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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 mo bonded to a central carbon atom of mass mc. You know from chemistry that the atomic masses of carbon and oxygen are, respectively, 12 and 16. Assume that 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.

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 mo bonded to a central carbon atom of mass mc. You know from chemistry that the atomic masses of carbon and oxygen are, respectively, 12 and 16. Assume that 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.

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NOTE: Parts A and Part B and Part C are already included. I need help with Part D. That is why I included Parts A, B, and C. What is the solution to Part D?

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 mo bonded to a central carbon atom of mass mc. You know from chemistry that the atomic masses of carbon and oxygen are, respectively, 12 and 16. Assume that 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 mo k 2 mc 1x₂ k < 1 of 1 > 3 mo X3 Part A Let x₁, x2, and 3 be the atoms' positions measured from their equilibrium positions. First, use Hooke's law to write the net force on each atom. Pay close attention to signs! For each oxygen, the net force equals mod²x/dt². Carbon has a different mass, so its net force is mcd²x/dt². Define a² = k/mo and ß² = k/mc. Find an equation for the second derivative of ₁ coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, ß. d²x₁ dt² Submit Previous Answers = -a²x₁ + a²x₂ ✓ Correct Part B Find an equation for the second derivative of 2 coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. d²x2 dt² = Submit -23²x₂ + ß²x₁ + ²x3 Previous Answers Review Correct
Transcribed Image Text: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 mo bonded to a central carbon atom of mass mc. You know from chemistry that the atomic masses of carbon and oxygen are, respectively, 12 and 16. Assume that 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 mo k 2 mc 1x₂ k < 1 of 1 > 3 mo X3 Part A Let x₁, x2, and 3 be the atoms' positions measured from their equilibrium positions. First, use Hooke's law to write the net force on each atom. Pay close attention to signs! For each oxygen, the net force equals mod²x/dt². Carbon has a different mass, so its net force is mcd²x/dt². Define a² = k/mo and ß² = k/mc. Find an equation for the second derivative of ₁ coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, ß. d²x₁ dt² Submit Previous Answers = -a²x₁ + a²x₂ ✓ Correct Part B Find an equation for the second derivative of 2 coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. d²x2 dt² = Submit -23²x₂ + ß²x₁ + ²x3 Previous Answers Review Correct
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 mo bonded to a central carbon atom of mass mc. You know from chemistry that the atomic masses of carbon and oxygen are, respectively, 12 and 16. Assume that 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 O 1 mo X₁ k C 2 mc X2 k < 1 of 1 3 mo 1 X3 Part C Find an equation for the second derivative of 3 coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. d²x3 dt² = Submit a² (x₂-x3) Part D Previous Answers Correct Review The main task is to find three new variables whose differential equation is the SHM equation, meaning that they oscillate at a single frequency that you can identify. The first is w = X1 + (Mc/M0)X2 +£3. Notice that the mc/mo = a² /B². If you calculate d² w/dt², 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 mo 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 d² w/dt² = 0 is w = 0. A second new variable (call it u) can be formed from only ₁ and 3, the oxygen coordinates. You are looking for a combination such that d²u/dt² = -constant xu. Determine u. The solution to the equation for u is u = 2A cos(wt + a). Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the appropriate constants.
Transcribed Image Text: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 mo bonded to a central carbon atom of mass mc. You know from chemistry that the atomic masses of carbon and oxygen are, respectively, 12 and 16. Assume that 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 O 1 mo X₁ k C 2 mc X2 k < 1 of 1 3 mo 1 X3 Part C Find an equation for the second derivative of 3 coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. d²x3 dt² = Submit a² (x₂-x3) Part D Previous Answers Correct Review The main task is to find three new variables whose differential equation is the SHM equation, meaning that they oscillate at a single frequency that you can identify. The first is w = X1 + (Mc/M0)X2 +£3. Notice that the mc/mo = a² /B². If you calculate d² w/dt², 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 mo 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 d² w/dt² = 0 is w = 0. A second new variable (call it u) can be formed from only ₁ and 3, the oxygen coordinates. You are looking for a combination such that d²u/dt² = -constant xu. Determine u. The solution to the equation for u is u = 2A cos(wt + a). Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the appropriate constants.
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