Page:Elementary Text-book of Physics (Anthony, 1897).djvu/34

point move in a circle its velocity is equal to the product of its angular velocity and the radius of the circle; its acceleration in the circle is equal to the product of its angular acceleration and the radius of the circle. If its angular acceleration be constant, the relations between the distance traversed by it in the circle, its velocity, its acceleration in the circle and the time are the same as those expressed in equations (9), (9a), (10). Substituting for these quantities their equivalents in terms of the angular magnitudes involved, we obtain the following relations among these angular magnitudes:

$\phi -\phi _{0}={\frac {\omega -\omega _{0}}{2}}(t-t_{0});$

(11)

$\phi -\phi _{0}=\omega _{0}(t-t_{0})+{\tfrac {1}{2}}\alpha (t-t_{0})^{2};$

(12)

$\omega ^{2}=\omega _{0}^{2}+2\alpha (\phi -\phi _{0}).$

(13)

If the line describing the angle start from rest, $\omega _{0}=0$, and if we take the line in this position as the initial line from which to reckon $\phi$, and the time of starting as the origin of time, then $\phi _{0}=0$, $t_{0}=0$, and equations (8), (12), (13), become $\omega =\alpha t$, $\phi ={\tfrac {1}{2}}\alpha t^{2}$, and $\omega ^{2}=2\alpha \phi$.

21. Simple Harmonic Motion.—If a point move in a circle with a constant velocity, the point of intersection of a diameter and a perpendicular drawn from the moving point to this diameter will have a simple harmonic motion. Its velocity at any instant will be the projection of the velocity of the point moving in the circle at that instant upon the diameter. The radius of the circle is the amplitude of the motion. The period is the time between any two successive recurrences of a particular condition of the moving point. The position of a point executing a simple harmonic motion can be expressed in terms of the interval of time which has elapsed since the point last passed through the middle of its path in the positive direction. This interval of time, when expressed as a fraction of the period, is the phase.

We further define rotation in the positive direction as that rotation in the circle which is contrary to the motion of the hands of