But this is precisely the same result as we got by applying the principle of the "triangle of forces." Hence, the "principle of work" agrees with the "triangle of forces" in respect to the inclined plane.

Cases 2 and 3 may be treated by the student in exactly the same way, and the correct results will be the same as those found by the "triangle of forces."

EXAMPLE IV.-An inclined plane is used for withdrawing barrels from a cellar by securing two ropes to the top of the incline at B, then passing them down the incline, half round

b.

RAISING BARRELS BY
RETURN ROPE AND IN-
CLINED PLANE.

the barrel, and up to the horizontal platform at the top of the incline, where two men pull on the ropes in a direction parallel to the plane. If the weight, W, of the barrel is 200 lbs., the length, l, of the incline 20 ft., and the height 10 ft., find, by the principle of work, the least force which must be exerted by the two men, and the work expended, neglecting friction, in drawing the barrel from the cellar.

Let the accompanying figure represent a vertical cross section through the middle of the barrel and the inclined plane. Then a statical force, P, applied at the c.g. of the barrel, would just balance its weight, W, and the reaction from the plane (not shown). By the principle of work, neglecting friction

The work put in = The work got out.
Px its distance = W x its distance.

But by passing the rope round the barrel, as explained in the question, this force P is halved on the ropes (see Lecture V. on the pulley and snatch-block). Therefore the least force which the two men must exert in order just to move the barrel will beР 100

50 lbs.

But this force acts through a distance 2 7=40' ft.; therefore the work expended will be—

Р

2

× 2 l = 50 lbs. × 40′ = 2000 ft.-lbs.

Or, work got out = W x h=200 lbs. x 10' = 2000 ft.-lbs.

In this question we have a combination of the pulley and the inclined plane. The inner ends of the two ropes being fixed at the top of the inclined plane, the force with which the men act on the free ends is communicated throughout the ropes, so that the stress in the ropes on each side of the barrel balances the force P, that would be required to move the barrel up the incline if applied at its centre of gravity.

Or, the theoretical advantage due to the pulley part of the

Then for the inclined plane part we have by the "triangle of forces," or by the "principle of work," a theoretical advantage of —

Therefore, the total theoretical advantage is the product of the two separate advantages, viz.

Consequently, a force of 1 lb. applied at the free end of the rope would balance a weight of 4 lbs. on the incline. Or, as in the question, and, neglecting friction, a barrel weighing 200 lbs. requires a pull of 50 lbs. to move it up the inclined plane.

We have simply split up the total advantage in this way to show the student that the combined advantages of the several parts of a compound machine must equal the advantage of the whole. We might have said at once, as we have done before in other cases

NOTE.-I have this day (Sept. 9, 1892) witnessed the interesting operation of lowering four very large 25-ton steam boilers of the marine type, down an incline of about 100 feet in length by the method described in the foregoing question. One man, by aid of an ordinary block and tackle, supplied the requisite restraining force on the free end of the rope.

LECTURE IX.-QUESTIONS.

I. Prove by the triangle of forces (drawn to scale) the relation betwee the weight W of a body resting on a smooth inclined plane, the reaction, R, from the plane, and the force, P, necessary to just balance the weight— (1) when the force, P, acts parallel to the plane; (2) when it acts parallel to the base; (3) when it acts at an angle, 0, to the plane.

2. A ball, weighing 100 lbs., rests on an inclined plane, being held in position by a string which is fastened to a bracket so as to be parallel to the plane. The height of the plane being of the length, find the tension of the string and the pressure perpendicular to the plane. Establish your results by reasoning on known principles, such as the principle of work or that of the parallelogram of forces. Ans. P=33.3 lbs., and R=94.3 lbs.

3. Prove the relation between W, P and R, acting on a body resting on a smooth inclined plane by the "principle of work" for cases 1, 2 and 3 in this Lecture. An incline is I ft. in vertical height for 15 in length. A weight of 100 lbs, rests on the plane and is held up by friction; make a diagram for estimating the pressure on the plane, and find its amount. Ans. 99.7 lbs.

4. Friction being neglected, find the force, acting parallel to the plane, which will support 1 ton on an incline of 1 ft. vertical and 10 ft. along the ncline. Prove the formula which you employ. If the incline were 1 ft. vertical and 280 ft. along the incline, find the force in pounds which would support I ton. (S. and A. Exam. 1890.) Ans. 224 lbs., and 8 lbs.

5. A smooth incline plane has a vertical side of 1 ft., and a length of 10 ft.; what work is done in pulling 10 lbs. up 8 ft. of the incline? Ans. 8 ft.-lbs.

6. When a body is raised through a given height, how is the work done estimated? A body weighing 8 cwt. is drawn along 100 ft. up an incline, which rises 2 ft. in height for every 5 ft. along the incline; the resistance of friction being neglected, find the work done. Ans. 35,840 ft.-lbs.

7. A smooth incline is 8 ft. long, and the total vertical rise from the bottom to the top thereof is 2 ft. What amount of work is performed in drawing a weight of 100 lbs. up 4 ft. of the incline, and what is the least force which will do this work? Ans. 100 ft.-lbs. ; 25 lbs.

8. Friction is neglected, and it is found that a force acting horizontally will move 10 lbs. up 5 ft. of an incline rising I in 4. Find the work done, and find also the force parallel to the plane which will just support the weight of 10 lbs. Ans. 12.5 ft.-lbs. ; 2.58 lbs.

