Understanding Critical Damping Coefficient Calculation

This solution explains how to determine the critical damping coefficient for a mechanical vibrating system. The critical damping coefficient ($c_c$) is the minimum value of damping required to prevent oscillations in a system after a disturbance.

Given System Parameters

We are provided with the following information about the vibrating system:

• Mass ($m$): 50 kg
• Spring Stiffness ($k$): 30 kN/m = $30 \times 10^3$ N/m

Calculating Natural Frequency ($\omega_n$)

Before calculating the critical damping coefficient, we first need to find the natural frequency ($\omega_n$) of the undamped system. The formula for natural frequency is:

$$ \omega_n = \sqrt{\frac{k}{m}} $$

Substituting the given values:

$$ \omega_n = \sqrt{\frac{30 \times 10^3 \, \text{N/m}}{50 \, \text{kg}}} $$

$$ \omega_n = \sqrt{600 \, \text{s}^{-2}} $$

$$ \omega_n \approx 24.495 \, \text{rad/s} $$

Calculating Critical Damping Coefficient ($c_c$)

The critical damping coefficient ($c_c$) is calculated using the formula that relates it to the mass ($m$) and the natural frequency ($\omega_n$):

$$ c_c = 2m\omega_n $$

Now, we plug in the values for mass and the calculated natural frequency:

$$ c_c = 2 \times (50 \, \text{kg}) \times (24.495 \, \text{rad/s}) $$

$$ c_c = 100 \, \text{kg} \times 24.495 \, \text{s}^{-1} $$

$$ c_c \approx 2449.5 \, \text{kg/s} $$

The unit kg/s is equivalent to N/(m/s) or Ns/m for the damping coefficient.

Conclusion

Based on the calculations, the critical damping coefficient for the given system is approximately 2449.5 N/m/s. This value is very close to option 3.

Comparison with Options

Let's compare our calculated value with the given options:

The calculated value closely matches option 3.