LECTURE XX.-QUESTIONS.

1. Draw a section through a hydrostatic press, showing the cylinder, ram, and force pump, together with the valves. Why is the base of the cylinder of a large press rounded instead of being flat as in a steam cylinder? If the diameter of the ram is 9 times that of the force pump, and if Q be the pressure on the pump, what is the pressure exerted by the ram, neglecting friction? Ans. 81 P.

2. Explain by aid of a sketch the mode of packing the ram of a hydraulic press and explain how it acts. The force which actuates the force pump is applied at the end of a lever giving a mechanical advantage of 14 to 1, and the area of the plunger of the pump is I square inch. What pressure must be applied to the end of the lever to produce a pressure of 1 ton per square inch on the water enclosed in the press? Ans. 160 lbs.

3. In the force pump of a press the area of the plunger is of a square inch, the distance from the fulcrum of the lever handle to the plunger is 2 inches, and the distance from the fulcrum to the other end of the lever is 2 feet; what pressure per square inch is exerted on the water underneath the plunger, when a weight of 20 lbs. is hung at the end of the lever handle? Ans. 720 lbs. per square inch.

4. In what way do you estimate the theoretical advantage gained by the use of the hydraulic press? In a small press the ram is 2 inches and the plunger inch in diameter; the length of the lever handle is 2 feet, and the distance from the fulcrum to the plunger is 1 inches. Find the pressure exerted on the ram when 10 lbs. is hung at the end of the lever. Ans. 2560 lbs.

5. In an hydraulic press with two pumps the plungers are 24 and I inch in diameter, and each is worked by a similar lever, which is acted on by the same force. When the larger pump alone is at work the pressure on the ram is 40 tons; what will it be when the smaller plunger is only working? Ans. 250 tons.

6. An hydraulic press, which is used for making lead pipes, has a ram 20 inches in diameter, while the ram which presses the lead is 5 inches in diameter. Find the pressure per square inch on the lead when the hydraulic gauge indicates I ton per square inch. Sketch a sectional elevation of the press, and show the packing of the hydraulic ram. (S. and A. Exam. 1891.) Ans. 16 tons.

7. How is the pressure taken off the object under compression when required, in a hydraulic press? Sketch the arrangement. What is the proportion of the diameters of the plunger and ram when the theoretical advantage gained thereby is 100 to 1, neglecting friction? (S. and A. Exam. 1888.) Ans. I to 10.

8. Make a rough sketch, and write a short description of the hydraulic lifting jack. It may be arranged on any system that you are acquainted with. Show clearly how the valves act and how the jack is lowered.

9. Sketch and describe the hydraulic bear or portable punching machine. Explain how the punch is raised and how the tool is handled.

Io. Sketch and describe the construction of a vessel suitable for storing up a supply of water under pressure, and intended for actuating hydraulic machinery. If the plunger of this vessel be 17 inches in diameter, what load will bring the pressure of the water to 700 lbs. per square inch? Ans. 158,950 lbs.

11. Šketch and describe the hydraulic accumulator for storing up water

under pressure. If the ram of the accumulator be 6 inches in diameter, what load will be required to produce a water pressure of 500 lbs. on the square inch? To what head of water would this pressure correspond? (S. and A. Exam. 1887.) Ans. 14,142.8 lbs. and 1152 feet.

12. An hydraulic accumulator, with a ram of 16 inches in diameter, is connected with an hydraulic press whose ram is 26 inches in diameter. The load on the accumulator is 80 tons; what force would the press exert? Make a vertical section through the accumulator, showing its construction. (8. and A. Exam. 1889.) Ans. 211*25 tons.

13. Make a sectional sketch of a hydrostatic press suitable for giving a pressure of 100 tons, showing the valves and pump and by what contrivance the leakage of water is prevented.

