Final answer:
The mechanical energy of a mass-spring system with a 0.25 kg mass and a spring constant of 12.25 N/m that has dropped 80cm vertically can be calculated by adding the spring potential energy and gravitational potential energy.
Step-by-step explanation:
To calculate the mechanical energy of a mass-spring system we need to consider both the potential energy stored in the spring and the gravitational potential energy.
The potential energy stored in a compressed or stretched spring is given by the formula U = ½kx², where k is the spring constant and x is the displacement from the spring's equilibrium position. In this case, since the spring is held vertically and the mass has dropped, the displacement x would be equivalent to the distance the mass has fallen: 0.80 m.
Additionally, the gravitational potential energy, Ug = mgh, must be considered where m is the mass, g is the acceleration due to gravity (9.81 m/s²), and h is the height which, in this case, also corresponds to the displacement (0.80 m).
However, since the mass stops moving, its kinetic energy is zero, and the mechanical energy of the system will just be the sum of the potential energies. Now, we can calculate the total mechanical energy:
- Spring potential energy: U = ½(12.25 N/m)(0.80 m)²
- Gravitational potential energy: Ug = (0.25 kg)(9.81 m/s²)(0.80 m)
Adding these two values gives us the total mechanical energy of the system.