Final answer:
The system is overdamped if the damping ratio (ζ) is greater than 1, and (ζ) can be calculated using ζ = {c} / {2 {mk}^1/2, where c is the damping coefficient, m is the mass, and k is the spring constant, with k = Force / Displacement and c determined from the damping force equation F_d = c . v, given F_d = 1 lb and v = 4 \ ft/sec.
Step-by-step explanation:
To analyze the damping in the system, we need to consider the damping ratio (ζ). The damping ratio is a dimensionless quantity that characterizes the level of damping in a system. For an overdamped system, the damping ratio is greater than 1.
The damping ratio (ζ) can be calculated using the following formula:
ζ = {c} / {2{mk}^1/2}
where:
- c is the damping coefficient,
- m is the mass of the system,
- k is the spring constant.
In this case, we are given that a mass of 24 lbs stretches a spring by 10 inches. We can use this information to find the spring constant (k). The force exerted by the spring F can be calculated using Hooke's Law:
F = kx
where:
- k is the spring constant,
- x is the displacement.
Given that the spring stretches by 10 inches (converted to feet for consistency), we have:
F = k . 10 inches
Also, we are told that the mass experiences a viscous resistance of 1 lb when the velocity is 4 ft/sec. The damping force F_d can be expressed as:
F_d = c . v
where:
- c is the damping coefficient,
- v is the velocity.
Given that F_d = 1 lb and v = 4 ft/sec, we can find the damping coefficient c.
Once we have k and c, we can use them to calculate the damping ratio (ζ) using the earlier formula. If \(ζ > 1\), the system is overdamped.