59. System Shown For the system shown in Fig. 16-25, the...

59. System Shown For the system shown in Fig. $16-25$, the block has a mass of $1.50 \mathrm{~kg}$ and the spring constant is $8.00 \mathrm{~N} / \mathrm{m}$. The damping force is given by $-b(d y / d t)$, where $b=230 \mathrm{~g} / \mathrm{s}$. Suppose that the block is initially pulled down a distance $12.0 \mathrm{~cm}$ and released. (a) Calculate the time required for the amplitude of the resulting oscillations to fall to one-third of its initial value. (b) How many oscillations are made by the block in this time?

Key Concepts

Damped Harmonic Oscillator

This concept involves a mass-spring system where the restoring force is opposed by a damping force proportional to the velocity. The system is described by a second?order differential equation that includes both the spring force and the damping force. Such systems exhibit oscillatory behavior where the amplitude of oscillation gradually decreases over time due to energy losses, and this behavior depends on the relative values of the mass, spring constant, and damping coefficient.

Exponential Decay of Amplitude

In underdamped harmonic motion, the oscillation amplitude decreases exponentially over time. The decay is characterized by an envelope function, A(t) = A0 exp(–?t), where ?, often expressed in terms of the system parameters (such as b/2m), determines how quickly the energy is dissipated. This concept is key for determining the time at which the oscillation amplitude reaches a specific fraction of its initial value.

Natural Frequency

The natural frequency of an undamped oscillator is determined by the mass and spring constant and is given by ?0 = ?(k/m). It represents the frequency at which the system would oscillate if there were no damping. This term is fundamental in understanding the inherent oscillatory behavior of the system and acts as a basis for comparing the effects of damping.

Damped Frequency

When damping is present, the frequency of the actual oscillations is slightly lower than the natural frequency. The damped angular frequency, ?' = ?(?0² – (b/(2m))²), reflects the system’s oscillatory response under the influence of damping. This concept is essential for calculating the period of the damped oscillations, which in turn is used to determine how many oscillations occur within a given time period.

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