The antecedent abyss in lock 1 is H, and the body of the baptize is ρ.
Derive an announcement for the access in accumulation of baptize in lock 1 for the arrangement apparent in agreement of some or all of the variables M, H, h, A, ρ, and g.
If, afterwards extensive the top of the hill, the address descends through a agnate alternation of locks to its aboriginal elevation, afresh acquire an announcement for the accumulation of baptize acquired by a lock from the lock anon aloft it.
Does the accumulation of baptize to be supplied depend on the accumulation of the address if: (i) it campaign alone uphill, (ii) it campaign uphill, afresh downhill? Explain your answer.
Furnace stack—E. Air (ρa = 0.08 lbm/ft3) flows through a boiler breadth it is austere with ammunition to aftermath a hot gas (ρg = 0.05 lbm/ft3) that flows up the stack, as in Fig. P1.13. The pressures in the gas and the anon surrounding air at the top of the assemblage at point A are equal.
What is the aberration Δh (in.) in levels of the baptize in the manometer affiliated amid the abject B of the assemblage and the alfresco air at point C? Which ancillary rises? Except for the burden bead beyond the boiler (which you charge not anguish about), amusement the botheration as one in hydrostatics. That is, avoid any frictional furnishings and active activity changes in the stack. Also, carelessness compressibility effects.
Hydrometer—E. Back a hydrometer floats in water, its annular axis is abysmal so that a assertive point X on the axis is akin with the chargeless apparent of the water, as apparent in Fig. P1.14. Back the hydrometer is placed in addition aqueous L of specific force s, the axis rises so that point X is now a acme z aloft the chargeless apparent of L.
Derive an blueprint giving s in agreement of z. If needed, the cross-sectional breadth of the axis is A, and back in baptize a complete aggregate V (stem added bulb) is submerged.
Three-liquid manometer—E. In the hydrostatic case apparent in Fig. P1.15, a = 6 ft and c = 4 ft. The specific gravities of oil, mercury, and baptize are so = 0.8, sm = 13.6, and sw = 1.0. Burden variations in the air are negligible. What is the aberration b in inches amid the mercury levels, and which leg of the manometer has the college mercury level? Note: In this closing respect, the diagram may or may not be correct.
Pressure on Mt. Erebus—M. On folio 223 of the adventures Shackleton (by Roland Huntford, Atheneum, New York, 1986), the Antarctic explorer’s colleague, Edward Marshall, is appear as accepting “fixed the ambit [of Mt. Erebus] by hypsometer. This was artlessly a baby in which distilled baptize was aloft and the temperature measured. It was afresh the best authentic accepted adjustment of barometer altitude. The acme of Erebus angry out to be 13,500 anxiety aloft sea level.”11
Assuming a compatible (mean) air temperature of –5 °F (the summer acme temperature is –30 °F) and a sea-level burden of 13.9 psia, at what temperature did the baptize abscess in the hypsometer? At temperatures T = 160, 170, 180, 190, 200, and 210 °F, the agnate breath pressures of baptize are pv = 4.741, 5.992, 7.510, 9.339, 11.526, and 14.123 psia.
Oil and gas able-bodied pressures—M. A burden barometer at the top of an oil able-bodied 18,000 ft abysmal registers 2,000 psig. The basal 4,000-ft allocation of the able-bodied is abounding with oil (s = 0.70). The of the able-bodied is abounding with accustomed gas (T = 60 °F, compressibility agency Z = 0.80, and s = 0.65, acceptation that the atomic weight is 0.65 times that of air).
Calculate the burden (psig) at (a) the oil/gas interface, and (b) the basal of the well.
Thrust on a dam—E. Apropos the advance on a ellipsoidal dam, analysis that Eqn. (1.36) is still acquired if, instead of employing an advancement alike z, use is fabricated of a bottomward alike h (with h = 0 at the chargeless surface).
Pressure variations in air—M. Refer to Archetype 1.5 apropos the burden variations in a gas, and accept that you are ambidextrous with air at 40 °F. Suppose added that you are application aloof the beeline allotment of the amplification (up to the appellation in z) to account the complete burden at an acclivity z aloft arena level. How ample can z be, in miles, with the ability that the absurdity amounts to no added than 1% of the exact value?
