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In Section 5.2 we discussed four equivalent ways to
Chapter 5, Problem 5.5(choose chapter or problem)
In Section 5.2 we discussed four equivalent ways to represent simple harmonic motion in one dimension:
\(\begin{aligned}
x(t) & =C_{1} e^{i \omega t}+C_{2} e^{-i \omega t} \quad \quad \quad \quad (I) \\
& =B_{1} \cos (\omega t)+B_{2} \sin (\omega t) \quad (II)\\
& =A \cos (\omega t-\delta) \quad \quad \quad \quad \quad (III)\\
& =\operatorname{Re} C e^{i \omega t} \quad \quad \quad \quad \quad \quad \quad (IV)
\end{aligned}\)
To make sure you understand all of these, show that they are equivalent by proving the following implications: \(\mathrm{I} \Rightarrow \mathrm{II} \Rightarrow \mathrm{III} \Rightarrow \mathrm{IV} \Rightarrow \mathrm{I}\). For each form, give an expression for the constants (\(C_{1}, C_{2}\), etc.) in terms of the constants of the previous form.
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QUESTION:
In Section 5.2 we discussed four equivalent ways to represent simple harmonic motion in one dimension:
\(\begin{aligned}
x(t) & =C_{1} e^{i \omega t}+C_{2} e^{-i \omega t} \quad \quad \quad \quad (I) \\
& =B_{1} \cos (\omega t)+B_{2} \sin (\omega t) \quad (II)\\
& =A \cos (\omega t-\delta) \quad \quad \quad \quad \quad (III)\\
& =\operatorname{Re} C e^{i \omega t} \quad \quad \quad \quad \quad \quad \quad (IV)
\end{aligned}\)
To make sure you understand all of these, show that they are equivalent by proving the following implications: \(\mathrm{I} \Rightarrow \mathrm{II} \Rightarrow \mathrm{III} \Rightarrow \mathrm{IV} \Rightarrow \mathrm{I}\). For each form, give an expression for the constants (\(C_{1}, C_{2}\), etc.) in terms of the constants of the previous form.
ANSWER:Step 1 of 5
General expressions for simple harmonic motion are:
\(x\left( t \right) = {C_1}{e^{i\omega t}} + {C_2}{e^{ - i\omega t}}.....\left( i \right)\)
\(x\left( t \right) = {B_1}\cos \left( {\omega t} \right) + {B_2}\sin \left( {\omega t} \right)...\left( {ii} \right)\)
\(x\left( t \right) = A\cos \left( {\omega t - \delta } \right)....\left( {iii} \right)\)
\(x\left( t \right) = {\mathop{\rm Re}\nolimits} C{e^{i\omega t}}......\left( {iv} \right)\)
Here, \({C_1},{C_2},{B_1}\;{\rm{and}}\;{B_2}\) are the constants, A is the amplitude,\( \omega\) is the angular frequency, t is the time, \(\delta\) is the phase constant, and Re is the real part.
Step 2 of 5
a)
From the complex analysis, \({e^{i\omega t}}\) and \({e^{ - i\omega t}}\) can be written as,
\({e^{i\omega t}} = \cos \omega t + i\sin \omega t\)
\({e^{ - i\omega t}} = \cos \omega t - i\sin \omega t\)
Substitute the values in equation (i), and we get,
\(x\left( t \right) = {C_1}\left( {\cos \omega t + i\sin \omega t} \right) + {C_2}\left( {\cos \omega t - i\sin \omega t} \right)\)
\(x\left( t \right) = \left( {{C_1} + {C_2}} \right)\cos \omega t + i\left( {{C_1} - {C_2}} \right)\sin \omega t...\left( v \right)\)
Let \(\left( {{C_1} + {C_2}} \right) = {B_1}\) and \(i\left( {{C_1} - {C_2}} \right) = {B_2}\). Substitute the values in equation (v), and we get,
\(x\left( t \right) = {B_1}\cos \left( {\omega t} \right) + {B_2}\sin \left( {\omega t} \right)\)
Here, \({B_2}\) is a complex number.
This equation is the same as equation (ii). Thus, equation (ii) is obtained from equation (i).
H
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