for a damped harmonic oscillator of mass 5kg supported by a spring of constant 5N/m if the natural frequency is 2/√3 times the damped oscillation frequency then find out s and r

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).

---

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.

Solve for the damping constant and damping factor in a damped harmonic oscillator with a mass of 5kg and spring constant of 5N/m. Given the natural frequency is 2/√3 times the damped frequency, this solution breaks down the steps using key formulas like ω₀ = √(k/m) and γ = b/(2m). Ideal for advanced high school and undergraduate physics students.

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