Page:Elementary Text-book of Physics (Anthony, 1897).djvu/32

direction are given. It is therefore a vector quantity and may be represented by a straight line. Two or more accelerations may be compounded by the rules for the composition of vectors.

18. Angular Velocity and Acceleration.—The angle contained by the line passing through two points, one of which is in motion, and any assumed line passing through the iixed point, will, in general, vary. The rate of its change is called the angular velocity of the moving point. If and 0„ represent the angles made by the moving line with the fixed line at the instants $t$ and $t_{0}$, then the angular velocity, if constant, is measured by

$\omega ={\frac {\phi -\phi _{0}}{t-t_{0}}}\cdot$

(7)

If variable, it is measured by the limit of the same expression, ${\frac {d\phi }{dt}}={\frac {\phi -\phi _{0}}{t-t_{0}}}$, as the interval $t-t_{0}$ becomes indefinitely small.

The angular acceleration is the rate of change of angular velocity. If constant, it is measured by

$\alpha ={\frac {\omega -\omega _{0}}{t-t_{0}}}\cdot$

(8)

If variable, it is measured by the limit of the same expression, ${\frac {d\omega }{dt}}={\frac {\omega -\omega _{0}}{t-t_{0}}}$, as the interval $t-t_{0}$ becomes indefinitely small.

If the radian be taken as the unit of angle, the dimensions of angle become $\left[{\frac {\text{arc}}{\text{radius}}}\right]={\frac {L}{L}}=1.$ Hence the dimensions of angular velocity are $T^{-1}$, and of angular acceleration $T^{-2}$.

If any point be revolving about a fixed point as a centre, its velocity in the circle is equal to the product of its angular velocity and the length of the radius of the circle.

19. Linear Motion with Constant Acceleration.—The space $s-s_{0}$ traversed by a point moving with a constant acceleration $a$, during a time $t-t_{0}$, is determined by considering that, since the acceleration is constant, the average velocity ${\frac {v+v_{0}}{2}}$ for the time