Problem: Objective: To understand how the two standard ways to write the general solution to a harmonic oscillator are related.There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:x(t)=A cos(ωt+ϕ) andx(t)=C cos(ωt)+S sin(ωt).Either of these equations is a general solution of a second-order differential equation (F⃗ =ma⃗ ); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (Some texts refer to these arbitrary constants as boundary values.)A)Find analytic expressions for the arbitrary constants C and S in Equation 2 (found in Part B) in terms of the constants A and ϕ in Equation 1 (found in Part A), which are now considered as given parameters.Give your answers for the coefficients of cos(ωt) and sin(ωt), separated by a comma. Express your answers in terms of A and ϕ.B)Find analytic expressions for the arbitrary constants A and ϕ in Equation 1 (found in Part A) in terms of the constants C and S in Equation 2 (found in Part B), which are now considered as given parameters.Express the amplitude A and phase ϕ (separated by a comma) in terms of C and S.

FREE Expert Solution

cos (a + b) = cos a·cos b - sin a·sin b

cos2 θ+ sin2 θ=1

tan θ=sin θcos θ

A) 

x(t) = Acos(ωϕ

x(t)=A cos (ωt + ϕ)=A(cos ωt·cos ϕ-sin ωt·sin ϕ=(Acos ϕ)cos ωt+(-Asin ϕ)sin ωt

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Problem Details

Objective: To understand how the two standard ways to write the general solution to a harmonic oscillator are related.

There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:

x(t)=cos(ωt+ϕ) and

x(t)=cos(ωt)+sin(ωt).

Either of these equations is a general solution of a second-order differential equation (F⃗ =ma⃗ ); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (Some texts refer to these arbitrary constants as boundary values.)

A)

Find analytic expressions for the arbitrary constants C and S in Equation 2 (found in Part B) in terms of the constants A and ϕ in Equation 1 (found in Part A), which are now considered as given parameters.

Give your answers for the coefficients of cos(ωt) and sin(ωt), separated by a comma. Express your answers in terms of A and ϕ.

B)

Find analytic expressions for the arbitrary constants A and ϕ in Equation 1 (found in Part A) in terms of the constants C and S in Equation 2 (found in Part B), which are now considered as given parameters.

Express the amplitude A and phase ϕ (separated by a comma) in terms of C and S.