Fig. 167. is perfectly manifest that both forces are satisfied by the pendulum reaching the point e, exactly opposite the centre d, in half a second. To reach this point, it can be shown that it must describe the circular arc b e, and it will pursue its way along the continuation of the same arc, to a, and then pass round to b. Thus, by the rectangular impulse the rectilinear oscillation is converted into a rotation, the pendulum describing a circle, as shown in Fig. 167.
If the force applied at b be sufficient to urge the weight in half a second through a greater distance than b c, the pendulum will describe an ellipse, with the lines a b for its smaller axis; if, on the contrary, the force applied at b urge the pendulum in half a second through a distance less than b c, the weight will describe an ellipse, with the line a b for its greater axis.
Let us now inquire what occurs when the rectangular impulse is applied at the moment the ball is passing through its position of rest at d.
Supposing the pendulum to be moving from a to b, Fig. 168, and that at d a shock is imparted to it sufficient of itself to carry it in half a second to c; it is here manifest that the resultant motion will be along the straight line d g lying between b d and d c. The pendulum will return along this line to d, and pass on to h. In this case, therefore, the pendulum will describe a straight line, g h, oblique to its original direction of oscillation.
Supposing the direction of motion at the moment the push is applied to be from b to a, instead of from a to b, it is manifest that the resultant here will also be a straight line oblique to the primitive direction of oscillation; but its obliquity will be that shown in Fig. 169.
| Fig. 168. | Fig. 169. |
When the impulse is imparted to the pendulum neither at the centre nor at the limit of its swing, but at some point between both, we obtain neither a circle nor a straight line, but something between both. We have, in fact, a more or less elongated ellipse with its axis oblique to a b, the original direction of vibration. If, for example, the impulse be imparted at d′, Fig. 170, while the pendulum is moving toward b, the position of the ellipse will be that shown in Fig. 170; but if the push at d′ be given when the motion is toward a, then the position of the ellipse will be that represented in Fig. 171.
| Fig. 170. | Fig. 171. |
