VELOCITY IN SIMPLE HARMONIC MOTION 



257 



turbances communicated by the rod to the air in Fig. VII-2 are 

 longitudinal waves. The end of the rod is vibrating in its pistonlike 

 motion in the same direction and in phase with the air particles. The 

 resulting longitudinal air vibrations may be experienced as acoustical 

 sensations. 



Velocity of a Paeticle 

 Simple Harmonic Motion 



Executing 



An air particle swinging in a simple 

 harmonic motion develops its maximum 

 velocity as it passes through its midpoint; 

 it decelerates as it moves away from this 

 point until at maximum displacement it 

 comes to rest. At this point it reverses its 

 direction of motion and accelerates to at- 

 tain its maximum velocity again at its 

 midpoint. It continues until it comes to 

 rest once more through deceleration at its 

 opposite extreme displacement, where it 

 again changes its direction, and acceler- 

 ates to regain its maximum velocity at the 

 middle of its swing. 



The instantaneous velocity at any point 

 on this path may be obtained with the aid 

 of the velocity of the particle moving 

 around the reference circle. 



In Fig. VII-3 let v be the linear velocity 

 of the reference particle Pi. The projec- 

 tion of v on the y axis is the simple-har- 

 monic-motion velocity desired, which is the 

 y component v y of the linear velocity. From 

 similar triangles cos 6 = cos u>t = v y /v or 



v v — v cos oit = coA cos o)t = — A cos — 



Note that, while the displacement is rep- 

 resented by a sine-wave curve, the accom- 

 panying velocity is a cosine-wave curve, identical except for a phase 

 shift of 7r/2 radians. The maximum velocity of the particle is attained 

 when it passes through the midpoint, i.e., when t = T/i, such that 



2tt it 2ttA 



*W = —Acos- = —^- = 2*nA 



Fig. VII-3. 



