Final answer:
The total energy of a 3 kg object attached to a spring oscillating with a 4 cm amplitude and a 2 s period is calculated using the formulas for simple harmonic motion, converting the given amplitude to meters, calculating the spring constant from the period and mass, and finally using these values to find the energy.
Step-by-step explanation:
To calculate the total energy of a 3 kg object attached to a spring oscillating with an amplitude of 4 cm and a period of 2 s, we need to understand the dynamics of simple harmonic motion (SHM). In SHM, the total energy (E) of the system is the sum of its potential and kinetic energy, which remains constant if we ignore any damping forces such as friction. Since the object is oscillating, its maximum potential energy occurs at the maximum displacement (amplitude), and its maximum kinetic energy occurs at the equilibrium position.
The total energy of the object in SHM can be given by the formula E = 1/2 kA², where k is the spring constant and A is the amplitude. But we are not given the spring constant directly. Instead, we can find k using the formula for the period of the oscillation (T), which is T = 2π√(m/k) where m is the mass of the object. Rearranging the formula gives us k = 4π²m/T². Using the given values of mass m = 3 kg and period T = 2 s, we find k = 4π²(3 kg)/(2 s)².
Once we have k, we can calculate the total energy. The amplitude A needs to be in meters, so we convert 4 cm to 0.04 m. Inserting the values into the energy formula, we calculate the total energy E.