31 



Figure 20. 



\ 



\ 



N./ 



/ 



Variation in length of resultant vector. 



If then in figure 20 A, Oj)i is laid off equal to Ai, the angle MpiPs 

 equal to bt, and pip^ equal to A2, the triangle OprPs in figure 20A 

 reproduces the triangle CP1P3 in figure 20, and Op^ is the length of 

 the resultant vector CP3. As the time increases the point jh evi- 

 dently describes a circle of radius A2 around pi as a center. The 

 length of the resultant vector fluctuates between 0M=^i+^2 and 

 0N=Ai—A2. The period in which the point p^ completes the circuit 

 around pi as a center obviously is 360°/6. This interval is called the 

 synodic period of the two components, and, without regard to the 

 algebraic sign of b, is the interval between the successive times at 

 which the resultant vector reaches its maximum length, and also 

 between the times at which this vector has its minimum length. 



55. Variation in the speed of the resultant vector. — A consideration 

 of figure 20A makes it apparent that the resultant vector alternately 

 leads and lags behind the radius vector of the major component; and 

 that its mean speed is the speed of the major component. A study 

 of the figure shows fin-ther that when b is positive, the point ^3 rotates 

 around pi in the same direction that CPi rotates around C, and the 

 speed of the resultant is a maximum at the point M, when the length 

 of the resultant is a maximum. When b is negative, ps i^otates in the 

 opposite direction and the speed of the resultant is a maximum at 

 the point N, when the length of the resultant is a minimum. 



56. Speed of the resultant of two components of equal amplitudes. — If 

 the amplitudes of the two components are equal, the resultant CP^ 

 (fig. 20), evidently bisects the angle PiCPo, and the angle YCP3 is 

 therefore equal to (ai + }^6)f. The speed of the resultant therefore 

 has the constant value of ai-\-%b, the average of the speeds of the two 

 components. 



57. Form of resultant of two components whose speeds are nearly 

 equal. — If the difference between the speeds of two components is 

 relatively small, so that the length of their resultant vector changes 



