CCCXXVIII.

"OVERHAULING" OF A MECHANICAL POWER.

BY J. BURKITT WEBB, HOBOKEN, N. J.

(Member of the Society.)

IN Professor Ball's "Experimental Mechanics," page 118, the following statement is made:

"The principle which we have here established* extends to other mechanical powers, and may be stated generally. Whenever rather more than half of the applied energy is uselessly consumed by friction, the load will remain suspended without overhauling."

It might also be inferred that the converse of this statement would be true and that, to prevent overhauling, more than half of the applied energy must be wasted in friction. In fact the converse statement is the one most needed, so that we may know how to

make a non-overhauling "power" which will waste the minimum amount of energy in friction.†

I propose to show that there is no such law as that proposed by Professor Ball, or its converse.

To show the fallacy of the statement we will examine one of the simplest of the mechanical powers,-the inclined plane.

Fig. 80 shows the relations existing between the weight, power and reaction of the plane when the weight is being pulled up the plane. The weight necessarily acts downwards, while the reaction. * With respect to a differential pulley block.

The converse of the statement is assumed, in other parts of the book, to be true.

makes an angle o with the normal to the plane such that tan p = coefficient of friction between the body and the plane. The power may be applied in any direction within certain limits, that is; ẞ may have any value between - (90°-a-p) and 90°. At the lower limit P will be infinite, i.e., it will be impossible to move the weight

because the value of ẞ is such as to cause all the power to be wasted in friction; at the upper limit P = W and no power is thus wasted. Anywhere inside of these limits the power will cause the body to move up the plane, and we have at once two cases which will serve to test the proposed law and its converse.

Fig. 81 illustrates the first case, the dotted arrow indicating the position of P at its lower limit and the full arrow a position of P inside of this limit and therefore causing motion up the plane. At this lower limit all, and near it nearly all, of the applied energy disappears in friction and yet the "power" will overhaul or not

Fig.82.

Norma

W

according as we make a greater or less than p. It is evident, therefore, that the proposed law itself does not hold, and that the amount of applied energy wasted in friction has no connection with the overhauling or non-overhauling property.

Fig. 82 illustrates the second case, the dotted arrow indicating the upper limit for P and the full arrow a position causing motion up the plane. At this upper limit none of the applied energy is wasted

in friction, and near it but little is so wasted, and yet overhauling and non-overhauling are dependent upon the value of a, just as in the previous case. It is evident, therefore, that the converse of the proposed law is equally false, and that overhauling may be prevented without any such waste of applied energy.

We will now examine the law governing the loss of energy by friction.

If we assume that P is applied parallel to the plane, or that ß = α, we create a case, it is true, where non-overhauling depends upon the lost energy being more than half of that applied, but this is so only because we have assumed P to be so applied as not to affect the amount of friction between the body and the plane, or, because by making a we have made the lost energy, which depends naturally upon 6, to depend also upon a.

Were this direction of P the most economical the proposed law might, perhaps, be saved by conditioning it to apply to this direction of P as the direction to be adhered to in practice, but less energy will be lost and the apparatus will be otherwise improved if ẞ be made somewhat greater than a, or, as we shall show, greater than a + . We shall, therefore, dismiss the proposed law from further consideration and examine the relation between B, P and the lost energy.

Fig. 83 illustrates the cases of an inclined plane, overhauling or non-overhauling according to the value of a. The figure is drawn with P removed because the conditions of overhauling and the reverse are conditions occurring when P is absent. In the case of overhauling the reaction R will make an angle with the up-hill side of the normal, but, being less than a, this will leave R on the down-hill side of the vertical, and thus Wand R together will

have a resultant to move the body down the plane. In the other case the angle of R with the normal will be 0 < p on account of the fact that there will only be as much friction as the tendency

down the plane produced by W, and the exact counterbalance of this tendency by the friction will leave R exactly vertical and equal to W, so that the resultant of W and R will be zero and the body will be at rest.

The equilibrium between the forces (see Fig. 80) is expressed by the two equations

WP sin ẞ + R cos (a + 4)

P cos B R sin (a + q)

which by the elimination of R becomes

or

W sin (a + q) = P [sin ẞ sin (a + q) + cos ẞ cos (a + p)]

If now we desire to know that value of ẞ which will make P the least possible, we seek the value which makes W÷P a maximum,

that is to say, in order to have P the least possible, we must pull at an angle above the plane, as shown in Fig. 84, and this will, therefore, be the best direction if it is desired to have P as small as it can be made.

If, however, it be desirable to consider also the lost energy, then we have only to remember that this decreases with the increase of B, as already indicated, so that by making ẞ somewhat greater than a + we may reduce the lost energy still further without

seriously increasing P, and thus attain the best practical direction for the pull.

Dr. Coleman Sellers, of Stevens Institute, in a lecture delivered before the last Senior class, called attention to the failure of this proposed law for a train of wheel work, but I am not aware of any proof having been given before this that there is no such general law as that proposed, except that in an article by myself before the last meeting of the A. A. A. S. a lever with a large pivot was described, to which the proposed law will not apply.

APPENDIX.*

To make the foregoing paper more conclusive, if possible, I have calculated the lost energy in certain definite cases of the inclined plane and also in the case of a windlass, which embodies the principle of a lever with a large pivot, described in my paper at the

last meeting of the A. A. A. S. Models have also been constructed, by means of which the calculations have been verified and the falsity of the proposed law experimentally demonstrated.

* Added since the meeting. This appendix is intended principally as a reply to some of the objections made to the paper by Sir Robert Stawell Ball, Astronomer Royal of Ireland. These objections were inclosed in a letter to me and contributed as a part of the discussion upon the paper, where they are to be found. To make the paper more complete, part of my reply to these objections is put in this form of an appendix; the rest appears in proper place closing the discussion.