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 CO2strongly absorbs infrared radiation when its vibrationalnormal modes are excited by light at the normal-modefrequencies. CO₂ is a linear triatomic molecule, asshown in (Figure 1), with oxygen atoms of mass mobonded to a central carbon atom of mass mc. Youknow from chemistry that the atomic masses of carbonand oxygen are, respectively, 12 and 16. Assume thatthe bond is an ideal spring with spring constant k.There are two normal modes of this system for whichoscillations take place along the axis. (You can ignoreadditional bending modes.) In this problem, you will findthe normal modes and then use experimental data todetermine the bond spring constant.Figuremok2mc1x₂k3moX3Part ALet x₁, x2, and 3 be the atoms' positions measured from their equilibrium positions. First, use Hooke's law to write the net forceon each atom. Pay close attention to signs! For each oxygen, the net force equals mod²x/dt². Carbon has a different mass, so itsnet 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.d²x₁dt²Submit Previous Answers= -a²x₁ + a²x₂✓ CorrectPart BFind 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.d²x2dt²=Submit-23²x₂ + ß²x₁ + ²x3Previous AnswersReviewCorrect The greenhouse-gas carbon dioxide molecule CO2strongly absorbs infrared radiation when its vibrationalnormal modes are excited by light at the normal-modefrequencies. CO2 is a linear triatomic molecule, asshown in (Figure 1), with oxygen atoms of mass mobonded to a central carbon atom of mass mc. Youknow from chemistry that the atomic masses of carbonand oxygen are, respectively, 12 and 16. Assume thatthe bond is an ideal spring with spring constant k.There are two normal modes of this system for whichoscillations take place along the axis. (You can ignoreadditional bending modes.) In this problem, you will findthe normal modes and then use experimental data todetermine the bond spring constant.FigureO1moX₁kC2mcX2k

Question

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 CO2strongly absorbs infrared radiation when its vibrationalnormal modes are excited by light at the normal-modefrequencies. CO₂ is a linear triatomic molecule, asshown in (Figure 1), with oxygen atoms of mass mobonded to a central carbon atom of mass mc. Youknow from chemistry that the atomic masses of carbonand oxygen are, respectively, 12 and 16. Assume thatthe bond is an ideal spring with spring constant k.There are two normal modes of this system for whichoscillations take place along the axis. (You can ignoreadditional bending modes.) In this problem, you will findthe normal modes and then use experimental data todetermine the bond spring constant.Figuremok2mc1x₂k<1 of 1 >3moX3Part ALet x₁, x2, and 3 be the atoms' positions measured from their equilibrium positions. First, use Hooke's law to write the net forceon each atom. Pay close attention to signs! For each oxygen, the net force equals mod²x/dt². Carbon has a different mass, so itsnet 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.d²x₁dt²Submit Previous Answers= -a²x₁ + a²x₂✓ CorrectPart BFind 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.d²x2dt²=Submit-23²x₂ + ß²x₁ + ²x3Previous AnswersReviewCorrect The greenhouse-gas carbon dioxide molecule CO2strongly absorbs infrared radiation when its vibrationalnormal modes are excited by light at the normal-modefrequencies. CO2 is a linear triatomic molecule, asshown in (Figure 1), with oxygen atoms of mass mobonded to a central carbon atom of mass mc. Youknow from chemistry that the atomic masses of carbonand oxygen are, respectively, 12 and 16. Assume thatthe bond is an ideal spring with spring constant k.There are two normal modes of this system for whichoscillations take place along the axis. (You can ignoreadditional bending modes.) In this problem, you will findthe normal modes and then use experimental data todetermine the bond spring constant.FigureO1moX₁kC2mcX2k< 1 of 13mo1 X3Part CFind 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.d²x3dt²=Submita² (x₂-x3)Part DPrevious AnswersCorrectReviewThe main task is to find three new variables whose differential equation is the SHM equation, meaning that they oscillate at a singlefrequency that you can identify. The first is w = X1 + (Mc/M0)X2 +£3. Notice that the mc/mo = a² /B². If you calculated² 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 thatmultiplying 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 tod² 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 combinationsuch 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.

Accepted Answer

Given: 
∂2x1∂t2 = -α2x1+α2x2∂2x3∂t2 = α2x2-α2x3