Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/369

 

through an angle $\pi -\alpha$ into the position $SP^{\prime \prime }$, the acceleration of $P$ will be equal in magnitude and direction to

${\frac {\omega ^{2}}{\sin ^{2}\alpha }}SP^{\prime \prime }$,

where $SP^{\prime \prime }$ is equal to $SP$ turned through an angle $2\pi -2\alpha$. If we draw $PF$ equal and parallel to $SP^{\prime \prime }$, the acceleration will be ${\frac {\omega ^{2}}{\sin ^{2}\alpha }}PF$, which we may resolve into

${\frac {\omega ^{2}}{\sin ^{2}\alpha }}PS$ and ${\frac {\omega ^{2}}{\sin ^{2}\alpha }}PK$.

The first of these components is a central force towards $S$ proportional to the distance.

The second is in a direction opposite to the velocity, and since

$PK=2\cos \alpha P^{\prime }S=-2{\frac {\sin \alpha \cos \alpha }{\omega }}v$,

this force may be written

$-2{\frac {\omega \cos \alpha }{\sin \alpha }}v$.

The acceleration of the particle is therefore compounded of two parts, the first of which is an attractive force $\mu r$, directed towards $S$, and proportional to the distance, and the second is $-2kv$, a resistance to the motion proportional to the velocity, where

$\mu ={\frac {\omega ^{2}}{\sin ^{\alpha }}}$, and $k=\omega {\frac {\cos \alpha }{\sin \alpha }}$.

If in these expressions we make $\alpha ={\frac {\pi }{2}}$, the orbit becomes a circle, and we have $\mu _{0}=\omega _{0}^{2}$, and $k=0$.

Hence, if the law of attraction remains the same, $\mu =\mu _{0}$, and

$\omega =\omega _{0}\sin \alpha$

or the angular velocity in different spirals with the same law of attraction is proportional to the sine of the angle of the spiral.

732.] If we now consider the motion of a point which is the projection of the moving point $P$ on the horizontal line $XY$, we shall find that its distance from $S$ and its velocity are the horizontal components of those of $P$. Hence the acceleration of this point is also an attraction towards $S$, equal to $\mu$ times its distance from $S$, together with a retardation equal to $k$ times its velocity.

We have therefore a complete construction for the rectilinear motion of a point, subject to an attraction proportional to the distance from a fixed point, and to a resistance proportional to the velocity. The motion of such a point is simply the horizontal