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A mass m is connected to a fixed wall by a spring of stiffness k and is also connected to a viscous damper (damping coefficient c). The damper is attached between the mass and a moving base whose displacement is y(t). The displacement of the mass is x(t). Write the correct equation of motion for the mass.

1. m*x_ddot + c*x_dot + k*(x - y) = 0
2. m*(x_ddot - y_ddot) + c*(x_dot - y_dot) + k*x = 0
3. m*x_ddot + c*(x_dot - y_dot) + k*x = 0
4. m*(x_ddot - y_ddot) + c*(x_dot - y_dot) + k*(x - y) = 0

Correct answer: m*x_ddot + c*(x_dot - y_dot) + k*x = 0

Solution

The spring is fixed to the wall (ground), so its restoring force depends on absolute displacement x. The damper connects the mass to the moving base, so its force depends on the relative velocity (x_dot - y_dot). Newton's second law gives m*x_ddot = -c*(x_dot - y_dot) - k*x.

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