Simple harmonic motion and circular motion

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Discussion Overview

The discussion explores the relationship between simple harmonic motion (SHM) and uniform circular motion, examining how one can be viewed as a projection or component of the other. The scope includes theoretical comparisons and mathematical representations of both types of motion.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant questions why simple harmonic motion is often compared to uniform circular motion.
  • Another participant asserts that uniform circular motion can be viewed as "two dimensional simple harmonic motion," using the analogy of a shadow cast by an object moving in a circle.
  • A different participant explains that in a Cartesian coordinate system, the x and y coordinates of a particle in uniform circular motion can be expressed as simple harmonic motion equations, specifically x=R*cos(wt) and y=R*sin(wt), indicating a relationship between the two motions.
  • It is noted that uniform circular motion can be seen as a combination of two simple harmonic motions with the same frequency but a phase difference of π/2, and that these motions are perpendicular to each other.
  • Another participant mentions that the usual form of the solution for SHM can be interpreted as the x component of a body in uniform circular motion.

Areas of Agreement / Disagreement

Participants generally agree on the mathematical relationship between simple harmonic motion and uniform circular motion, but the discussion remains exploratory without a definitive consensus on the implications of this relationship.

Contextual Notes

The discussion relies on specific mathematical representations and interpretations, which may depend on the definitions and assumptions regarding the coordinate systems used.

gunparashar
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why simple harmonic motion is projected as or compared with uniform circular motion ?
 
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Because uniform circular motion is essentially "two dimensional simple harmonic motion."

Consider the shadow of an object moving in a circle when the light is shined "edge on". (This essentially "collapses" or "ignores" one of the dimensions of the motion)
 
Well if one considers a cartesian coordinate system with origin at the center of the circle of the uniform circular motion, then can prove fairly easily that the x and y coordinates (of the particle that does circular motion), are doing simple harmonic motion. That is it will be x=R*cos(wt) and y=R*sin(wt) where w the angular velocity, R the radius of the circle.

So you can view uniform circular motion as composition of two simple harmonic motions of the same frequency, with phase differnce pi/2, and (thats important) of direction perpendicular to each other.
 
The usual form the solution is given in, that is in terms of a shifted cosine, can be interpreted as the x component of a body in uniform circular motion.
 

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