In the context of a damped harmonic oscillator, we are typically concerned with the relationship between the natural frequency $\omega_0$ (the frequency of the undamped oscillator) and the damped frequency $\omega_d$. The key equations for this are:

1. Natural frequency $\omega_0 = \sqrt{\frac{k}{m}}$, where:

   • $k$ is the spring constant,
   • $m$ is the mass of the oscillator.

2. Damped frequency $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$, where:

   • $\gamma = \frac{b}{2m}$ is the damping coefficient per mass, with $b$ being the damping constant.

From the problem, we are told: $\omega_0 = \frac{2}{\sqrt{3}} \cdot \omega_d$

Let's go step by step:

Step 1: Find the natural frequency $\omega_0$

We know: $\omega_0 = \sqrt{\frac{k}{m}} = \sqrt{\frac{5}{5}} = 1 \, \text{rad/s}$

Step 2: Express the damped frequency $\omega_d$

Using the relation: $\omega_0 = \frac{2}{\sqrt{3}} \cdot \omega_d$ Substitute $\omega_0 = 1$: $1 = \frac{2}{\sqrt{3}} \cdot \omega_d$ Solve for $\omega_d$: $\omega_d = \frac{\sqrt{3}}{2}$

Step 3: Use the relationship between $\omega_0$ and $\omega_d$

The damped frequency is related to the natural frequency by: $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$ Substitute the values of $\omega_d$ and $\omega_0$: $\frac{\sqrt{3}}{2} = \sqrt{1^2 - \gamma^2}$ Square both sides: $\frac{3}{4} = 1 - \gamma^2$ $\gamma^2 = 1 - \frac{3}{4} = \frac{1}{4}$ $\gamma = \frac{1}{2}$

Step 4: Find the damping constant $b$

Recall that $\gamma = \frac{b}{2m}$, so: $\frac{1}{2} = \frac{b}{2 \times 5}$ $b = 5$

Thus, the damping constant $b = 5 \, \text{kg/s}$.

Conclusion

• $s = b = 5 \, \text{kg/s}$ (the damping constant),
• $r = \gamma = \frac{1}{2} \, \text{s}^{-1}$ (the damping factor per mass).

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Would you like more details on any step? Here are some related questions to deepen your understanding:

1. What is the physical meaning of the natural frequency $\omega_0$?
2. How does damping affect the motion of a harmonic oscillator?
3. What is the difference between underdamped, critically damped, and overdamped systems?
4. How is the quality factor $Q$ related to damping in oscillatory systems?
5. Can the damping constant $b$ be experimentally measured?

Tip: Damped oscillators lose energy over time due to friction or resistance, which slows down the oscillation. Understanding how the damping factor $\gamma$ affects the frequency can help predict the system's behavior over time.