MECHANICS. 



from the centre of gravity of the cylinder in a 

 vertical direction, represent the whole weight ol 



the cylinder. 

 This force 

 may be re- 

 solved into 

 two others, ce 

 perpendicular 

 to the plane, 

 and eg paral- 

 lel to it (see 

 No. 13). But 

 ce is counter- 

 acted by the 

 resistance of 



Fig. 24. 



the plane, and can produce no effect ; therefore 

 eg or ef represents the force with which the 

 cylinder is urged in the direction of BA ; fe will 

 therefore represent the force of P which keeps W 

 in equilibrium. Now, it can be shewn that the 

 triangle cfe is similar to the triangle ABD ; there- 

 fore^ is to efsis AB to BD that is, W is to P as 

 the length of the plane to its height. If AB is 

 twice the length ofBD, one hundredweight at P 

 will sustain two hundredweight at W ; and if 

 AB is six times the length of BD, W will be six 

 times P. 



The proportion of P to W now stated is true 

 only when the direction of the force P is parallel 

 to the plane ; in any other direction, part of the 

 effect of P is lost. 



In speaking of sloping roads, they are said to 

 have an inclination or rise of one foot in ten, one 

 in thirty, one in a hundred, and so on. The 

 degree of slope in a railway is called the gradient, 

 and seldom exceeds one in 50 or 60. The annexed 

 cut represents a cart drawn by a horse up a corn- 



Fig. 25. 



mon road with a rise of one in ten. On a perfectly 

 level road, a certain force of draught would be re- 

 quired to overcome friction (see page 222); on the 

 incline there is required, in addition to this, a con- 

 stant pull to counteract the tendency of the cart 

 to run down the slope. That tendency is, in this 

 case, equal to a tenth of the weight of the load ; 

 and if the load is a ton, the horse has to pull with 

 a force of one-tenth of a ton, or 224 pounds, above 

 what would have sufficed to draw it along a level. 

 In going over ten feet of the road, the horse has 

 raised the cart one foot perpendicularly, but he 

 has done it by instalments ; the exertion has been 

 spread over a movement of ten feet. Intensity of 

 exertion is saved at the expense of time. 



THE WEDGE. 



A common form of the wedge is represented 

 in fig. 26. By forcing such a body below the 

 bottom of an upright post, for instance, which 

 can move only vertically, we may raise the post a 

 few inches. It obviously acts on the principle of 



Fig. 26. 



the inclined plane; it is, in fact, the inclined 

 plane made movable. 



The proportion of the power to the 

 resistance in the wedge is calculated 

 theoretically in the same way as in the 

 inclined plane. But the theory is of 

 little or no value. The power em- 

 ployed being percussion or blows, can- 

 not be rightly compared with the resist- 

 ance ; it cannot be estimated in pounds, 

 like other forces. The friction, too, is so 

 great as to render any precise state- 

 ment of proportion impossible. In a general way, 

 however, the wedge is more powerful as the angle 

 is more acute ; just as the advantage of 

 the inclined plane increases with the 

 smallness of its height. 



The wedge is used in splitting timber, 

 stones, &c. Ships are raised in docks 

 by driving wedges under them. In 

 expressing oil from seeds, the seeds are 

 placed in bags between solid pieces of 

 wood, and these are forced together by 

 means of wedges, till the seeds become 

 a mass as compact as wood. 



Cutting and piercing instruments all 

 act on the principle of the wedge. The plough is 

 also an instance. 



THE SCREW. 



The screw is an inclined plane wound round 

 a cylinder. Take a cylindrical ruler AB, and 

 cut a slip of paper in the form of a right-angled 

 triangle abf, having ab equal to the length of 

 the cylinder, and bf equal, say, to four times its 

 circumference. If the edge ab is applied to the 

 cylinder lengthwise, and the triangle is then 



Fig. 27. 



Fig. 28. 



wrapped round and round, the slanting side, af f 

 will form a spiral going four times round the 

 cylinder before it reach the bottom. In the fig. 

 to the left of the cut, it is represented as half 

 wrapt on. 



If a ridge or projection were raised on the 

 cylinder along the spiral line, it would form a 

 screw. The several turns or coils of the spiral 

 are called the threads of the screw ; ed represents 

 the length of one thread, and ec the distance 

 between the threads. In moving over a complete 

 round of the spiral, a perpendicular ascent or 

 descent is made equal to ec. As in the inclined 

 plane, therefore, the mechanical advantage of the 

 screw depends on the proportion between the 

 length of a thread and the distance between two 

 contiguous threads. 



In using the screw as a machine, the resistance 

 is not applied directly to the spiral surface, as in 

 the wedge or the straight inclined plane ; the 

 screw is made to work in a hollow cylinder with 

 spiral grooves cut out to correspond to the pro- 

 jecting threads. This concave screw is technically 



216 



