9/(2 × 24 × 60 × 60 × 8) feet or 1/12800th inch.
Whence, if the clock weight is 10 lbs., the impulse received by the clock at each beat is equivalent to a weight of 10 lbs. falling through 1/12800th of an inch, or to the fall of six grains through an inch.
The power thus expended goes in friction of the wheels and hands, and in maintaining the pendulum in spite of the friction of the air.
The work therefore that is put into the clock by the operation of winding is gradually expended during the week in movement against friction. The work is indestructible. The friction of the parts of the clock develops heat, which is dissipated over the room and gradually absorbed in nature. But this heat is only another form of work. Amounts of work are estimated in pressures acting through distances. Thus, if I draw up a weight of 1 lb. against the accelerative force of gravity through a distance of one foot, I am said to do a foot-pound of work.
One pound of coal consumed in a perfect engine would do eight millions foot-pounds of work. Hence, if the energy in a pound of coal could be utilized, it would keep about 100,000 grandfather’s clocks going for a week. As it is consumed in an ordinary steam engine it will do about half a million foot-pounds of work. One pound of bread contains about three million foot-pounds of energy. A man can eat about three pounds of bread in a day, and, as he is a very good engine, he can turn this into about three-quarters of a million foot-pounds of work. The rest of the work contained in the bread goes off in the form of heat.
Fig. 54.
As has been previously said, the power of the action of gravity in drawing back a pendulum that has been pushed aside from its position of rest becomes less in proportion as the pendulum is longer, and hence as the pendulum is longer the time of vibrations increases. In the [appendix] to this chapter a short proof will be given showing that the length of a pendulum varies as the square of the time of its vibration. A pendulum which is 39·14 inches in length vibrates at London once in each second. Of course at other parts of the earth, where the force of gravity is slightly different, the time of vibration will be different, but, since the earth is nearly a globe in shape, the force of gravity at different parts of it does not vary much, and therefore the time of vibration of the same pendulum in different parts of the earth does not vary very much. The length of a pendulum is measured from its point of suspension down to a point in the bob or weight. At first sight one would be inclined to think that the centre of gravity of the pendulum would be the point to which you must measure in order to get its length. So that if B were a circular bob, and the rod of the pendulum were very light, the distance A B to the centre of the bob would be the length of the pendulum. But if we were to fly to this conclusion, we should be making the same error that Galileo made when he allowed a ball to roll down an inclined plane. He forgot that the motion was not a simple one of a body down a plane, but was also a rolling motion. The pendulum does not vibrate so as always to keep the bob immovable with the top side C always uppermost. On the contrary, at each beat the bob rotates on its centre and makes, as it were, some swings of its own. Therefore in the total motions of the pendulum this rotation of the bob has to be taken into account. Of course, if the pendulum were so arranged that the bob did not rotate, and the point C were always uppermost, as, for instance, if the pendulum consisted of two parallel rods, A B and C D, suspended from A and C, then we might consider the bob as that of a pendulum suspended from E, and the pendulum would swing once in a second if A B = C D = E F were equal to 39·14 inches, for by this arrangement there would be no rotation of the bob. But as pendulums are generally made with the bob rigidly fixed to the rod E F, the rotation must be taken into account.
Fig. 55.