Final answer:
To find the mass's speed halfway to equilibrium, we use the conservation of mechanical energy. The sum of the potential energy in the spring and the kinetic energy of the mass remains constant. By equating these for initial and halfway positions, the mass's speed can be calculated.
Step-by-step explanation:
The student's question involves calculating the speed of a mass attached to a spring when it is halfway to its equilibrium position, having been released from a displaced position with an initial speed. To find this speed, the conservation of mechanical energy principle is used, as there are no non-conservative forces like friction to consider in this scenario. The total mechanical energy of the system (kinetic energy plus potential energy) remains constant.
Since the mass is attached to a spring, it executes Simple Harmonic Motion (SHM). The potential energy stored in the spring when the mass is displaced can be found using the spring potential energy formula PEspring = (1/2)kx2, where k is the spring constant and x is the displacement from equilibrium. The kinetic energy (KE) of the mass is given by KE = (1/2)mv2, where m is the mass and v is the velocity.
Using energy conservation, the total mechanical energy at the initial point is equal to the total mechanical energy at the halfway point. Therefore, (1/2)kxi2 + (1/2)mvi2 = (1/2)kxf2 + (1/2)mvf2. By substituting the known values and solving for vf, the final velocity at the halfway point can be determined.
For the given problem with an initial displacement of 8.0 cm (0.08 meters), spring constant of 60 N/m, mass of 2.0 kg, and initial speed of 0.22 m/s, we would use the above approach to calculate the mass's speed halfway to equilibrium. The specific calculation would not be detailed within the constraints of this word count.