Final answer:
To find the spring constant of the system, use the formula k = 2E/A². With the given energy E = 38.61 joules and the maximum displacement A = 0.284 meters, the spring constant is calculated to be approximately 957.74 N/m.
Step-by-step explanation:
To determine the spring constant for a horizontal block-spring system with a given mechanical energy and maximum displacement from equilibrium, we can use the relationship between the total mechanical energy (E), spring constant (k), and amplitude of motion (A).
The total energy (E) of the system is given by the formula: E = (1/2)kA², where A is the maximum displacement from equilibrium. We're given that E = 38.61 joules and A = 0.284 meters. Rearranging the formula to solve for the spring constant (k) gives us: k = 2E/A².
Plugging the values we have into this equation:
k = 2(38.61 joules) / (0.284 meters)²
k = 77.22 / 0.080656
k ≈ 957.74 N/m
Therefore, the spring constant of the system is approximately 957.74 N/m.