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)=Acos(ωt+ϕ) and

x(t)=Ccos(ωt)+Ssin(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.

Question

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Respuesta :

moya1316 moya1316 · Answer 1 · 2020-01-03T18:32:34+01:00

Answer:

a) C = A cos φ , S = A sin φ, b) φ = tan⁻¹ (S / C) , A² = C² + S²

Explanation:

a) The two forms given are equivalent, let's start by developing the double angle

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

Call us

a = wt and b = φ

x = A (cos wt cosφ- sin wt sin φ

the second equation is

x = C cos wt + S sin wt

Let's match

C cos wt + S sin wt = A cos φ coswt - a sin φ sin wt

The coefficients of the cosine and breasts must be equal,

C = A cos φ

S = A sin φ

b) Divide the last two expressions

tan φ = S / C

φ = tan⁻¹ (S / C)

Let's square the two equations

C² = A² cos² φ

S² = A² sin² φ

Let's add

C² + S² = A² (cos² φ + sin² φ)

The part in brackets vouchers1

A² = C² + S²