Proving Simple Harmonic Motion: Amplitude and Frequency Analysis

[lmedin02]

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.

 

[rock.freak667]

expand out Acos(ct-φ) and then equate coefficients. You should get two equations. Just try to relate them.

Hint: sin²x+cos²x=1

 

[lmedin02]

Got it. I expanded using the trig sums of angles formula for cosine. Thank you.

 

[Gregg]

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.

 

[rock.freak667]

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