472 APPLIED MECHANICS 
Let y= a (dan eye? 
dy _* (0 sin 6)\(1- cos 8)6* .' 0 *(6—sin 0)? 
3/8 —sin OA 1/0 — sin 0™\3% 
coo (1-208 6) — (==) 
y is a maximum when @ q 7 = 0, that is, when 
3/6 — sin 0\4 1/0 —sin 0\% 
(g(t 008 0) —3(“— 5 
which reduces to sin 6 = 6(3 cos 6 — 2). 4 
The value of 6, found by trial or by plotting, which satisfies this , 
equation, is 308° to the nearest degree. 
The depth of water in the channel is then 
+r cos (180 — °) = r(1 + cos 26°) = 1°8997, 
or practically 0°95 of the diameter. 
Exercises XXVIIIb. 
1. A horizontal pipe 3 inches in diameter suddenly enlarges to a pipe 4 inches _ 
in diameter. Water is flowing from the smaller to the larger pipe at the rate — 
of 90 gallons per minute. What is the loss of energy at the enlargement, in 
ft.-lbs. per minute ? 4 
2. If the water flows through the pipes of the preceding exercise at the same 
rate as before, but in the opposite direction. What is the loss of energy at the — 
sudden contraction, in ft.-lbs. per minute? Assume that the coefficient of — 
contraction (%) of the stream on entering the smaller pipe is 0°7. 
3. A pipe of 3 inches diameter conveying water is suddenly enlarged to 
5 inches diameter. A U-tube containing mercury is connected to two points, 
one on each side of the enlargement, at points where the flow is steady. Find 
the difference in level in the two limbs of the U when water flows at the rate of 
3 cubic foot per second from the small to the large section, and vice versd. The 
specific gravity of the mercury is 13°6. [U.L.] 
4. State briefly the laws of fluid friction deduced from the experiments « 
Froude. ‘Taking skin friction to be 0°4 1b. per square foot at 10 feet per second, 
find the skin resistance in pounds of a ship of 12,000 square feet immersed 
surface, at 15 knots. Also the horse-power to overcome skin friction. _ 
[Inst.C.E. 
5. The friction of a thin plate when moved edgewise throngh water is foun 
by experiment to be 4 Ib, per square foot of surface in contact with the water, 
when the velocity of rubbing is 600 feet per minute, and that it varies as the 
square of the velocity of rubbing. How many ft.-lbs. of work per minute will 
be expended in overcoming the skin friction in the case of a ship steaming at 
183 knots, if the immersed surface of the ship when floating at her load line is 
27,620 square feet? If this skin friction is 70 per cent. of the total resistances 
encountered by the ship, what is the total horse-power usefully expended in 
propelling the ship? [B.E.] 
6. Calculate the hydraulic mean depth for (1) a channel having a bottom 
width of 6 feet, side slopes of 2 vertical to 1 horizontal, and a depth of water 
5 feet; (2) a channel whose section is an arc of a circle of 4 feet radius, the 
greatest depth of water being 2 feet. 
7. Ina water main 3 feet in diameter the velocity of the water is 3 feet per 
second; Find the head lost in friction in feet per mile, using 0°005 as the 
coefficient of friction, 
