To find the mass that would produce critical damping, we can follow these steps using physics and mathematical equations.
1. Determine the spring constant (k): The spring's force can be determined using Hooke's Law, which is given by F = kx, where F is the force applied to the spring, k is the spring constant, and x is the displacement from the spring's equilibrium position.
Given that a force of 5 N is needed to hold the spring 1 meter beyond its natural length, we can solve for k:
F = kx
5 N = k * 1 m
5 N/m = k
So the spring constant k is 5 N/m.
2. Use the critical damping condition: Critical damping occurs when the damping coefficient (b) is equal to the critical damping coefficient (bc), which is given by the formula:
bc = 2 * sqrt(m * k)
Where bc is the critical damping coefficient, m is the mass, and k is the spring constant.
Given that the damping constant b is 8 kg/s, for critical damping, b should be equal to bc:
b = bc
8 kg/s = 2 * sqrt(m * k)
3. Solve for the mass (m): Now we can rearrange the above equation to solve for m by squaring both sides and then dividing by 4k:
(8 kg/s)^2 = (2 * sqrt(m * 5 N/m))^2
64 kg^2/s^2 = 4 * m * 5 N/m
64 kg^2/s^2 = 20 m N
64 kg^2/s^2 = 20 m kg/s^2 (since N = kg*m/s^2)
(64/20) kg = m
3.2 kg = m
Thus, a mass of 3.2 kg will produce critical damping when the spring is stretched 2 meters beyond its natural length and then released with zero velocity.