‘EFFORT, ACCELERATION, AND VELOCITY DIAGRAMS 297 
The mean velocity of a between a and a, is }(v,+v), and the time taken 
to travel from a to a is iw ae °) =" =), If f is the mean accelera- 
_ 9) 2e—%) _ vi -v 
tion of a between aand a,, f=(v,—v) + sae Ha —n me Ve +2). 
Now if AA, is made indefinitely small, x, mere equal to #, and i 
becomes the acceleration of a ata, Hence f vad i 
If f be plotted on the space base BOC, the straight line bOc is the 
result, the maximum values of f being at B and C wherex=r. At the 
centre O, where x=0, f=0. At B and C, v=0, and at O, v has its 
maximum value, and is there equal to V. 
When A is at B or C, A and a coincide, and f for a becomes the 
radial acceleration of A, namely, — Ly =a result which has been proved in 
another way in Art. 21, p. 17. 
Since the acceleration of the point a is directly proportional to its 
displacement from its middle position, this property may be used as a 
test of simple harmonic motion. In fact, the 
definition of simple harmonic motion is better > aaa st 
given as the motion which a point has when its B 
acceleration is proportional to its displacement ares 40 
| from its middle position, because this includes Sead si 
| the case of a point oscillating in a curved path (Fig. 460), where the are 
OB or the are OC =7, and the are Oa=z. 
A complete oscillation or vibration is a movement from one end of 
| the path to the other and back again. The time of a complete oscillation 
| is called the periodic time. If V is in feet per second, f in feet per 
second per second, 7 in feet, and ¢, the periodic time, in seconds, then 
_ Referring again to Fig. 459, Aa=rsin 6, but v= V sin 0, therefore if 
the velocity scale be chosen so that 7 represents V, then Aa will represent 
Fig. 461. 
v on that scale, and the circle BACD will be the velocity diagram on 
the space mn BOC for the point which has simple harmonic motion. 
Again, f Be; = ; therefore if the acceleration scale be so chosen that r 
