107. A cylindrical log, 11 in. in diameter, is sawed off on such a slant that the pieces are 8 in. longer on the longest than on the shortest side. Find the dimensions of the ellipse thus made, and its area.

452. The number of vibrations that pendulums make in a given time is inversely as the square root of their lengths.

A pendulum passing its central point of rest once every mean solar second is 39.138 in. long.

108. Find the length of a pendulum beating half-seconds; of a pendulum beating quarter-seconds.

109. How many centimeters long is a pendulum swinging 80 times a minute? a pendulum swinging 30 times a minute?

110. If a cannon-ball be suspended by a fine wire 176 ft. long in the central well of the Bunker Hill Monument, how many times a minute will it swing?

453. If a plunger fits tightly in a small cylinder, and by it water is forced into a large cylinder, the plunger in the large cylinder is lifted with a force nearly equal to the product of the force with which the little plunger is driven in multiplied by the square of the ratio of the diameters of the two cylinders.

111. Find the lifting-power of a hydraulic press, the plunger being 1cm in diameter and driven with a force of 100kg, if the lifting-piston is 1m in diameter.

112. If the plunger is in. in diameter, and is driven with a force of 1000 lbs., how much can it lift with a liftingpiston 4 ft. in diameter?

113. If the plunger is 2 in. in diameter, and is driven with a force of 1000 lbs., how much can it lift with a lifting-piston 2 ft. in diameter?

114. The water stands in a fissure in a rock 10m high and

12m long. What pressure is exerted to split the rock

on the lowest meter's width? on the highest meter's width? in the whole fissure?

NOTE. This pressure is found by multiplying the surface upon one side by the height of water above the centre-line and counting the product as volume of water, and then finding the weight of this volume of water. The principle is precisely the same as in the hydraulic press.

115. A dam is 100 ft. long and is just flowing over it.

10 ft. deep, and the water

What pressure is exerted

over the lowest two feet of the dam?

454. A body falling in a vacuum falls 4.903m in the first second; it then has acquired a velocity of 9.806m.

A falling body increases its velocity in proportion to the time it is falling; and the distance fallen is in proportion to the square of the number of seconds of time it is falling.

Thus, a body falling from a state of rest in a vacuum will in half a second have fallen 1.2257m, and have acquired a velocity of 4.903TM; in 3 sec. it will have fallen 44.127", and have acquired a velocity of 29.418 per second.

The velocity of heavy bodies falling short distances in air will not be much less.

116. What velocity in meters a second will a cannon-ball acquire in falling three-quarters of a second? in falling three and a quarter seconds?

117. How long will it take a leaden ball, rolling off a table 29 in. high, to reach the floor?

118. What velocity will a crowbar attain in falling endwise from a balloon 2000m high? How long will it be in coming down?

119. What velocity will a crowbar attain in falling endwise from a balloon one mile and a quarter high? How long will it be coming down?

120. If Carisbrook Well is 210 ft. deep, how long after a pebble is dropped will it be before it is heard to strike the bottom, if its velocity is reckoned at 32 ft. at the end of a second, and the velocity of sound is 1120 ft. a second?

121. On the same suppositions as in Ex. 120, how long after a pebble is dropped will it be heard to strike the bottom of a ventilating shaft 1600 ft. deep?

122. If a pebble is dropped over a precipice, and is heard to strike the bottom in 7 sec., how far has it fallen, on the same suppositions as in Ex. 120?

123. A pebble dropped down a shaft is heard to strike the bottom in 3 sec. after it begins to fall. Find the depth of the shaft.

124. How long will it take a ball, rolling off a table, to drop 1cm? 1 in.? 10cm? 6 in.?

455. The velocity with which water will flow out of a hole in the side of a reservoir is nearly proportional to the square root of the depth of the hole below the surface of the water; and is about 32 ft. a second at the depth of 16 ft.

125. With what velocity will water flow through a hole 9 ft. below the surface?

126. With what velocity will water leave a fountain having free play, and a head of 25 ft.? a head of 100 ft.? 127. If a hole in the side of a cistern 4 ft. below the surface of the water is delivering 10 gals. an hour, how many gallons would it deliver with 5 ft. more head?

128. If a pipe 2 in. in diameter, and 1 ft. long, inserted in a dam, the head of water being kept constant, delivers 4 gals. a minute, how many gallons a minute may be expected when another pipe of the same length, but 2 in. in diameter, is substituted for the two-inch pipe?

129. If a one-inch pipe, 20 in. long, is substituted for the two-inch pipe, 1 ft. long, in Ex. 128, and the flow is found to be 5 pts. a minute, what part of the diminution is due to the smaller area of the orifice, and what part to the increased friction on the sides of the longer pipe?

456. The quantity of water issuing from a hole is in proportion to the square root of the head; and the velocity is in proportion to the square root of the head.

The work which the water can do is in proportion to the quantity multiplied by the square of the velocity; that is, The work is in proportion to the square root of the cube of the head.

130. A miller is using water flowing through the gate-way under 4 ft. head. How much more work could he do if the head was raised to 9 ft. ? how much more if the head was raised to 25 ft. ?

457. When a body is moving in a circle, the centrifugal force is about 1.227 of the continued product of the weight of the body, the number of feet in the radius of the circle, and the square of the number of revolutions in a second.

1

Thus, a body going round a circle of 5 ft. radius once a minute, presses away from the centre with a force equal to 1.227 × 5 × of the weight of the body.

602

NOTE. When the radius is measured in meters, the multiplier 4.025 must be used in place of 1.227.

131. If a top 3 in. in diameter is making 200 revolutions. a second, with what force does the outer layer pull away from the centre ?

132. If a sling 30 in. long contains a stone, and is whirled round 80 times a minute, what is the force pulling on the string?

133. With what force does a locomotive running at 30 mi. an hour, on a curve of 800 ft. radius, bear against the outer rail?

134. It washed wool is put wet into a wire basket 1.2m in diameter, and the basket is set to spinning at the

rate of 180 revolutions a minute, with what force is water wrung out of the wool?

135. If steel pens are revolved in a basket 32cm in diameter, 17 revolutions a second, with what force is the oil drained from them?

When a chain of uniform thickness hangs from two points not in the same vertical line, it hangs in a curve called a common catenary.

The length of the chain from the lowest point to any point selected may be called half-chain. The height of the point selected above the lowest point is called sag. The horizontal distance of the point selected from the lowest point is called half-span. The horizontal force with which the point selected is drawn inward is called tension. The radius of the circle which will fit the curve at its lowest point is called radius. The straight line which touches the curve at the point selected is called tangent.

The following propositions have been proved by higher mathematics:

I. Tension = weight of a piece of the same chain as long as the radius.

II. Radius sum of half-chain and sag multiplied by difference of half-chain and sag, and divided by twice sag.