NOTE: Parts A and Part B are already included. I need help with Part C. That is why I included Parts A and B.
NOTE: Parts A and Part B are already included. I need help with Part C. That is why I included Parts A and B.
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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
**Task:**
Find an equation for the second derivative of the \( x_3 \) coordinate.
**Instructions:**
Express your answer in terms of some, all, or none of the variables \( x_1, x_2, x_3 \), and the constants \( \alpha, \beta \).
**Equation Input:**
\[ \frac{d^2 x_3}{dt^2} = 2 \beta x_3 \]
**Response:**
- Result: Incorrect; Try Again
- Attempts Remaining: 5
**Explanation:**
The main task is to find three new variables whose differential equation is the Simple Harmonic Motion (SHM) equation, meaning that they oscillate at a single frequency that you can identify. The first is:
\[ w = x_1 + \left(\frac{m_c}{m_o}\right)x_2 + x_3. \]
Notice that
\[ \frac{m_c}{m_o} = \frac{\alpha^2}{\beta^2}. \]
If you calculate
\[ \frac{d^2 w}{dt^2}, \]
you'll find that it equals zero. (Check this to verify that your work is correct to this point.) This seems odd, but notice that multiplying \( w \) by \( m_o \) gives an expression for the center of mass. So what you've learned is that the center of mass doesn't accelerate. It could be drifting through space, but for this problem you can let the center of mass be at rest at the origin. So the solution to
\[ \frac{d^2 w}{dt^2} = 0 \]
is
\[ w = 0. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fd2a550-2608-4552-aa22-8731021ca9ca%2Ff5815279-5979-43a6-bdef-6e79451787dc%2Fzm9ww7k_processed.png&w=3840&q=75)
Transcribed Image Text:## Part C
**Task:**
Find an equation for the second derivative of the \( x_3 \) coordinate.
**Instructions:**
Express your answer in terms of some, all, or none of the variables \( x_1, x_2, x_3 \), and the constants \( \alpha, \beta \).
**Equation Input:**
\[ \frac{d^2 x_3}{dt^2} = 2 \beta x_3 \]
**Response:**
- Result: Incorrect; Try Again
- Attempts Remaining: 5
**Explanation:**
The main task is to find three new variables whose differential equation is the Simple Harmonic Motion (SHM) equation, meaning that they oscillate at a single frequency that you can identify. The first is:
\[ w = x_1 + \left(\frac{m_c}{m_o}\right)x_2 + x_3. \]
Notice that
\[ \frac{m_c}{m_o} = \frac{\alpha^2}{\beta^2}. \]
If you calculate
\[ \frac{d^2 w}{dt^2}, \]
you'll find that it equals zero. (Check this to verify that your work is correct to this point.) This seems odd, but notice that multiplying \( w \) by \( m_o \) gives an expression for the center of mass. So what you've learned is that the center of mass doesn't accelerate. It could be drifting through space, but for this problem you can let the center of mass be at rest at the origin. So the solution to
\[ \frac{d^2 w}{dt^2} = 0 \]
is
\[ w = 0. \]
![**Carbon Dioxide Vibrational Modes and Bond Spring Constant Determination**
The greenhouse-gas carbon dioxide molecule, CO₂, strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. CO₂ 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. The bond is considered an ideal spring with spring constant \( k \).
There are two normal modes of this system for which oscillations occur along the axis. Ignoring additional bending modes, this problem focuses on finding the normal modes and utilizing experimental data to determine the bond spring constant.
**Figure Explanation:**
The figure depicts a linear triatomic molecule:
- Two oxygen atoms (O) with masses \( m_0 \).
- A central carbon atom (C) with mass \( m_C \).
- These atoms are interconnected by springs with spring constant \( k \).
- The positions \( x_1 \), \( x_2 \), and \( x_3 \) denote the displacement from equilibrium for each atom.
**Part A**
Let \( x_1 \), \( x_2 \), and \( x_3 \) be the atoms' positions measured from their equilibrium positions. Using Hooke's law, the net force on each atom is defined:
- For oxygen: \( m_0 \frac{d^2x}{dt^2} \)
- For carbon: \( m_C \frac{d^2x}{dt^2} \)
Define \( \alpha^2 = \frac{k}{m_0} \) and \( \beta^2 = \frac{k}{m_C} \). An equation for the second derivative of \( x_1 \) is found:
\[ \frac{d^2x_1}{dt^2} = -\alpha^2 x_1 + \alpha^2 x_2 \]
**Part B**
Find the equation for the second derivative of \( x_2 \) coordinate:
\[ \frac{d^2x_2}{dt^2} = -2\beta^2 x_2 + \beta^2 x_1 + \beta^2 x_3 \]
Both parts require expressing the solutions in terms of the variables \( x_1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fd2a550-2608-4552-aa22-8731021ca9ca%2Ff5815279-5979-43a6-bdef-6e79451787dc%2Fv541l78_processed.png&w=3840&q=75)
Transcribed Image Text:**Carbon Dioxide Vibrational Modes and Bond Spring Constant Determination**
The greenhouse-gas carbon dioxide molecule, CO₂, strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. CO₂ 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. The bond is considered an ideal spring with spring constant \( k \).
There are two normal modes of this system for which oscillations occur along the axis. Ignoring additional bending modes, this problem focuses on finding the normal modes and utilizing experimental data to determine the bond spring constant.
**Figure Explanation:**
The figure depicts a linear triatomic molecule:
- Two oxygen atoms (O) with masses \( m_0 \).
- A central carbon atom (C) with mass \( m_C \).
- These atoms are interconnected by springs with spring constant \( k \).
- The positions \( x_1 \), \( x_2 \), and \( x_3 \) denote the displacement from equilibrium for each atom.
**Part A**
Let \( x_1 \), \( x_2 \), and \( x_3 \) be the atoms' positions measured from their equilibrium positions. Using Hooke's law, the net force on each atom is defined:
- For oxygen: \( m_0 \frac{d^2x}{dt^2} \)
- For carbon: \( m_C \frac{d^2x}{dt^2} \)
Define \( \alpha^2 = \frac{k}{m_0} \) and \( \beta^2 = \frac{k}{m_C} \). An equation for the second derivative of \( x_1 \) is found:
\[ \frac{d^2x_1}{dt^2} = -\alpha^2 x_1 + \alpha^2 x_2 \]
**Part B**
Find the equation for the second derivative of \( x_2 \) coordinate:
\[ \frac{d^2x_2}{dt^2} = -2\beta^2 x_2 + \beta^2 x_1 + \beta^2 x_3 \]
Both parts require expressing the solutions in terms of the variables \( x_1
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