Problem $5$

$($Adapted from Rowell $\backslash$& Wormley, System Dynamics$)$

The figure below is a schematic of an experimental apparatus used to determine the friction characteristics of various test materials. A mass is released with a known initial velocity and slides on the test sample. The velocity decay due to friction resistance is measured and used to determine the equivalent linear damping constant B of the friction between the mass and the test material.

a$.$ Derive the system state equation $($this is a first$-$order system so there is only one state equation$).$ Determine the system time constant, which will be a function of M and B$.($Tip: See the "DEQs $-$ Classical Method" lecture notes for an equivalent problem.$)$

b$.$ Solve the equation to find an expression for the velocity of the mass, assuming that it starts with an initial velocity of $\backslash ($ v$\_\{$o$\}\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ at time $\backslash (\backslash$mathrm$\{$t$\}=0\backslash ).$

c$.$ Assume that the mass is $10$ kg and that after a time of $5\mathrm{.}0$ seconds the velocity of the mass is at $\backslash (50\backslash \%\backslash )$ of its initial velocity. What is the value of $\backslash ($ B $\backslash ),$ the equivalent linear damping constant of the test material? Include the units for B in your answer.