For the free vibrations of a mass on a spring, we have m=2,  b=4 and  k=10. Is the motion of the mass underdamped, overdamped, or critically damped

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?

---

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.

For the free vibrations of a mass on a spring, we have m=2, b=4 and k=10. Is the motion of the mass underdamped, overdamped, or critically damped?

Analyze the damping condition of a mass-spring system with m=2, b=4, and k=10 to determine if the motion is underdamped, overdamped, or critically damped. Learn step-by-step how to calculate the damping ratio and interpret the results. Suitable for college-level physics or engineering studies.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation