Hi guys, I'm stuck on a problem that states:

Two equal masses oscillate in the vertical direction. Show that the frequences of the normal modes of oscillation are given by:

[tex]\omega^2 = (3 +- \sqrt{5})\frac{s}{2m} [/tex]

and that in the slower mode the ratio of the amplitude of the upper mass to that of the lower mass is [tex] \frac{1}{2}(\sqrt{5}-1) [/tex] whilst in the faster mode this ratio is [tex] \frac{-1}{2}(\sqrt{5}+1) [/tex]. The figure in the problem is basically:

______

s

s

s

M

s

s

s

M

Where s is the spring and M are the masses (both springs have equal stiffness s).

Basically, I'm not sure if I have the equations of motion down correctly. So far, I have:

1) [tex]\frac{md^2y_{1}}{dt^2} = -sy_{1} + s(y_{2} - y_{1}) => y_{1}'' = -\omega_{0}^2y_{1} + \omega_{0}^2(y_{2} - y_{1}) [/tex]

2) [tex]\frac{md^2y_{2}}{dt^2} = -s(y_{2} - y_{1}) => y_{2}'' = -\omega_{0}^2(y_{2} - y_{1})[/tex]

Where [tex]y_{1}[/tex] and [tex]y_{2}[/tex] are the displacements of the first and second mass, respectively.

However when I use the solutions [tex] y_{1} = A_{1}\cos{wt}[/tex] and [tex]y_{2} = A_{2}\cos{wt} [/tex], find the derivatives, plug back in, etc, I cannot cleanly solve for the normal modes in terms of [tex]\omega[/tex]. I'm suspecting my equations of motion are incorrect, help?

Thanks

Two equal masses oscillate in the vertical direction. Show that the frequences of the normal modes of oscillation are given by:

[tex]\omega^2 = (3 +- \sqrt{5})\frac{s}{2m} [/tex]

and that in the slower mode the ratio of the amplitude of the upper mass to that of the lower mass is [tex] \frac{1}{2}(\sqrt{5}-1) [/tex] whilst in the faster mode this ratio is [tex] \frac{-1}{2}(\sqrt{5}+1) [/tex]. The figure in the problem is basically:

______

s

s

s

M

s

s

s

M

Where s is the spring and M are the masses (both springs have equal stiffness s).

Basically, I'm not sure if I have the equations of motion down correctly. So far, I have:

1) [tex]\frac{md^2y_{1}}{dt^2} = -sy_{1} + s(y_{2} - y_{1}) => y_{1}'' = -\omega_{0}^2y_{1} + \omega_{0}^2(y_{2} - y_{1}) [/tex]

2) [tex]\frac{md^2y_{2}}{dt^2} = -s(y_{2} - y_{1}) => y_{2}'' = -\omega_{0}^2(y_{2} - y_{1})[/tex]

Where [tex]y_{1}[/tex] and [tex]y_{2}[/tex] are the displacements of the first and second mass, respectively.

However when I use the solutions [tex] y_{1} = A_{1}\cos{wt}[/tex] and [tex]y_{2} = A_{2}\cos{wt} [/tex], find the derivatives, plug back in, etc, I cannot cleanly solve for the normal modes in terms of [tex]\omega[/tex]. I'm suspecting my equations of motion are incorrect, help?

Thanks

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