Proving Simple Harmonic Motion: Amplitude and Frequency Analysis

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SUMMARY

The discussion focuses on proving the existence of a number A > 0 and φ such that the equation acos(ct) + bsin(ct) can be expressed as A*cos(ct - φ). The constants a, b, and c are predetermined, with c > 0. The solution involves deriving relationships between A, φ, a, and b using trigonometric identities and properties of derivatives, ultimately leading to the conclusion that φ = arctan(b/a) and A = a/cos(φ), with conditions ensuring A and φ remain positive.

PREREQUISITES
  • Understanding of trigonometric identities, specifically sin and cos functions.
  • Familiarity with derivatives and their properties.
  • Knowledge of simple harmonic motion and its mathematical representation.
  • Ability to manipulate and solve equations involving trigonometric functions.
NEXT STEPS
  • Study the derivation of trigonometric identities, particularly the angle sum formulas.
  • Learn about the properties of simple harmonic motion and its mathematical modeling.
  • Explore the application of derivatives in proving function equivalences.
  • Investigate the implications of amplitude and phase shift in oscillatory systems.
USEFUL FOR

Students studying physics or mathematics, particularly those focusing on mechanics and oscillatory motion, as well as educators looking for clear explanations of simple harmonic motion principles.

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Homework Statement


Prove that there exists a number A>0 and \phi such that acos(ct)+bsin(ct)=Acos(ct-\phi).


Homework Equations


a,b,c are predermined constants where c>0. From this equation I can justify conclusions regarding the amplitude, frequency, and so forth of a simple harmonic ocillator.


The Attempt at a Solution


Obviously if a (or b) is 0, then A is equal b (or a, respectively) and \phi is 0. Thus, I can now assume that a and b are not 0. I try defining two different functions and proving that they are equal for every t using properties of the derivatives.
 
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expand out Acos(ct-φ) and then equate coefficients. You should get two equations. Just try to relate them.

Hint: sin2x+cos2x=1
 
Got it. I expanded using the trig sums of angles formula for cosine. Thank you.
 
how do you finish this?

do you get b=-Asin(phi) and a = Acos(phi)

then phi = arctan(b/a)

and A = a/(cos(phi))

do you say that there exists phi = arctan(b/a) > 0 which implies cos(phi) > 0 for 0<phi<pi/4. provided that a > 0 A > 0. etc? I don't see how you 'prove' this.
 
Gregg said:
how do you finish this?

do you get b=-Asin(phi) and a = Acos(phi)

then phi = arctan(b/a)

and A = a/(cos(phi))

do you say that there exists phi = arctan(b/a) > 0 which implies cos(phi) > 0 for 0<phi<pi/4. provided that a > 0 A > 0. etc? I don't see how you 'prove' this.


you'd get b=Asinφ and a = Acosφ

consider what a2+b2, gives. Since tanφ=b/a, then φ exists since a,b≠0
 

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