First slide

Damped and forced oscillations

Question

A 21.2kg object oscillates at the end of a vertical spring that has a spring constant 20500 Nm^–1. The effect of air resistance is represented by the damping coefficient b = 2 Nsm^–1. Find the time interval that elapses while the energy of the system drops to 10% of its initial value (Given, ln 10 = 2.302)

Moderate

A

34.4s
B

12.2s
C

24.4s
D

8.9s

Solution

$\begin{array}{l} τ = \frac{m}{b} = \frac{21.2}{2} = 10.6 s \\ E = E_{0} e^{- t / τ} \\ \Rightarrow 0.1 = e^{- t / 10.6} \\ e^{t / 10.6} = 10 \\ \Rightarrow \frac{t}{10.6} = \ln 10 \\ t = (10.6) (2.302) = 24.4 s \end{array}$

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