In the device shown in the Fig. 8.15 the bumper of mass \(M=26.2 \mathrm{~g}\) is subject, in addition to the action of the spring of spring constant \(k=3.40 \mathrm{~N} / \mathrm{m}\), also to a braking frictional force with magnitude \(F_{A}=-b V\), which is proportionally opposed to the speed of \(M\) with constant \(b=6.0 \times 10^{-3} \mathrm{~kg} / \mathrm{s}\). If the initial speed of the bumper is \(V_{0}=1.73 \mathrm{~m} / \mathrm{s}\) :

1. show that the amplitude \(A\) of the oscillations decreases with time as with exponential attenuation \(A(t)=A_{0} \exp (-t / \tau)\)

2. determine \(\tau\) and draw the function \(A(t)\);

3. determine after how much time the amplitude decreases by \(90 \%\) from the maximum.

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h 18.0 g