Hello all,

I've been trying to understand SHM, especially with a spring. Suppose we exert a force and pull the spring by a displacement x. The spring then takes t seconds (let's say 4 secs) to return to equilibrium. However at equilibrium, although the restoring force -kx is 0, the object moves past the equilibrium point making x negative, and oscillates. My source of information says that the momentum of the spring is non-zero at equilibrium point and thus continues past.

This makes physical sense to me but when I try to apply mathematics I'm not sure of what to do. I've tried to analyse it like this:

t=0 when the spring is displaced by x. The spring takes four seconds to return to equilibrium. This is when the restoring force kicks in.

At t = 0, the velocity of the mass m is 0, and so is the momentum p.

dp/dt = F the force, so p = the integral from 0 to t (Sorry about this. Still trying to figure out how to insert math symbols) Fdt, where 0 is when t = 0, so at and t is the time we are interested in. The whole is the interval over which we apply the force. therefore ∆p = Ft, the impulse momentum theorem.

However, we have F = -kx, so ∆p = -kxt. This gives p = -kxt + u, where u is the initial momentum and p is the actual momentum.

Using this, we can say that at t = 3, p = -3kx[t], where x[t] is the position at t = 3. The initial momentum would be zero since at t= 0, the momentum was 0.

However at x = 0, which is the equilibrium point at t = 4, applying the above formula gives p = 0. This seems impossible, so there must be something wrong with the approach, or the calculations.

Could anyone please tell me what went wrong and what would be a correct approach to analyse SHM in terms of momentum?

Thanks