Force & Potential Energy Practice Problems

The binding energy of the single electron in He ⁺ is 24.6 eV and the mass of He ⁺ is 4.001506. Calculate the percentage decrease in mass when a He²⁺ ion gains an electron to form the He ⁺ ion. Express the result in amu and as a percentage of He ²⁺ mass.

Imagine a ball rolling along a horizontal track under the influence of potential energy given by U(x) = Ax² + Bcos((π/2L)x), where A, B, and L are constants. Determine the magnitude and direction of the force acting on the ball when it reaches (i) the starting point (x = 0), (ii) the midpoint of the track (x = L/2), and (iii) the end of the track (x = L).

Imagine a spaceship cruising along a space-time continuum, traveling on the x-axis. As it moves through space, it experiences a potential energy U = x⁵ - x³ J, where x represents the spaceship's position in meters. Determine if the spaceship is in stable or unstable equilibrium at each of its equilibrium points.