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Elastic potential energy is the energy stored within a spring when it is compressed or stretched. In this particular exercise, when the cat attached to the spring reaches the highest point of its motion, the spring is at its natural length, which means there is no stretch. Thus, the elastic potential energy is zero at this point. However, when the cat is at the lowest point of its oscillation, the spring is fully stretched to the extent of the amplitude, creating the maximum elastic potential energy. The formula for elastic potential energy is given by \[ U_e = \frac{1}{2} k x^2 \] where \( k \) is the spring constant, and \( x \) is the displacement (0.050 m in this case). At this lowest point, we calculate it to be 1.96 J when \( k \) is found to be 1568 N/m.

Kinetic energy refers to the energy an object possesses due to its motion. In the scenario of the oscillating cat, the kinetic energy is zero at both the highest and the lowest points of the motion. That is because at these points, the cat momentarily comes to rest before reversing its direction. At the equilibrium position, however, all the system's energy is in the form of kinetic energy because the net forces are balanced, and the cat is moving at its maximum speed. In this scenario, \[ KE = 1.96 \, \text{J} \] This ensures energy is conserved as the potential energies at equilibrium are zero.

Gravitational potential energy is the energy an object possesses because of its position in a gravitational field, usually Earth's. For the cat-spring system, this energy is maximum when the cat is at its highest point above the lowest position. The formula used to determine gravitational potential energy is: \[ U_g = mgh \] where \( m = 4.00 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \), and \( h = 0.050 \, \text{m} \). This calculation results in a gravitational potential energy of 1.96 J at the highest point. At the lowest point, the gravitational potential energy is zero because this position is considered the reference level.

Energy conservation is a crucial principle in physics, asserting that the total energy in an isolated system remains constant. In the context of the thrill-seeking cat, the sum of elastic potential energy, kinetic energy, and gravitational potential energy equals 1.96 J at any point in the motion. Hence, even as energy shifts between different types, the total remains the same: - At the highest point: all energy as gravitational - At the equilibrium position: all energy as kinetic - At the lowest point: all energy as elastic potential This consistency in total energy highlights energy conservation in Simple Harmonic Motion.

The spring constant \( k \) is a measure of the stiffness of the spring. A higher spring constant indicates a stiffer spring that requires more force to stretch or compress it by a certain distance. In the case at hand, we use the condition of energy conservation to determine the spring constant of 1568 N/m, ensuring that the total energy remains 1.96 J. By using the formula: \[ k = \frac{2 \times 1.96}{(0.050)^2} \] This calculated value aids in predicting how the spring will store and release energy throughout the motion of the cat, ultimately allowing us to better understand the dynamics of the SHM system.