Particle Simple harmonic motion particle problem

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SUMMARY

The discussion focuses on deriving the relationship between the maximum speed (vmax) and the average speed (vaverage) of a particle undergoing simple harmonic motion. It is established that vmax equals (π/2) times vaverage, where vaverage is calculated by determining the total distance traveled in one cycle divided by the period (T). The equations used include v(t) = -vmax sin(wt + φ) and vmax = 2πA/T = ωA, emphasizing the importance of amplitude (A) and angular frequency (ω) in the calculations.

PREREQUISITES
  • Understanding of simple harmonic motion principles
  • Familiarity with trigonometric functions and their applications in physics
  • Knowledge of the concepts of period (T) and amplitude (A)
  • Basic calculus for integrating motion equations
NEXT STEPS
  • Calculate average speed in simple harmonic motion using total distance and period
  • Explore the derivation of maximum speed in simple harmonic motion
  • Investigate the role of angular frequency (ω) in harmonic motion
  • Study the differences between average speed and average velocity in oscillatory systems
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and oscillatory motion, as well as educators seeking to clarify concepts related to simple harmonic motion.

<3Science
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Homework Statement



For a particle in simple harmonic motion, show that vmax = (pi/2) vaverage, where vaverage is the average speed during one cycle of the motion.

Homework Equations



v(t) = -vmaxsin(wt + phi)
vmax = 2piA/ T = wA

The Attempt at a Solution



I'm really not sure at all how to start this, but I am guessing there really wouldn't be a numerical value just theory based?
 
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<3Science said:
I'm really not sure at all how to start this, but I am guessing there really wouldn't be a numerical value just theory based?
Normally I would insist that show more of your efforts first, but this one gave me a little trouble at the beginning too.

To start with, determine the average speed over one period. The only tricky part is figuring out what "average speed" really means. The average velocity is zero, so that certainly doesn't help. It's also not the root mean square (rms) velocity. Don't go that way.

To find the average speed, first determine the total distance the object travels in one period (not to be confused with displacement). It's a simple function of the amplitude A in,
x = Acos(ωt + Φ). But don't over-think it. How many As are traversed during a single period? Once you have the total distance, divide that by the period T. That's the average speed.
 

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