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Learning Goal 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 1. z(t)= A cos (wt+) and 2. z(t)=C cos (wt) + Ssin (wt). Either of these equations is a general solution of a second-order differential equation (F=ma); 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.) Part C Find analytic expressions for the arbitrary constants C and Sin Equation 2 (found in Part B) in terms of the constants A and in Equation 1 (found in Part A), which are now considered as given parameters Give your answers for the coefficients of cos(ut) and sin(at), separated by a comma. Express your answers in terms of A and View Available Hint(s) VAL C.S= Submit Previous Answers 0? X Incorrect, Try Again; 5 attempts remaining Part D Find analytic expressions for the arbitrary constants A and in Equation 1 (found in Part A) in terms of the constants C and Sin Equation 2 (found in Part B), which are now considered as given parameters Express the amplitude A and phase (separated by a comma) in terms of C and S View Available Hint(s)

Learning Goal 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 1. z(t)= A cos (wt+) and 2. z(t)=C cos (wt) + Ssin (wt). Either of these equations is a general solution of a second-order differential equation (F=ma); 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.) Part C Find analytic expressions for the arbitrary constants C and Sin Equation 2 (found in Part B) in terms of the constants A and in Equation 1 (found in Part A), which are now considered as given parameters Give your answers for the coefficients of cos(ut) and sin(at), separated by a comma. Express your answers in terms of A and View Available Hint(s) VAL C.S= Submit Previous Answers 0? X Incorrect, Try Again; 5 attempts remaining Part D Find analytic expressions for the arbitrary constants A and in Equation 1 (found in Part A) in terms of the constants C and Sin Equation 2 (found in Part B), which are now considered as given parameters Express the amplitude A and phase (separated by a comma) in terms of C and S View Available Hint(s)

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Learning Goal: 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: 1. x(t) = A cos (wt + p) and 2. x(t) = C cos (wt) + Ssin (wt). Either of these equations is a general solution of a second-order differential equation (F = mā); 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.) Part C Find analytic expressions for the arbitrary constants C and S in Equation 2 (found in Part B) in terms of the constants A and in Equation 1 (found in Part A), which are now considered as given parameters. Give your answers for the coefficients of cos(wt) and sin(wt), separated by a comma. Express your answers in terms of A and ▸ View Available Hint(s) C.S= Submit Part D 195) ΑΣΦ A. = Submit 4 Previous Answers X Incorrect; Try Again; 5 attempts remaining [ΕΙ ΑΣΦ A Find analytic expressions for the arbitrary constants A and 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 (separated by a comma) in terms of C and S. ▸ View Available Hint(s) ^ C A ? C ?
Transcribed Image Text:Learning Goal: 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: 1. x(t) = A cos (wt + p) and 2. x(t) = C cos (wt) + Ssin (wt). Either of these equations is a general solution of a second-order differential equation (F = mā); 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.) Part C Find analytic expressions for the arbitrary constants C and S in Equation 2 (found in Part B) in terms of the constants A and in Equation 1 (found in Part A), which are now considered as given parameters. Give your answers for the coefficients of cos(wt) and sin(wt), separated by a comma. Express your answers in terms of A and ▸ View Available Hint(s) C.S= Submit Part D 195) ΑΣΦ A. = Submit 4 Previous Answers X Incorrect; Try Again; 5 attempts remaining [ΕΙ ΑΣΦ A Find analytic expressions for the arbitrary constants A and 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 (separated by a comma) in terms of C and S. ▸ View Available Hint(s) ^ C A ? C ?
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