- #1

Dazed&Confused

- 191

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##m## that had a pair of horizontal identical springs of spring constant ##k## attached on either side and that the mass is free to move horizontally. The mass is also placed on a table that gives rise to an additional frictional force; the coefficient of sliding friction between the mass and the table is ## \mu ##.

You have to 'verify' that the following is a solution of the equation of motion:

$$ x_B(t) = B \sin( \omega_B t) + C( \cos( \omega_B t) - 1).$$

In my opinion this is wrong. The solution clearly should decay in time. It seems that the differential equation they wanted was of the form

$$ x''(t) = -2kx(t) - \mu mg.$$

I think it should be

$$ x''(t) = -2kx(t) - \mu mg \text{Sign}(x'(t)),$$

where ##\text{Sign}(x)## is defined to be 1 if ##x > 0## and -1 if ## x < 0. ##

Is there something wrong with my thinking? Keep in mind that Mathematica's numerical solution does show a particle approximately exhibiting SHM motion, but also decaying with time, however it does not ultimately stop moving at the equilibrium position.