Answered: NOTE: Parts A and Part B are already included. I need help with Part C. That is why I included Parts A and B. Part C's answer as seen in the screenshot is...

NOTE: Parts A and Part B are already included. I need help with Part C. That is why I included Parts A and B. Part C's answer as seen in the screenshot is incorrect, a expert here gave me incorrect answer.

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Transcribed Image Text:**Part C** Find an equation for the second derivative of the \( x_3 \) coordinate. **Express your answer in terms of some, all, or none of the variables \( x_1, x_2, x_3 \), and the constants \( \alpha, \beta \).** The equation entered: \[ \frac{d^2 x_3}{dt^2} = \alpha^2 x_2 - x_3 \] Options available for editing equations include various mathematical functions such as exponentiation, fractions, roots, and others. **Submission Options:** - **Submit**: Click to submit your current answer. - **Previous Answers**: Review past submitted answers. - **Request Answer**: Request the correct answer if needed. **Feedback:** - *Incorrect; Try Again; 4 attempts remaining*

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Transcribed Image Text:The greenhouse-gas carbon dioxide molecule, \( \text{CO}_2 \), strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. \( \text{CO}_2 \) is a linear triatomic molecule, as shown in Figure 1, with oxygen atoms of mass \( m_0 \) bonded to a central carbon atom of mass \( m_C \). The atomic masses of carbon and oxygen are 12 and 16, respectively. Assume the bond is an ideal spring with spring constant \( k \). There are two normal modes of this system for which oscillations take place along the axis. (You can ignore additional bending modes.) In this problem, you will find the normal modes and then use experimental data to determine the bond spring constant. **Figure 1 Explanation:** The figure illustrates a linear \( \text{CO}_2 \) molecule. Two oxygen atoms (\( O \)), each with mass \( m_0 \) and position \( x_1 \) or \( x_3 \), are connected to a carbon atom (\( C \)) of mass \( m_C \) at position \( x_2 \) by springs with spring constant \( k \). **Part A:** - Let \( x_1, x_2, \) and \( x_3 \) be the atoms' positions measured from their equilibrium positions. - Use Hooke's law to write the net force on each atom. - For each oxygen atom, the net force is \( m_0 \frac{d^2 x}{dt^2} \). - For the carbon atom, the net force is \( m_C \frac{d^2 x}{dt^2} \). - Define \( \alpha^2 = \frac{k}{m_0} \) and \( \beta^2 = \frac{k}{m_C} \). - Find the equation for the second derivative of \( x_1 \). \[ \frac{d^2 x_1}{dt^2} = -\alpha^2 x_1 + \alpha^2 x_2 \] **Part B:** - Find an equation for the second derivative of \( x_2 \). \[ \frac{d^2 x_2}{dt^2} = -2\beta^2 x_2 + \beta^2 x_1 + \beta

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