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A model of the carbon dioxide (CO

Two sliders A and C, each of mass M, represent the oxygen atoms and are connected by light springs, of force constant k, to a slider B of mass m, representing the carbon atom. All three sliders are placed on a linear air track. The two important modes of oscillation along the axis of the model molecule are as follows:

Mode 1: B remains stationary, and A and C oscillate so that the centre of mass of the model molecule remains stationary

Mode 2: A and C move equal distances in one direction, and B moves in the opposite direction in such a way that the centre of mass again remains stationary.

Show that the frequency f of mode 2 is given by [tex]f = (1/2\pi) [(k/M) + (2k/m)]^{1/2}[/tex]

Attempt:

Forgive me for my messy working:

[tex]\text{In\ equilibrium,} zm = M (2y)[/tex]

I came to me that the centre of mass of the system should remain unchanged throughout the motion, but how exactly do I justify the statement? Can I say "because there is no net force acting on the system, the centre of mass remains the same throughout the motion"?

Also, I am able to obtain the about equation f if I consider the motion in C. Could someone please show me how to obtain the same equation using B?

Can anybody explain to me how

Thank you!

_{2}) molecule is constructed as shown in Fig. SM10.1Two sliders A and C, each of mass M, represent the oxygen atoms and are connected by light springs, of force constant k, to a slider B of mass m, representing the carbon atom. All three sliders are placed on a linear air track. The two important modes of oscillation along the axis of the model molecule are as follows:

Mode 1: B remains stationary, and A and C oscillate so that the centre of mass of the model molecule remains stationary

Mode 2: A and C move equal distances in one direction, and B moves in the opposite direction in such a way that the centre of mass again remains stationary.

Show that the frequency f of mode 2 is given by [tex]f = (1/2\pi) [(k/M) + (2k/m)]^{1/2}[/tex]

Attempt:

Forgive me for my messy working:

[tex]\text{In\ equilibrium,} zm = M (2y)[/tex]

I came to me that the centre of mass of the system should remain unchanged throughout the motion, but how exactly do I justify the statement? Can I say "because there is no net force acting on the system, the centre of mass remains the same throughout the motion"?

Also, I am able to obtain the about equation f if I consider the motion in C. Could someone please show me how to obtain the same equation using B?

Can anybody explain to me how

actually came about?in equilibrium, [tex]zm = M (2y)[/tex]

Thank you!

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