To tackle your question, let's break it down into manageable parts. We're looking at a damped harmonic oscillator, where a mass attached to a spring is subject to a restoring force and a damping force due to friction. The key parameters you've given us are the mass (m = 1.52 kg), the spring constant (k = 8.13 N/m), and the damping constant (b = 227 g/s, which we need to convert to kg/s for consistency). The initial displacement is 12.5 cm (or 0.125 m).

Calculating the Time Interval for Amplitude Decay

First, we need to understand how the amplitude of a damped oscillator decreases over time. The amplitude A(t) of a damped oscillator can be expressed as:

A(t) = A₀ * e^(-bt/2m)

Where:

• A₀ is the initial amplitude (0.125 m).
• b is the damping coefficient (227 g/s or 0.227 kg/s).
• m is the mass (1.52 kg).
• t is time in seconds.

We want to find the time t when the amplitude A(t) is one-third of the initial amplitude A₀. Thus, we set up the equation:

A(t) = (1/3) * A₀

Substituting the values:

(1/3) * 0.125 = 0.125 * e^(-0.227t/(2 * 1.52))

We can simplify this to:

(1/3) = e^(-0.227t/(3.04))

Now, taking the natural logarithm of both sides:

ln(1/3) = -0.227t/(3.04)

Next, we solve for t:

t = -3.04 * ln(1/3) / 0.227

Calculating the right-hand side gives:

t ≈ -3.04 * (-1.0986) / 0.227

t ≈ 13.06 seconds

Determining the Number of Oscillations

Next, we need to find out how many oscillations occur in this time interval. The angular frequency ω of a damped oscillator is given by:

ω = sqrt(k/m) * sqrt(1 - (b/(2m))²)

First, let's calculate the undamped angular frequency:

ω₀ = sqrt(8.13/1.52) ≈ 2.31 rad/s

Now, we find the damping ratio:

α = b/(2m) = 0.227/(2 * 1.52) ≈ 0.0745 s⁻¹

The damped angular frequency is then:

ω_d = sqrt(ω₀² - α²) ≈ sqrt(2.31² - 0.0745²) ≈ 2.31 rad/s

Now, we can calculate the number of oscillations in the time interval we found:

Number of oscillations = t * (ω_d / 2π)

Substituting our values:

Number of oscillations = 13.06 * (2.31 / (2 * π)) ≈ 13.06 * 0.367 ≈ 4.80 oscillations

Final Summary

To summarize, the time interval required for the amplitude to fall to one-third of its initial value is approximately 13.06 seconds, and during this time, the block makes about 4.80 oscillations. This understanding of damped harmonic motion illustrates how energy is dissipated in oscillatory systems, leading to gradual reductions in amplitude over time.