Question

A 2kg mass is attached to a spring with a spring constant of k=40N/m. The spring is displaced 1 meter in the positive x direction on a horizontal frictionless surface. The equilibrium length of the spring is 2 meter. At what position x does the system have the most kinetic energy? What is the total mechanical energy of the system? At what position x are the potential and kinetic energies equal? Thank you in advance!

Answer 1

The system has the most kinetic energy at the point of equilibrium, or at x=0.

The total mechanical energy of the system is

$$E=\frac12 k\Delta x^2=20\text{ J}.$$

Find the position when the potential and kinetic energies are equal:

$$E=E_P+E_K,\\ E_P=E_K,\\ E=E_P+E_P=2E_P,\\\space\\ E=2·\frac12k\Delta x^2,\\\space\\ \Delta x=\sqrt{E/k}=0.7\text{ m}.$$