To solve this problem, we need to analyze the system's behavior in terms of its oscillations and damping characteristics. We have a mass-spring system where a box with a mass of 900g is attached to a spring, and we need to find the force constant (k), relaxation time (τ), and the quality factor (Q) of the system. Let's break this down step by step.
Understanding the System
The system consists of a spring with a box attached to it. The box has a mass of 900g, and the spring has a mass of 100g. The total mass (m) of the system is the sum of these two masses:
- Mass of the box = 900g = 0.9 kg
- Mass of the spring = 100g = 0.1 kg
- Total mass (m) = 0.9 kg + 0.1 kg = 1.0 kg
Calculating the Force Constant (k)
The frequency of oscillation (f) is given as 4 vibrations per second. The relationship between frequency and the force constant in a mass-spring system is given by the formula:
f = (1/2π) * √(k/m)
Rearranging this formula to solve for k gives us:
k = (2πf)² * m
Substituting the values:
Now, calculating k:
k = (2π * 4)² * 1.0
k = (8π)² * 1.0 ≈ 79.58 N/m
Finding the Relaxation Time (τ)
The relaxation time (τ) is related to the damping of the system. It can be calculated using the formula:
τ = m / (b)
Where b is the damping coefficient. To find b, we can use the change in amplitude over time. The amplitude decreases from 2 cm to 1 cm in 15 seconds. The logarithmic decrement (δ) can be calculated as:
δ = (1/T) * ln(A1/A2)
Where A1 is the initial amplitude, A2 is the final amplitude, and T is the period of oscillation:
- A1 = 2 cm = 0.02 m
- A2 = 1 cm = 0.01 m
- T = 1/f = 1/4 = 0.25 s
Now, calculating δ:
δ = (1/0.25) * ln(0.02/0.01) = 4 * ln(2) ≈ 4 * 0.693 = 2.772
Next, we can relate δ to the damping coefficient b using:
δ = bT/(2m)
Rearranging gives us:
b = (2m * δ) / T
Substituting the values:
b = (2 * 1.0 * 2.772) / 0.25 ≈ 22.176 kg/s
Now we can find τ:
τ = m / b = 1.0 / 22.176 ≈ 0.045 kg·s/N
Calculating the Quality Factor (Q)
The quality factor (Q) is a measure of the damping of the oscillator and is given by:
Q = (ω₀ * m) / b
Where ω₀ is the angular frequency:
ω₀ = 2πf
Now substituting the values:
ω₀ = 2π * 4 ≈ 25.13 rad/s
Now we can calculate Q:
Q = (25.13 * 1.0) / 22.176 ≈ 1.13
Summary of Results
To summarize, we have calculated the following:
- Force constant (k): 79.58 N/m
- Relaxation time (τ): 0.045 kg·s/N
- Quality factor (Q): 1.13
This analysis provides a comprehensive understanding of the oscillatory behavior of the system and its damping characteristics. If you have any further questions or need clarification on any part of the process, feel free to ask!
