Page:Elementary Text-book of Physics (Anthony, 1897).djvu/38

and are in the same line. Let their displacements be represented by $s=a\cos {\omega t}$, and $s'=a\cos {(\omega t-\epsilon )}$.

The resultant displacement is the sum of the displacements due to each; hence

$s+s'=a[\cos {\omega t}+\cos {(\omega t-\epsilon )}],$
$=a[\cos {\omega t}+\cos {\omega t}\cos {\epsilon }+\sin {\omega t}\sin {\epsilon )}],$
$=a[\cos {\omega t}(1+\cos {\epsilon })+\sin {\omega t}\sin {\epsilon )}].$

If for brevity we assume a value $A$A and an angle $\phi$ such that $A\cos {\phi }=a(1+\cos {\epsilon })$, and $A\sin {\phi }=a\sin {\epsilon }$, we may represent the last value of $s+s'$ by $A\cos {(\omega t-\phi )}$. From the two equations containing $A$, we obtain, by adding the squares of the values of $A\sin {\phi }$ and $A\cos {\phi }$, $A=(2a^{2}+2a^{2}\cos {\epsilon })^{\frac {1}{2}}$; and, by dividing the value of $A\sin {\phi }$ by that of $A\cos {\phi }$, we obtain $\phi =\tan ^{-1}{\frac {\sin {\epsilon }}{1+\cos {\epsilon }}}\cdot$ The displacement thus becomes

$s+s'=a(2+2\cos {\epsilon })^{\frac {1}{2}}\cos {\left(\omega t-\tan ^{-1}{\frac {\sin {\epsilon }}{1+\cos {\epsilon }}}\right)}\cdot$

(17)

This equation is of great value in the discussion of problems in optics.

The principle suggested by the result of the above discussion, that the resultant of the composition of two simple harmonic motions is a periodic motion of which the elements depend on those of the components, can be easily seen to hold generally.

A very important theorem, of which this principle is the converse, was given by Fourier. It may be stated as follows: Any complex periodic function may be resolved into a number of simple harmonic functions of which the periods are commensurable with that of the original function.

22. Force.— When we lift or sustain a weight, stretch a spring, or throw a ball, we are conscious of a muscular effort which we designate as a force. Since no change can be perceived in the weight if it be suspended from a cord, or in the spring if it be held stretched by being fastened to a hook, and since the ball moves in just the same way if it be projected from a gun, we conclude that