P6.10

Scuba divers carry ballast weights to have neutral buoyancy. At that condition,

the buoyancy force on the diver exactly balances weight, and there is no

tendency either to float toward the surface or to sink. In freshwater, a certain

diver carries 10 lb of lead alloy ballast of density 1.17 104

kg/m3

. During

an excursion in seawater, the diver must carry 50% more ballast to remain

neutrally buoyant. How much does this diver weigh?

P6.11

Examine the transition between the laminar and turbulent flows of water by

sketching the stream of water that exits from a faucet (without an aerator) or

hose (without a nozzle). You can control the velocity of the stream by adjusting

the water valve.

(a) Make sketches of the flow for four different fluid speeds: two below and

two above the laminar-turbulent transition point.

(b) Estimate the velocity of the water by determining the time Dt required

to fill a container of known volume, such as a beverage container. By

measuring the diameter of the stream with a ruler, calculate the average

speed of the water from Equations (6.9) and (6.10).

(c) For each speed, calculate the Reynolds number.

(d) Indicate the value of the Reynolds number where turbulence begins.

P6.12

Water flows through a 5 cm diameter pipe at the average velocity of 1.25 m/s.

(a) In the dimensions of L/s, what is the volumetric flow rate?

(b) If the diameter of the pipe is reduced by 20% at a constriction, by what

percentage does the water's velocity change?

P6.13

In a pipeline that connects a production oilfield to a tanker terminal, oil having

density 1.85 slug/ft3

and viscosity 6 10?3 slug/(ft s) flows through a 48 in.

diameter pipeline at 6 mph. What is the Reynolds number? Is the flow laminar

or turbulent

(1) (20 points total) (a) (5 points) Suppose X, Y are two univariate continuous random variables with Ex [x] = Ey ly] = 0, and let Vx,y[x + y] denote the variance of their sum. Show that Vx, y [x + y] = Vx[x] + Vy ly] + 2 Cov[x, y]. Please explain each step of your calculation. It may be helpful to use the formulas for the variance and covariance that we established in class: Vx(x] = Ex(x2] - (Ex [x])2 Vyly] = Ey ly2] - (Ex [y])2 Cov[x, y] = Ex,y [xy] - (Ex [x])(Exly]) (b) (5 points) Conclude that if X, Y are statistically independent, then Vx, y [x + y] = Vx[x] + Vyly]. (c) (5 points) Now suppose X = (X1, . . ., Xn) and Y = (Y1, . . ., Yn) E R" are two multivariate random variables whose components Xi, Y, are i.i.d. (independent and identically distributed), with distributions that are symmetric about 0 with mean 0. Show that . Ex. v [x . y] = 0 and . V x, v (x . y] = n V x., .[xigi]. (d) (5 points) Now suppose m |z . w|. (ii) It is more likely that (x . y) z . w| and |x . y) n E A. Show that for any set A, if A* is expressible in L, then there is a Gdel sentence for A