356 



PRACTICAL MATHEMATICS 



But OQ = OPsin(n + E), which is a simple harmonic motion, 

 of amplitude equal to the length of OP and the epoch E = angle 

 ROP. 



Then 



OP = AJ[ + A* + 2AiA 2 cos (6 2 - 



- A, Sin(TU+,) 



FIG. Il8. 



Let P^! and PR be drawn perpendicular to OR and P X S 

 parallel to OR 

 Then 



and 



Now 



tanE = 



OR = ORi + RiR 



= A x cos e x + A 2 cos e 2 

 PR = PS + SR 

 = PS + PxRi 



= A 2 sin e 2 + A x sin e x 

 PR 

 OR 



Aj sin ej + A 2 sin e 2 

 Aj cos e x + A 2 cos e 2 



180. The Vibration of a Spring. Let h be the stiffness of the 

 spring that is, a force of 1 lb., will elongate the spring h ft. If 



tvt 



the spring is elongated x ft. the force required will be -r lb. 



Let a body of mass m lb. be hung from this spring and then 

 displaced from its equilibrium position and then let go. 



If there are no frictional resistances to the motion of the body, 

 the only force acting on the body will be the resistance of the 

 spring, and when the body is at a distance x ft, from its equili- 



sy> 



brium position this resistance is r lb. 



