Question

in lecture 18 at 32 min - Prof. Lewin uses the conservation of mechanical energy at the spring. He writes "1/2 m vb^2 - 1/2 k xb^2 = 1/2 k xmax^2". Why is on the left side the potential energy from the spring negative but on the right side positive? Should not both energies be positive?

Answer 1

Yes. At equilibrium the displacement from the equilibrium position is 0, so we know all energy is kinetic, which he writes as\[1/2mv _{A}^{2}\]and at its fullest extension the mass is stopped, so all the energy is potential energy which he writes as \[1/2 k x _{\max}^{2}\]. In between these positions, the energy is part potential and part kinetic, but they still add up to the total energy, so he should have written\[1/2mv _{A}^{2} = 1/2mv _{B}^{2} + 1/2kx _{B}^{2} = 1/2kx _{\max}^{2}\]

Answer 2

Otherwise the energy at position A would also be greater than the energy at position B because the kinetic energy at A must be greater than at B so you have to add energy otherwise the conservation of energy would be broken.