Final answer:
The maximum energy stored in a mass-spring system is equal to the potential energy at maximum compression or extension, calculated with U = 1/2kx². The total mechanical energy is constant, allowing us to find kinetic energy as K = (1/2)mv² at different points.
Step-by-step explanation:
Understanding Potential Energy in a Mass-Spring System
When analyzing the maximum energy stored in a mass-spring system, we must consider the total mechanical energy of the system. The potential energy (U) stored in the spring at maximum compression or extension is calculated using the formula U = 1/2kx², where 'k' is the spring constant and 'x' is the displacement from the spring's equilibrium position.
Example Calculations of Potential Energy
(a) For the 250 g carts connected by a spring with a 120 N/m spring constant that compresses from 5.0 cm to 2.0 cm, the potential energy is found as follows: U = 1/2 * 120 N/m * (0.03 m)² = 0.054 J.
(b) A 300 g block attached to a 100 N/m spring and compressed by 12 cm would have a potential energy of: U = 1/2 * 100 N/m * (0.12 m)² = 0.72 J at the point of release.
The kinetic energy (K) of the block at any point can be found by knowing that the total mechanical energy (E) is constant. Thus, at the equilibrium position, the potential energy is zero, and the kinetic energy is equal to the maximum total energy. We use the formula K = (1/2)mv² to solve for the velocity (v).