How do you calculate potential energy given a force that is dependent on time?

The restoring force of a spring is F(x) = -k(x-x0)exp(-t/T) where k and T are constants, x is the position of a mass on the spring, and x0 is the position of equilibrium of the spring.

How do you find the potential energy of the mass-spring system?

How do you determine the amount of energy lost per time?

$$exp(-t/T) {kx^2\over 2}$$
• If T is very big compared to the natural oscillation frequency, you are in the adiabatic regime. Here, the energy divided by the frequency, the action variable J, is constant. The frequency goes as the square root of k, so it's time dependence is $\omega(t) = \omega_0 exp(-t/2T)$. The energy is therefore $E=E_0 exp(-t/2T)$. You can perturb away from this for smaller T, to find the small change in the action variable J with time.
If your force is explicitly time dependent, then the power (work per second) is given with a time-dependent expression:$W=F(t)v(t)dt$. Integrating this expression over all time gives you the total work $E=\int F(t)v(t)dt$. It does not mean you can introduce the corresponding potential difference $\Delta V$.