The pump for such a press has a cylindrical plunger I inch in diameter with a lever of 10 to 1, what should be the least diameter of the ram which would give 100 tons pressure when a force of 56 lbs. was applied at the end of the pump lever? What form is most suitable for the base of the ram cylinder, and for what reason is a special form adopted? (S. and A. Exam. 1893.) Ans. 20 inches.

14. Sketch and describe any tool used by riveters and worked by water pressure. (S. E. B. 1902.)

15. The pressure of water in a hydraulic company's main is 750 lb. per square inch, and the average flow is 25 cubic feet per minute. What horse-power does this represent? If the charge for the water is twopence per 100 gallons, what is the cost per horse-power hour? (S. E. B. 1902.) Ans. 818; 2.3d.

16. Distinguish between the velocity ratio and the mechanical advantage of a machine. In a hydraulic lifting jack the ram is 6" in diameter, the pump plunger is 3" diameter, the leverage for working the pump is 10 to 1. What is the velocity ratio of the machine? Experimentally we find that a force of 20 lbs. applied at the end of the lever lifts a weight of 8500 lbs. on the end of the ram. What is the mechanical advantage of the machine? (S. and A. 1899.) Ans. 470; 425.

17. A hydraulic crane is supplied with water at a pressure of 700 lbs. per square inch, and uses 2 cubic feet of water in order to lift 4 tons through a height of 12 feet. How much energy has been supplied to the crane? and how much has been converted into useful work? (S. and A. 1899.) Ans. 201,600 ft. -lbs. ; 107,520 ft.-lbs.

18. Sketch and describe the construction and working of any hydraulic accumulator with which you are acquainted. If an accumulator has a ram 20" diameter with a lift of 15', and the gross weight of the load lifted is 130 tons, what is the pressure of water per square inch and the maximum energy in ft.-lbs. stored in the accumulator, neglecting friction? (S. E. B. 1900.) Ans. (1) 927 lbs. (2) 4,368,000 ft.-lbs.

19. A single-acting hydraulic engine has three rams, each of 3 inches diameter: common crank 3 inches long; pressure of water above that of exhaust 100 lbs. per square inch; 100 revolutions per minute; no slip of water. What is the horse-power? If this engine does 2.15 horse-power usefully by means of a rope, what is the efficiency? (S. E. B. 1901.) Ans. 67.

LECTURE XXI.

CONTENTS.-Motion and Velocity-Uniform, Variable, Linear, and Angular Velocity-Unit of Velocity-Acceleration-Unit of AccelerationAcceleration due to Gravity-Graphic Representation of VelocitiesComposition and Resolution of Velocities-Newton's Laws of Motion -Formulæ for Falling Bodies-Formulæ for Linear Velocity-with Uniform Acceleration-Centrifugal Force due to Motion in a CircleExperiments I. II. III.—Example I.-Balancing High-speed Machinery -Centrifugal Stress in the Arms of a Fly-wheel-Example II.Energy Potential Energy-Kinetic Energy-Accumulated WorkAccumulated Work in a Rotating Body-The Fly-wheel-Radius of Gyration-Example III.-The Fly Press-Example IV.-Momentum -Examples V. VI. and VII.-Questions.

Motion and Velocity.—(1) Motion is the opposite of rest, for it signifies change of position.

(2) Velocity is the rate at which a body moves, or rate of motion. It is considered absolute when it is measured from some fixed point, and relative if it refers to another body in motion at the same time.

(3) Uniform Velocity takes place when the rate of motion does not change--i.e., when the body moves over equal distances in equal times.

(4) Variable Velocity takes place when the rate of motion changes-i.e., when a body moves with either a constantly increasing or decreasing velocity. For example, a stone pitched into the air rises with a gradually decreasing velocity, but falls with a gradually increasing rate of motion.

(5) The Unit of Velocity is the velocity of a body which moves through unit distance in unit time. The British unit of velocity is therefore I foot in 1 second. In physical problems velocity is generally expressed in feet per second, but for convenience the engineer reckons the piston speed of engines in feet per minute, and the public speak of the speed of a man walking, of a horse trotting, or of a train, in miles per hour.

