Chapter 4.4, Problem 22P

Referring to Problem $21$, find the logarithmic decrement of the system in Problem $10$.

Logarithmic Decrement

For the damped oscillation described by Eq. $\left(23\right)$, show that the time between successive maxima is $T_d=2\pi /v$.

Show that the ratio of the displacements at two successive maxima is given by $\exp \left(\gamma T_d/2m\right)$. Observe that this ratio does not depend on which pair of maxima is chosen. The natural logarithm of this ratio is called the logarithmic decrement and is denoted by $\Delta$.

Show that $\Delta =\pi \gamma /mv$. Since $m,v,$ and $\Delta$ are quantities that can be measured easily for a mechanical system, this result provides a convenient of the system, which is more difficult to measure directly. In particular, for the motion of a vibrating mass in a viscous of the fluid. For simple geometric shapes, the form of this dependence is known, and the preceding relation allows the experimental determination of the viscosity. This is one of the most accurate ways of determining the viscosity of a gas at high pressure.

A mass weighing $16$ lb stretches a spring $3\text{in}$. The mass is attached to a viscous damper with a damping constant of $2\text{lb}\cdot \text{s/ft}$. If the mass is set in motion from its equilibrium position with a downward velocity of $3\text{in}\text{./s}$, find its position $y$ at any time $t$. Plot $y$ versus $t$. Determine when the mass first returns to its equilibrium position. Also find the time $\tau$ such that $\left|y\left(t\right)\right|<0.01$ in for all $t>\tau$.