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Two masses and three springs*
Two identical masses M are hung between three identical springs. Each spring is massless and has spring constant k . The masses are connected as shown to a dashpot of negligible mass. Neglect gravity.
The dashpot exerts a force bv , where v is the relative velocity of its two ends. The force opposes the motion. Let x _{1 }and x _{2 }be the displacements of the two masses from equilibrium.
Find the equation of motion for each mass.
Show that the equations of motion can be solved in terms of the new dependent variables y _{1 }= x _{1 }+ x _{2 }and y _{2 }= x _{1 }− x _{2 }.
Show that if the masses are initially at rest and mass 1 is given an initial velocity v _{0 }, the motion of the masses after a suﬃ-
ciently long time is
Evaluate ω .
x _{1 }= x _{2}
_{= }v _{0}
2 ω
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