Page:Elementary Text-book of Physics (Anthony, 1897).djvu/89

61. Formula for Simple Pendulum.—The formula connecting the time of oscillation with the value of $g$ is obtained as follows: The acceleration of the bob at any point in the arc is, as we have seen, $g\sin {\phi }$, or $g\phi$ if the arc be very small. The acceleration in a simple harmonic motion is $-\omega ^{2}s=-{\frac {4\pi ^{2}}{T^{2}}}s$, where $s$ is the displacement.

Since the bob has a simple harmonic motion, we may set these two expressions for the acceleration equal, neglecting the minus sign, which merely expresses the fact that the acceleration is toward the centre of the path; hence $g\phi ={\frac {4\pi ^{2}}{T^{2}}}s.$

The displacement $s$ is equal to $l\phi$, if $l$ represent the length of the thread; hence $g={\frac {4\pi ^{2}l}{T^{2}}}$, from which $T=2\pi {\sqrt {\frac {l}{g}}}\cdot$

In this formula $T$ represents the time of a double oscillation. It is customary to observe the time of a single oscillation, when the formula becomes

$t=\pi {\sqrt {\frac {l}{g}}}\cdot$

(41)

62. Physical Pendulum.—Any pendulum fulfilling the requirements of the foregoing theory is, of course, unattainable in practice. We may, however, calculate from the known dimensions and mass of the portions of matter making up the physical pendulum, what would be the length of a simple pendulum which would oscillate in the same time. It is clear that there must be some point in every physical pendulum the distance of which from the point of suspension is equal to the length of the corresponding simple pendulum; for the particles near the point of suspension tend to oscillate more rapidly than those more remote, and the time of oscillation of the system, if it be rigid, will be intermediate between the times of oscillation which the particles nearest to, and most remote from the point of suspension would have if they were oscillating freely. There will, therefore, be some one particle of which the proper rate