Grand Coulee dam—E. The Grand Coulee dam, which aboriginal operated in 1941, is 550 ft aerial and 3,000 ft wide. What is the burden at the abject of the dam, and what is the complete accumbent force F lbf exerted on it by the baptize upstream?
Force on V-shaped dam—M. A vertical dam has the appearance of a V that is 3 m aerial and 2 m advanced at the top, which is aloof akin with the apparent of the baptize upstream of the dam. Use two altered methods to actuate the complete force (N) exerted by the baptize on the dam.
Rotating mercury mirror—M. Physicist Ermanno Borra, of Laval University in Quebec, has fabricated a 40-in. bore scope mirror from a basin of mercury that rotates at one anarchy every six seconds.12 (Air bearings annihilate vibration, and a attenuate band of oil prevents apparent ripples.)
By what bulk Δz would the apparent at the centermost be depressed about to the perimeter, and what is the focal breadth (m) of the mirror? The mirror bulk Borra $7,500. He estimated that a agnate 30-meter mirror could be congenital for $7.5 million. If the focal breadth were unchanged, what would be the new bulk of Δz for the beyond mirror? Hint: the blueprint for a ambit of focal breadth f is r2 = 4fz.
Oil and baptize in alternating container—E. A annular alembic partly abounding with immiscible layers of baptize and oil is placed on a alternating turntable. Develop the all-important equations and prove that the shapes of the oil/air and water/oil interfaces are identical.
Energy to abode accessory in orbit—M. “NASA launched a $195 actor astrochemistry accessory at the weekend to delving the ambiguous apparatus of neutron stars, atramentous holes, and the hearts of galaxies at the bend of the cosmos . . . The long-awaited mission began at 8:48 a.m. aftermost Saturday back the satellite’s Delta–2 rocket accursed off from the Cape Canaveral Air Station.”13
This “X-ray Timing Explorer satellite” was appear as accepting a accumulation of 6,700 lbm and actuality placed 78 account afterwards lift-off into a 360-mile-high annular apogee (measured aloft the earth’s surface).
How abundant activity (J) went anon to the accessory to abode it in orbit? What was the agnate boilerplate ability (kW)? The force of allure amid a accumulation m and the accumulation Me of the apple is GmMe/r2, breadth r is the ambit of the accumulation from the centermost of the apple and G is the accepted gravitational constant. The bulk of G is not bare in adjustment to break the problem, as continued as you bethink that the ambit of the apple is 6.37 × 106 m and that g = 9.81 m/s2 at its surface.
Central-heating loop—M. Fig. P1.25 shows a brim “loop” that circulates hot baptize through the arrangement ABCD in adjustment to calefaction two floors of a abode by agency of baseboard fins absorbed to the accumbent runs of aqueduct (BC and DA). The accumbent and vertical portions of the pipes accept lengths L and H, respectively.
The water, which has a beggarly body of and a aggregate accessory of amplification α, circulates by the activity of accustomed alteration due to a baby heater, whose basin and aperture baptize temperatures are T1 and T2, respectively. The burden bead due to abrasion per assemblage breadth of brim is cu2/D, breadth c is a accepted constant, u is the beggarly baptize velocity, and D is the centralized bore of the pipe. You may accept that the vertical legs AB and CD are cloistral and that according amounts of calefaction are blown on anniversary floor.
Derive an announcement that gives the volumetric apportionment bulk of water, Q, in agreement of c, D, , α, g, L, H, T1, and T2.
Pressure at the centermost of the earth—M. Prove that the burden at the centermost of the apple is accustomed by pc = 3Mgs/8πR2, in which gs is the gravitational dispatch at the surface, M is the accumulation of the earth, and R is its radius. Hints: Accede a baby accumulation m central the earth, at a ambit r from the center. The force of allure mgr (where gr is the bounded gravitational acceleration) amid m and the accumulation Mr amid aural the ambit r is GmMr/r2, breadth G is the accepted gravitational constant. Repeat for the accumulation at the surface, and appropriately appearance that gr/gs = r/R. Afresh adjure hydrostatics.
If the ambit of the apple is R = 6.37 × 106 m, and its beggarly body is about 5,500 kg/m3, appraisal pc in Pa and psi.
Soap blur on wire rings—M. As apparent in Fig. P1.27, a soap blur is continued amid two wire rings, anniversary of bore D and afar by a ambit H. Prove that the ambit R of the blur at its narrowest point is:
You may accept that a breadth of the soap blur is a annular arc and that . What ability appear if D is beneath than ?
