he mass of the system is approximately 0.63 kg and the spring constant is approximately 44.51 N/m.
What is the mass m and spring constant k?
Given
Mass: 10 lb ≈ 4.54 kg
String stretch: 6 inches ≈ 0.1524 m
Velocity: 6 ft/s ≈ 1.83 m/s
According to Newton's second law, the sum of forces acting on the mass equals its mass times its acceleration:
m * a = k * x - 5v - mg
where a is the acceleration of the mass.
At the moment of release:
Displacement (x) = 0.0762 m (3 inches below equilibrium)
Velocity (v) = 1.83 m/s (upward)
Acceleration (a) is unknown
When the mass initially stretches the string and comes to rest at equilibrium (x = 0 and v = 0), the equation of motion simplifies to:
0 = k * 0 - 5 * 0 - mg
Therefore, k = mg = 4.54 kg * 9.81 m/s² ≈ 44.51 N/m
m * a = 44.51 N/m * 0.0762 m - 5 * 1.83 m/s - 4.54 kg * 9.81 m/s²
a ≈ -14.64 m/s² (downward)
4.54 kg * (-14.64 m/s²) = 44.51 N/m * 0.0762 m - 5 * v - 4.54 kg * 9.81 m/s²
v ≈ 2.96 m/s (upward)
4.54 kg = 44.51 N/m * 0.0762 m / (-14.64 m/s²) + 5 * 2.96 m/s
m ≈ 0.63 kg
Therefore, the mass of the system is approximately 0.63 kg and the spring constant is approximately 44.51 N/m.