This question tests AP Physics C: Mechanics understanding of energy conservation in simple harmonic oscillators, focusing on the mathematical expression. In SHM without friction, total mechanical energy remains constant throughout oscillation, expressed as the sum of kinetic and potential energies equaling the maximum potential energy at amplitude. Choice A correctly states ½mv² + ½kx² = ½kA², showing that K + U equals the total energy E. Choice B incorrectly uses subtraction instead of addition. Choice C omits the ½ factors. Choice D rearranges incorrectly, suggesting potential energy equals total plus kinetic. Choice E incorrectly states total energy is zero. To help students: derive the energy conservation equation from first principles, practice identifying correct forms of conservation laws, and verify equations using dimensional analysis and limiting cases.

This question tests AP Physics C: Mechanics understanding of energy transformations in simple harmonic oscillators, specifically finding where kinetic and potential energies are equal. When K = U, each equals half the total energy since E = K + U. Setting U = E/2: ½kx² = ½(½kA²), which simplifies to x² = A²/2, giving |x| = A/√2 ≈ 0.707A. With A = 0.10 m, |x| = 0.10/√2 ≈ 0.0707 m ≈ 0.071 m. Choice B is correct because it matches this calculation. Choice C (0.050 m = A/2) is a common error from assuming linear rather than quadratic energy relationships. To help students: emphasize that energy varies as x², practice solving for positions where K and U have specific ratios, and use energy bar charts to visualize energy distribution at different positions.