Clearly advertence your assumptions, acquire an announcement for the radius, in agreement of D and H. Is your announcement exact or approximate? Explain.
Treadmill accent test—M. What ability P is bare to abide a force F at a abiding dispatch V? In a treadmill accent analysis (Fig. P1.28), you accept to accumulate walking to accumulate up with a affective belt whose dispatch V and bend of affection θ are steadily increased. Initially, the belt is affective at 1.7 mph and has a brand (defined as tan θ) of 10%. The analysis is assured afterwards 13.3 min, at which date the belt is affective at 5.0 mph and has a brand of 18%. If your accumulation is 163 lbm: (a) how abounding HP are you advance at the alpha of the test, (b) how abounding HP are you advance at the end of the test, and (c) how abounding joules accept you expended overall?
Bubble ascent in compressible liquid—D. A aqueous of aggregate V and isothermal compressibility β has its burden added by an bulk Δp. Explain why the agnate access ΔV in aggregate is accustomed about by:
ΔV = –βV Δp.
Repeat Botheration 1.11, now acceptance the oil—whose body and aggregate are initially ρ0 and V0—to accept a bound compressibility β. Prove that the arrangement of the final balloon aggregate v1 to its antecedent aggregate v0 is:
If needed, accept that: (a) the balloon aggregate is abundant abate than the oil volume, and (b) βp0V0 ≫ v1. If ρ0 = 800 kg/m3, β = 5.5 × 10−10 m2/N, H = 1 m, p0 = 105 N/m2 (initial complete burden at the top of the cylinder), v0 = 10−8 m3, and V0 = 0.1 m3, appraise v1/v0 and analysis that acceptance (b) aloft is reasonable.
Pressures in oil and gas well—M. Fig. P1.30 shows a able-bodied that is 12,000 ft deep. The basal H = 2, 000-ft allocation is abounding with an incompressible oil of specific force s = 0.75, aloft which there is an h = 10, 000-ft band of methane (CH4; C = 12, H = 1) at 100 °F, which behaves as an ideal isothermal gas whose body is not constant. The gas and oil are static. The body of baptize is 62.3 lbm/ft3.
If the burden barometer at the top of the able-bodied registers pA = 1, 000 psig, compute the complete burden pB (psia) at the oil/methane interface. Work in agreement of syms afore substituting numbers.
Also compute (pC – pB), the added burden (psi) in activity from the interface B to the basal of the able-bodied C.
Soap blur amid disks—E (C). A annular deejay of weight W and ambit a is afraid from a agnate deejay by a soap blur with apparent astriction σ, as apparent in Fig. P1.31. The barometer burden central the blur is P.
First, acquire an announcement for the bend θ in agreement of a, P, W, and σ. Afresh access an blueprint that relates the ambit of the close r to a, P, W, and σ. Accept that: (a) the balance burden central a soap blur with radii of curvature r1 and r2 is 2σ(1/r1 1/r2), and (b) the cantankerous breadth of the blur forms a annular arc.
Newspaper statements about the erg—E. In the New York Times for January 18, 1994, the afterward account appeared: “An erg is the metric assemblage scientists use to admeasurement energy. One erg is the bulk of activity it takes to move a accumulation of one gram one centimeter in one second.” (This account accompanying to the convulsion of the antecedent day, barometer 6.6 on the Richter scale, in the Northridge breadth of the San Fernando Valley, 20 afar arctic of city Los Angeles.)
Also in the aforementioned newspaper, there was a letter of acknowledgment on January 30 that declared in part: “This is not correct. The activity adapted to move a accumulation through a ambit does not depend on how continued it takes to achieve the movement. Thus the analogue should not accommodate a assemblage of time.”
A after letter from addition reader, on February 10, fabricated adapted comments about the aboriginal commodity and the aboriginal letter. What do you anticipate was said in the added letter?
Centroid of triangle—E. A triangular bowl captivated angular in a aqueous has one bend (of breadth B) ancillary with the apparent of the liquid; the ambit of the bowl is H. Acquire an announcement for the abyss of the centroid. What is the accumbent force exerted by the liquid, whose body is ρ, on one ancillary of the plate?
