Page:Elementary Text-book of Physics (Anthony, 1897).djvu/36

motion is proportional to the displacement, and that acceleration to the right of $O$ is negative, to the left of $O$ positive.

In these formulas the angular velocity $\omega$ may be replaced by an equivalent factor involving the period $T$. For, the line drawn from $O$ to the point moving in the circle sweeps out the angle $2\pi$ in the time $T$, so that $\omega ={\frac {2\pi }{T}}\cdot$

It is often convenient to reckon time from some other position than that of greatest positive elongation. In that case the time required for the moving point to reach its greatest positive elongation, from that position, or the angle described by the corresponding point in the circumference in that time, is called the epoch of the new starting-point. In determining the epoch, it is necessary to consider, not only the position, but the direction of motion, of the moving point at the instant from which time is reckoned. Thus, if $L$, corresponding to $K$ in the circumference, be taken as the starting-point, the epoch is the time required to describe the path $LX$. But if $L$ correspond to the point $K'$ in the circumference, the motion in the diameter is negative, and the epoch is the time required for the moving-point to go from $L$ through $O$ to $X'$ and back to $X$.

The epochs in the two cases, expressed in angle, are, in the first, the angle measured by the arc $KX$; and, in the second, the angle measured by the arc $K'X'KX$.

Choosing $K$ in the circle, or $L$ in the diameter, as the point from which time is to be reckoned, the angle $\phi$ equals angle $KOB$ — angle $KOX$, or $\omega t-\epsilon$, where $t$ is now the time required for the moving point to describe the arc $KB$, and $\epsilon$ is the epoch, or the angle $KOX$.

The formulas then become

$s=a\cos {(\omega t-\epsilon )};$

$v=-\omega a\sin {(\omega t-\epsilon )};$

$f=-\omega ^{2}a\cos {(\omega t-\epsilon )};$