To determine if the motion of the mass on a spring is underdamped, overdamped, or critically damped, we analyze the damping condition using the parameters provided.

Given:

• Mass, $m = 2$
• Damping coefficient, $b = 4$
• Spring constant, $k = 10$

The motion of the mass is described by a second-order differential equation of the form: $m \ddot{x} + b \dot{x} + k x = 0$

Step 1: Calculate the Damping Ratio

The damping ratio $\zeta$ is defined as: $\zeta = \frac{b}{2 \sqrt{m k}}$

Step 2: Substitute Values

Using the provided values: $\zeta = \frac{4}{2 \sqrt{2 \cdot 10}} = \frac{4}{2 \sqrt{20}} = \frac{4}{2 \cdot 4.472} \approx \frac{4}{8.944} \approx 0.447$

Step 3: Interpret the Damping Ratio

The criteria for damping conditions are:

• Underdamped if $\zeta < 1$
• Critically damped if $\zeta = 1$
• Overdamped if $\zeta > 1$

Since $\zeta \approx 0.447$, which is less than 1, the system is underdamped. This implies that the mass will oscillate with gradually decreasing amplitude due to the damping.

Would you like additional details on the derivation or further analysis?

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1. How would the motion change if the damping coefficient were increased?
2. What is the natural frequency of the system without damping?
3. How would we calculate the damped natural frequency for the underdamped system?
4. What happens to the damping ratio if the mass is doubled?
5. How does changing the spring constant affect the type of damping?

Tip: To reduce underdamped oscillations, increase the damping coefficient or the mass in the system.