(6) Linear Velocity is the rate of motion in a straight line, and is measured, as we have just stated, in feet per second or per minute, or in miles per

hour.

If v = the velocity; the distance; and t the time—

(7) Angular Velocity is the rate at which a body describes an angle about a given point-for example, the number of revolutions per minute of a pulley; but angular velocity may also be measured by the feet per second or per minute which a point at a known distance from the centre of motion moves,

(8) Acceleration.-In the case of variable velocity, the rate of change of the velocity is termed the acceleration, and may be either positive or negative-i.e., it may be an increasing or a decreasing

rate.

(9) The Unit of Acceleration is that acceleration which imparts unit change of velocity to a body in unit time; or in this country it is an acceleration of 1 foot per second in one second.

(10) The Acceleration due to Gravity is considerably greater than the above unit, and varies at different places on the earth's surface. At Greenwich it is 32.2 feet per second in one second. In Elementary Applied Mechanics questions we will indicate it by the symbol g, and consider g = 32 feet per second in one second. Graphic Representation of Velocities.-The linear velocity of a point (such as the c.g. of a body) may be represented in the same way as we have hitherto represented a force. A line drawn from a point with an arrow-head indicates the direction of motion, and the length of the line to scale the magnitude of the velocity. (See p. 3, Lecture I.)

Composition and Resolution of Velocities.-Velocities may be compounded and resolved in exactly the same way as we treated forces by the parallelogram and triangle of forces, &c., in Lecture VIII.

Newton's Laws of Motion.-I. A body in motion, and not acted on by any external force, will continue to move in a straight line and with uniform velocity.

II. When a force acts upon a body in motion, the change produced in the quantity of motion is the same, both in magnitude and direction, as if the force acted on the body at rest.

The change in the quantity of motion is therefore proportional to the force applied, and takes place in the direction of that force.* III. If two bodies mutually act upon each other, the quantities of motion developed in each in the same time are equal and opposite. Or, Action and reaction are equal and opposite.

These three laws were first stated clearly by Sir Isaac Newton as the result of inductive reasoning. Having observed certain facts, he set about investigating what would be the consequence if his conjectures as to these facts were applied to particular

Here" quantity of motion" means "momentum," or mass x velocity. ••. Quantity of motion or momentum=Wv/g.

cases. Finding that his estimate of the probable result came true, he formulated a general law in accordance with his observations and reasonings.

The student has already conceived the truth of the first and third laws in the reasonings and applications of force to matter, treated of in the previous Lectures. We will now give in as brief a form as possible the formulæ for falling bodies, because they naturally lead on to the formulæ for "centrifugal force" on a rotating body, and to the "energy stored " up in a moving body, both of which are of great interest and importance to the young engineer. The experimental and algebraical proofs of these formulæ are given in Elementary Manuals on Theoretical Mechanics, and we must either assume that the student has studied these, or ask him to assume their truth in the meantime. Formulæ for Falling Bodies.-If a body falls freely in vacuo under the action of gravity from rest through a height h feet; then (since gravity produces a constant acceleration in the velocity of the body) at the end of each successive second the velocity of the body will be increased by g, or 32 feet. Letv be the velocity of the body at the end of t seconds,

Formulæ for Linear Velocity with Uniform Acceleration. Suppose that instead of the uniform accelerating force of gravity we have any other constant force of F lbs. acting on a body, and if this force moves the body through a distance of 7 feet along a perfectly smooth horizontal plane, the above formulæ naturally become*—

* We intentionally use the letter for length or distance, and a for acceleration. Most writers use the word " space for distance and the symbols; but space is of three dimensions, and involves the idea of volume. It cannot therefore be, strictly speaking, used to represent distance or length, which is only of one dimension. The letter ƒ is also often used for acceleration; but ƒ naturally represents a force, so we prefer to use, a, for acceleration, in order to be consistent with our notation.