# What's the actuating force on a spring?

I'm trying to solve the following problem:
a) Consider a spring (of constant $$K$$) fixed on one end, $$F_1$$ is applied on the other such that $$x=x_0\ cos(\omega t)$$. What does $$F_1$$ have to be?
b) A force $$F_2$$ is applied to a body of mass $$m$$ so that its motion is as described by the spring in (a). What does $$F_2$$ have to be?
c) The body is attached to the spring and a force $$F_3$$ acts on the system so that $$x=x_0\ cos(\omega t)$$. What's $$F_3$$?
d) If $$\omega _0 = \sqrt{\frac{K}{m}}$$, What is $$F_3$$ to maintain that oscillation?

In (a) I'm not sure how to write the sum of forces since it's not clear whether the spring is massless or not. I assume it is and so you end up having $$0=F_1-Kx$$ This way of presenting it implies the force is always acting, but I think the spring should be able to oscillate without an external acting force.
I also thought of solving it with energy, but I cannot compute the work done by the force since I don't know for how much distance it is applied.

In (b) I think it's clear that the force is always acting and turns out to be the force the spring would exert on the body. $$F_2 = m \frac{d^2x}{dt^2}$$ $$F_2 = m [-\omega ^2 x_0\ cos(\omega t)]$$
In (c) I find the same problem that in a, How to solve it without knowing for how long or for how much distance the force is applied. I know the energy required is $$E_k=\frac{1}{2}Kx_0^2$$

In (d) I don't understand why you would need an external force $$F_3$$ acting on the system to keep the oscillatory motion since there are no non-conservative forces acting in the system.

(a) The spring on its own will not oscillate without an external force being applied because it has zero mass and hence zero inertia. For the same reason, the net force acting on the spring on its own must always be zero. So you can find $$F_1$$ as a function of time $$t$$ from $$F_1=Kx$$.

(c) There are two forces acting on the mass $$m$$. One is the applied force $$F_3$$. The other the force is the restoring force from the spring, which we know is $$-Kx$$. So we have

$$\displaystyle F_3 - Kx = m \frac {d^2x}{dt^2}$$

Since you are given $$x$$ as a function of $$t$$, you can find $$F_3$$ as a function of $$t$$.

(d) You don't need an external force to produce oscillations - the system will oscillate on its own at it natural frequency if it displaced from its equilibrium position. However, if an external force (and, in particular, a sinusoidal external force) is applied then you get driven or forced oscillations.

It seems to me you are dealing with a damped oscillator because that is where you need an external force of the form $$\mathbf{F = F_0 sin(\omega t)}$$ for the system to not reach exctinction of the motion.

Here the blue is the system of the mass and spring after it was excited but it is a viscous medium and it is governed by an equation of the form $$m\frac{d^2x}{dt^2} - Kx - b\frac{dx}{dt}=0$$, where the term involving b is the one introduced by the viscous medium. Then the external force it's applied and it reaches a steady motion after some time as shown in by the purple graph, which is governed by the following equation $$m\frac{d^2x}{dt^2} - Kx - b\frac{dx}{dt}+ F_0sin(\omega t)=0$$, which has a solution of the form $$y(t) = A sin(wt - \phi)$$, where the amplitude has the form $$A = \frac{F_0}{\sqrt{m(\omega^2 - \omega^2_0)^2 + (b\omega)^2}}$$, where $$\omega_0$$ is the natural frequency of the system or in other words the frequency of the system before the external force was applied.