THE INCLINED PLANE WITHOUT FRICTION.
258. The mechanical powers now to be considered are often used for other purposes beside those of raising great weights. For example: the parts of a structure have to be forcibly drawn together, a powerful compression has to be exerted, a mass of timber or other material has to be riven asunder by splitting. For purposes of this kind the inclined plane in its various forms, and the screw, are of the greatest use. The screw also, in the form of the screw-jack, is sometimes used in raising weights. It is principally convenient when the weight is enormously great, and the distance through which it has to be raised comparatively small.
Fig. 41.
259. We shall commence with the study of the inclined plane. The apparatus used is shown in [Fig. 41]. a b is a plate of glass 4' long, mounted on a frame and turning round a hinge at a; b d is a circular arc, with its centre at a, by which the glass may be supported; d c is a vertical rod, to which the pulley c is clamped. This pulley can be moved up and down, to be accommodated to the position of a b; the pulley is made of brass, and turns very freely. A little truck r is adapted to run on the plane of glass. The truck is laden to weigh 1 lb., and this weight is unaltered throughout the experiments; the wheels are very free, so that the truck runs with but little friction.
260. But the friction, though small, is appreciable, and it will be necessary to measure the amount and then endeavour to counteract its effect upon the motion. The silk cord attached to the truck is very fine, and its weight is neglected. A series of weights is provided; they are made from pieces of brass wire, and weigh 0·1 lb. and 0·01 lb.: these can easily be hooked into the loop on the cord at p. We first make the plane a b horizontal, and bring down the pulley c so that the cord shall be parallel to the plane; we find that a force must be applied by the cord in order to draw the truck along the plane: this force is of course the friction, and by a suitable weight at p the friction may be said to be counterbalanced. But we cannot expect that the friction will be the same when the plane is horizontal as when the plane is inclined. We must therefore examine this question by a method analogous to that used in [Art. 207].
261. Let the plane be elevated until b e, the elevation of b above a d, is 20"; let c be properly adjusted: it is found that when p is O·45 lb. r is just pulled up; and on the other hand, when p is only 0·40 lb. the truck descends and raises p; and when p has any value intermediate between these two, the truck remains in equilibrium. Let us denote the force of gravity acting down the plane by r, and it follows that r must be 0·425 lb., and the friction 0·025 lb. For when p raises r, it must overcome friction as well as r; therefore the power must be 0·025 + 0·425 = 0·45. On the other hand, when r raises p, it must also overcome the friction 0·025, therefore p can only be 0·425 - 0·025 = 0·40; and r is thus found to be a mean between the greatest and least values of p consistent with equilibrium. If the plane be raised so that the height b e is 33", the greatest and least values of p are 0·66 and 0·71; therefore r is 0·685 friction 0·025, the same as before. Finally, making the height b e only 2", the friction is found to be 0·020, which is not much less than the previous determinations. These experiments show that we may consider this very small friction to be practically constant at these inclinations. (Were the friction large, other methods are necessary, [see Art. 265].) As in the experiments r is always raised we shall give p the permanent load of 0·025 lb., thus sufficiently counteracting friction, which we may therefore dismiss from consideration. It is hardly necessary to remark that, in afterwards recording the weights placed at p, this counterpoise is not to be included.
262. We have now the means of studying the relation between the power and the load in the frictionless inclined plane. The incline being set at different elevations, we shall observe the force necessary to draw up the constant load of 1 lb. Our course will be guided by first making use of the principle of energy. Suppose b e to be 2'; when the truck has been moved from the bottom of the plane to the top, it will have been raised vertically through a height of 2', and two units of energy must have been consumed. But the plane being 4' long, the force which draws up the truck need only be 0·5 lb., for 0·5 lb. acting over 4' produces two units of work. In general, if l be the length of the plane and h its height, R the load, and P the power, the number of units of energy necessary to raise the load is R h, and the number of units expended in pulling it up the plane is P l: hence R h = P l, and consequently P : h :: R : l; that is, the power is to the height of the plane as the load is to its length. In the present case R = 1 lb., l = 48"; therefore P = 0·0208 h, where h is the height of the plane in inches, and P the power in pounds.
263. We compare the powers calculated by this formula with the actual observed values: the result is given in Table XIII.
Glass Plane 48" long, truck 1 lb. in weight, friction counterpoised; formula P = 0·0208 × h".
| Number of Experiment. | Height of plane. | Observed power in lbs. | P. Calculated power in lbs. | Differences of the observed and calculated powers. |
|---|---|---|---|---|
| 1 | 2" | 0·04 | 0·04 | 0·00 |
| 2 | 4" | 0·08 | 0·08 | 0·00 |
| 3 | 6" | 0·13 | 0·12 | -0·01 |
| 4 | 8" | 0·16 | 0·17 | +0·01 |
| 5 | 10" | 0·21 | 0·21 | 0·00 |
| 6 | 15" | 0·31 | 0·31 | 0·00 |
| 7 | 20" | 0·42 | 0·42 | 0·00 |
| 8 | 33" | 0·71 | 0·69 | -0·02 |
Thus for example, in experiment 6, where the height b e is 15", it is observed that the power necessary to draw the truck is 0·31 lb. The truck is placed in the middle of the plane, and the power is adjusted so as to be sufficient to draw the truck to the top with certainty; the necessary power calculated by the formula is also 0·31 lbs., so that the theory is verified.
264. The fifth column of the table shows the difference between the observed and the calculated powers. The very slight differences, in no case exceeding the fiftieth part of a pound, may be referred to the inevitable errors of experiment.