ON CONFINED MOTION. 33 



obvious and well known ; and we may be convinced of the retardation at- 

 tendant on the production of rotatory motion, by allowing two cylinders, 

 of equal dimensions, to roll down an inclined plane : the one being covered 

 with sheet lead, the other having an equal weight of lead in its axis, and 

 being covered with paper, and both having similar projecting surfaces at 

 the ends, which come into contact with the plane, we may easily observe that 

 in the first cylinder, much more of the force is consumed in producing 

 rotatory motion, than in the second, and that it therefore descends much 

 more slowly. (Plate II. Fig. 20.) 



When a body is placed on an inclined plane, the force urging it to 

 descend, in the direction of the plane, is to the whole force of gravity as 

 the height of the plane is to its length. This is demonstrable from the prin- 

 ciples of the composition of motion, and may also be shown experimentally 

 with great accuracy, when we consider the doctrine of the equilibrium of 

 forces. But the interference of friction will only allow us to observe, with 

 respect to the velocities produced, that they nearly approach to those which 

 the calculation indicates. Thus, if a plane be inclined one inch in 32, a 

 ball will descend on it in two seconds, instead of 64 feet, somewhat less 

 than two feet. 



It may be deduced from the laws of accelerating forces, that when 

 bodies descend on any inclined planes of equal heights, but of different in- 

 clinations, the times of descent are as the lengths of the planes, and the 

 final velocities are equal. Thus a body will acquire a velocity of 32 feet in a 

 second, after having descended 16 feet either in a vertical line, or in an oblique 

 direction ; but the time of descent will be as much greater than a second as 

 the oblique length of the path is greater than 16 feet. This may be shown 

 by experiment, as nearly as the obstacles already mentioned will permit, 

 the times being measured by a pendulum or by a stop watch. (Plate II. 

 Fig. 21.) 



There is an elegant proposition, of a similar nature, which is still more 

 capable of experimental confirmation ; that is, that the times of falling 

 through all chords drawn to the lowest point of a circle are equal. If two 

 or more bodies are placed at different points of a circle, and suffered to 

 descend at the same instant along as many planes which meet in the lowest 

 point of the circle, they will arrive there at the same time. (Plate II. 

 Fig. 22.) 



The velocity of a body descending along a given surface, is the same as 

 that of a body falling freely through an equal height, not only when the 

 surface is a plane, but also when it is a continued curve, in which the 

 body is retained by its attachment to a thread, or is supported by any 

 regular surface, supposed to be free from friction.* We may easily show, 

 by an experiment on a suspended ball, that its velocity is the same when 

 it descends from the same height, whatever may be the- form of its path, by 

 observing the height to which it rises on the opposite side of the lowest 

 point. We may alter the form of the path in which it descends, by placing 

 pins at different points, so as to interfere with the thread that supports the 

 ball, and to form in succession temporary centres of motion ; and we shall 

 * Principia, i. 40. 



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