The coefficient of friction of a carriage, looked at as forming an element of a sliding pair, is generally called the Draught, and its value is usually expressed in lbs. per ton weight. It varies of course considerably with the state of the roads, so only mean values will be given. These may be taken at

55 lbs. on a macadamised road

20 lbs. on a stone tramway

8 lbs. on a railroad (very slow speeds).

In the case of a train, however, the air offers a great resistance which increases very rapidly with the speed, so that at 60 miles per hour the resistance is believed to exceed 50 lbs. per ton. The laws of this resistance are

outside our limits.

Friction of Rest Limiting Friction.-The friction we have dealt with is that existing between moving bodies, or the friction of motion. When, however, we consider the process of starting a body, a different law of frictional resistance exists. There is then, at any instant, as we gradually apply the effort, an amount of friction called into play just sufficient to balance the effort. As the effort increases so does the friction, until it reaches a certain limiting value, beyond which it cannot go; any further increase of effort then causes motion. This limiting value is called the Friction

of Rest or Limiting Friction, and its value follows the same laws as the Friction of Motion, with, however, a slightly greater coefficient in most cases. Its value is not of importance to us; more especially because any slight jar, during the starting, causes the body to start directly the effort is greater than the value found for the friction of motion, so it would never be safe to reckon on the friction of rest to prevent motion.

EXAMPLES.

1. The diameter of the piston of an indicator is in., the steam pressure under it is 30 lbs. absolute; the atmosphere presses on the top, and it is kept down by a spring which requires a force of 32 lbs. to compress it I inch. Find how much the spring is compressed.

Ans. .147".

2. The piston of a steam cylinder is 90 ins. diameter, the piston rod diameter is 8 ins., and there is no tail rod; the cylinder is horizontal. Find the effective effort of the steam1st, when the pressure at the back of the piston is 16 lbs. absolute, and that in front 3 lbs. absolute; 2d, when these are reversed. Ans. 79746 and 78654 lbs.

3. The steam pressure in a boiler is 120 lbs. by gauge. One safety valve is 3 ins. diameter, and the spring keeping it in place is compressed 3 ins. from its original length. Find the increase of pressure necessary to lift the valve in., which is the ordinary lift allowed. Also if a stop be fitted which prevents the valve rising more than one fourth of its diameter. Find what pressure would force it up against the stop.

Ans. 4 and 150 lbs. per sq. in.

4. The diameter of a piston is 54 ins., stroke 3 ft. The piston approaches within 1 in. of the end of the cylinder when at the end of its stroke, and the steam is cut off at half stroke. The boiler pressure is 130 lbs. by gauge, and there is a drop of 10 per cent on the absolute pressure between the boiler and the cylinder. Find the pressure in the cylinder at each of the stroke, assuming the simple hyperbolic law of expansion.

Ans. 130.5 to half stroke, 109.7, 94.6, 83.2, 76.5, 67. lbs. per sq. in.

5. The spring ring in the preceding is 6 ins. wide, and the pressure between it and the cylinder is 3 lbs. per square inch.

Find the frictional resistance to motion, the surfaces being well lubricated. Ans. 214 lbs.

6. Draw by graphic construction a curve of effort for (4), and show how to represent on it the effect of (5) in reducing the effective effort.

Ans. The friction is equivalent to a loss of .1 lbs. per sq. in. on the piston, therefore the curve is lowered by 229 lbs.

7. A horse walking at 2 miles per hour can exert a pull of 166 lbs., and at 4 miles per hour a pull of 83 lbs. Find the total load he can move at those speeds on a road.

Ans. 3 tons and 1 ton.

CHAPTER III

WORK AND ENERGY

THE effect produced by the movement of a pair is the overcoming of a resistance through a certain distance, and it is for the production of this effect that the pair is, in the great majority of cases, required. This effect is spoken of as doing Work, and we say work is done against the resistance.

The general meaning of the word Work is of course well understood; but it is not in this general sense we use it in Mechanics, but strictly in the limited sense defined above.

Let us consider a simple case of sliding, viz. the raising of a weight, then the weight slides relatively to the earth against the resistance of gravity. Work then is done against gravity.

motion is caused; may take this to

Now let us consider how the some effort is required, and we be the muscular effort of a man. The man then exerts an effort through the distance the weight is raised. In ordinary language we say "the man is doing work," but in Mechanics we say he is exerting energy, which causes work to be done against the resistance.

The action of lifting the weight or generally of moving a sliding pair against a resistance has then two descriptions; according, we may say, to the point of view from which we regard it. Looked at from the effort side it is

called exerting energy, from the resistance side, doing work.

In order that energy be exerted, or work done, we must have the combination of force and motion. For example, no work is done when the man stands simply holding the weight, although it may be his duty to do so, and he would in ordinary language undoubtedly be working since he would suffer fatigue; but there is no motion, and hence, according to our definition, no work done, or energy exerted. Again, in the sliding of a weight along a perfectly smooth horizontal table, if such a thing could be, no work would be done, because there would be no resistance, although there is motion.

In all cases of doing work, we shall find there are at least three bodies to be considered.

One supplies the effort causing the motion, and thus exerting energy—e.g. the man lifting the weight.

One is moved, by the effort, against the resistance— e.g. the weight itself.

The third resists the motion, being the source of resistance-e.g. the earth, which is the source of the gravitation resistance.

In the case we have considered, the first and third are natural sources of effort and resistance, but we shall see as we advance that this is not necessary; but what is necessary is, that the first be connected in some way to a natural source of effort, and the third to one of resistance.

We have defined doing work, or working, but we cannot give any particular definition of the term Work by itself, but of Energy this is not the case.

A source of effort in nature can exert energy, and we say it possesses energy; now what it does possess is the power of causing work to be done, so that by using the term energy we make it mean power of causing work to be done; and this therefore is the definition of energy. The source then exerts energy when it puts forth this power.