9. A car laden with 20 passengers is drawn up an incline, one end of which is 160 ft. above the other; the car, when empty, weighs 3 tons, and the average weight of each passenger is 140 lbs. Find the number of ft.-lbs. of work done in ascending the incline, neglecting friction. Ans. 1,523,200 ft.-lbs., or 680 ft.-tons.

10. It will be observed that draymen sometimes lower heavy casks into cellars by means of an inclined plane and a rope. One end of the rope is secured to the upper end of the inclined plane, and is then passed under and over the cask, the men holding back by means of the loose end. Now, supposing the incline to be at an angle of 45 degrees, explain the mechanical principles that are here applied, and find the advantage. Ans. 2 √2: 1.

11. A barrel weighing 5 cwt. is lowered into a cellar down a smooth slide inclined at an angle of 45 degrees with the vertical. It is lowered by means of two ropes passing under the barrel, one end of each rope being fixed, while two men pay out the other ends of the ropes. What pull in lbs. must each man exert in order that the barrel may be supported at any point? (S. and A. Exam. 1889.) Ans. 99 lbs. nearly.

LECTURE X.

CONTENTS.-Friction-Heat is Developed when Force overcomes Friction -Laws of Friction-Apparatus for Demonstrating First and Second Laws of Friction-Experiment I.-Example I.-Angle of Repose or Angle of Friction-Experiment II.-Diagram of Angles of ReposeLimiting Angle of Resistance-Experiment III.-Apparatus for Demonstration of the Third Law of Friction-Experiment IV.Lubrication-Anti-Friction Wheels-Ball Bearings-Work done on Inclines, including Friction-Example II.-Questions.

Friction.-Whenever a body is caused to slide over another body, an opposing resistance is at once experienced. This natural resistance is termed friction.* The true cause of friction is the roughness of the surfaces in contact. The smoother the sliding surfaces are made the less will be the friction. Friction cannot, however, be entirely eliminated by any known means, for even the most microscopical protuberances on the smoothest of surfaces seem to fit into corresponding hollows on other equally smooth places, so that some force is required to make the one body slide over the other.

Friction has its advantages as well as its disadvantages. For example, if it were not for friction we could not walk, neither could a locomotive start from a railway station, nor could it be brought to rest in the usual speedy manner. Friction is also essential to the utility of nails, screws, wedges, driving belts, &c. On the other hand, power is often expended in overcoming friction with the result of much wear and tear in machinery. For example, in the case of working the slide valves of locomotive engines as much as twenty horse-power is required in moving these essential parts when running at full speed.†

It is the duty of the engineer to reduce friction to a minimum in the case of the bearings of engines, shafting, and machines generally, in order that a minimum of work may be expended in moving them. He has, however, also to devise means of producing a maximum of friction in the case of certain pulleys, grips, clutches, brakes, and such like appliances, where motion has to be transmitted by aid of friction, or bodies in motion (such as a * French writers call friction a passive resistance, because it is only apparent when one body tends to move or pass over another.

See the Author's "Elementary Manual on Steam and the Steam Engine," page 182, for an arithmetical example.

moving train) have to be brought to rest quickly when nearing a station.

Heat is Developed when Force overcomes Friction.— When a body is kept moving by a force, part (or in certain cases it may be the whole) of the mechanical force is expended in overcoming frictional resistance. This lost work is directly transformed into heat in the act of overcoming the frictional resistance through a distance. For example :-A person slips down a vertical rope by holding it between his hands and his legs. The force of gravity impels him downwards, overcoming the frictional resistance between his hands and limbs and the rope, with the consequence that they become severely heated, especially if he happens to slip down quickly. A boy takes a run, and then slides along a level piece of ice. The foot-pounds of work stored up in him just before he begins to slide are expended partly in overcoming the frictional resistance between the soles of his boots and the ice, and partly in the frictional resistance between his clothes and the air. As a consequence, he will find that by the time he gets to the end of the slide his soles are considerably warmed. If the ice were perfectly level, infinitely long, and if there were absolutely no friction between it and his boots, and if there were no frictional resistance between him and the air, then he would slide on for ever! If we could diminish the frictional resistance between the skin of a ship and the water, and between the exposed parts of the ship and the air, to nothing, then all that would be required to transport her across the Atlantic would be a strong force applied at the start until she attained the desired speed, when she would proceed forward, and arrive at her destination with undiminished velocity! In reality, however, we find it necessary to employ steam engines of 10,000 horse-power continuously in order to propel an Atlantic "greyhound" of 5000 tons at twenty knots an hour in the calmest of weather. About one-half of this power is absorbed in overcoming the frictional resistance of the ship through the water and air, and the other half in the frictional and other losses due to the working of the propelling machinery. Examples of the conversion of mechanical work into heat are so familiar to you all, being in fact brought prominently before your notice every day of your existence, that we need not further enlarge upon this question except to remind the student of Dr. Joule's discovery of the rate of exchange between heat and work. He found by experiment that if work is transformed into heat, every 772 ft.-lbs. of work will produce 1 heat unit, or that quantity of heat which would raise 1 lb. of water 1° Fahr.*

* For further examples and an explanation of Dr. Joule's experiments see the Author's Treatise on Steam and the Steam Engine.