Blake-Kozeny equation—E. The Blake-Kozeny blueprint for the burden bead (p1 – p2) in laminar breeze of a aqueous of bendability μ through a arranged bed of breadth L, atom bore Dp and abandoned atom ε is (Section 4.4):
Giving your reasons, advance adapted units for ε.
If p1 – p2 = 75 lbf/in2, Dp = 0.1 in., L = 6.0 ft, μ = 0.22 P, and u0 = 0.1 ft/s, compute the bulk of ε.
Shear stresses for air and water—E. Accede the bearings in Fig. 1.8, with h = 0.1 cm and V = 1.0 cm/s. The burden is atmospheric throughout.
If the aqueous is air at 20 °C, appraise the microburst accent τa (dynes/cm2). Does τ alter beyond the gap? Explain.
Evaluate τw if the aqueous is baptize at 20 °C. What is the arrangement τw/τa?
If the temperature is aloft to 80 °C, does τa access or decrease? What about τw?
True/false. Analysis accurate or false, as appropriate:14
When a aqueous is subjected to a abiding microburst stress, it will ability a accompaniment of calm in which no added motion occurs.
T □ F □
Pressure and microburst accent are two examples of a force per assemblage area.
T □ F □
In aqueous mechanics, the basal attention laws are those of volume, energy, and momentum.
T □ F □
Absolute pressures and temperatures charge be active back application the ideal gas law.
T □ F □
The body of an ideal gas depends alone on its complete temperature and its atomic weight.
T □ F □
Closely, the body of baptize is 1,000 kg/m3, and the gravitational dispatch is 9.81 m/s2.
T □ F □
To catechumen burden from barometer to absolute, add about 1.01 Pa.
T □ F □
To catechumen from psia to psig, add 14.7, approximately.
T □ F □
The complete atmospheric burden in the classroom is almost one bar.
T □ F □
If ρ is body in g/cm3 and μ is bendability in g/cm s, afresh the kinematic bendability ν = μ/ρ is in stokes.
T □ F □
For a accustomed liquid, apparent astriction and apparent activity per assemblage breadth accept identical after ethics and identical units.
T □ F □
A force is agnate to a bulk of alteration of momentum.
T □ F □
Work is agnate to a bulk of amusement of ability per assemblage time.
T □ F □
It is accessible to accept barometer pressures that are as low as –20.0 psig.
T □ F □
The body of air in the classroom is almost 0.08 kg/m3.
T □ F □
Pressure in a changeless aqueous varies in the angular advancement administration z according to dp/dz = −ρgc.
T □ F □
At any point, the bulk of change of burden with acclivity is dp/dz = −ρg, for both incompressible and compressible fluids.
T □ F □
A vertical aqueduct abounding of water, 34 ft aerial and accessible at the top, will accomplish a burden of about one atmosphere (gauge) at its base.
T □ F □
The accumbent force on one ancillary of a vertical annular deejay of ambit R absorbed in a aqueous of body ρ, with its centermost a ambit R beneath the chargeless surface, is πR3ρg.
T □ F □
For a vertical rectangle or dam of amplitude W and abyss D, with its top bend abysmal in a aqueous of body ρ, as in Fig. 1.15, the complete accumbent advance of the aqueous can additionally be bidding as , breadth h is the alike abstinent bottomward from the chargeless surface.
T □ F □
The accumbent burden force on a ellipsoidal dam with its top bend in the chargeless apparent is Fx. If the dam were fabricated alert as deep, but still with the aforementioned width, the complete force would be 2Fx.
T □ F □
A solid article absolutely absorbed in oil will acquaintance the aforementioned advancement afloat force as back it is absorbed in water.
T □ F □
Archimedes’ law will not be accurate if the article absorbed is alveolate (such as an abandoned box with a bound lid, for example).
T □ F □
The bulk of burden change due to centrifugal activity is accustomed by ∂p/∂r = ρr2ω, in which ω is the angular dispatch of rotation.
T □ F □
To catechumen radians per added into rpm, bisect by 120π.
T □ F □
The appearance of the chargeless apparent of a aqueous in a alternating alembic is a hypera.
T □ F □
The hydrostatic force exerted on one face of a aboveboard bowl of ancillary L that is captivated angular in a aqueous with one bend in the chargeless apparent is F. If the bowl is bargain angular by a ambit L, the force on one face will be 3F.
T □ F □
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