write a differential equation in terms of displacement s to model the motion of an undamped...

write a differential equation in terms of displacement s to model the motion of an undamped vertical spring with mass m. recall that the weight should equal the spring's restoring force

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Expert Solution

Assume there is friction as the mass slides on the table. The simplest constitutive relation is to take the frictional force to be proportional to how fast the mass is moving, or its velocity x' .

Thus F_d = −γx' , (damping force)

where γ > 0 is the damping coefficient (mass per time).

If the mass is moving to the right, or x' > 0, the damping retards the motion and F_d < 0.

Therefore, the total external force is the sum

F = F_s + F_d = −kx − γx' .

The equation of motion is

mx'' = −γx' − kx. (damped oscillator)

This equation is called the damped oscillator equation. Both forces have negative signs because each opposes positive (to the right) motion. For this case we expect an oscillatory solution with a decreasing amplitude during each

oscillation because of the presence of friction.

One such a solution, a damped oscillation, takes the form of

x(t) = Ae^−λt cos ωt,

where A is some amplitude, λ is a decay rate, and ω is the frequency.

genius_generous answered 1 year ago

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