Solution

Problem

$3.5.23.$ Provide an example of a system in which the action of one of its parts on another is described by a force that varies harmonically with time.

Solution

As an example, let us consider a classical mechanical system: a horizontal mass-spring system attached to a fixed wall.

The system consists of three elements: a fixed wall (support), a spring with stiffness $k$, and a block of mass $m$.

Let us introduce the necessary physical idealizations:

1. The system is on a smooth horizontal surface (there is no friction).
2. Gravity is strictly compensated by the normal reaction force of the support and does not affect the motion along the axis of deformation.
3. The spring is ideal and massless, and its deformation strictly obeys Hooke's law.

If the system is displaced from its equilibrium position, the mass will begin to undergo free, undamped harmonic oscillations. In general form, the coordinate of the mass (displacement from the equilibrium position) will vary with time according to the following law:
$$x(t) = A \cos(\omega t + \varphi_0)$$
where $A$ is the amplitude of the oscillations, $\omega = \sqrt{\frac{k}{m}}$ is the angular frequency, and $\varphi_0$ is the initial phase, which depends on exactly how we started the system (it determines whether the graph initially starts as a sine, a cosine, or something in between).

Since the spring is massless, the force with which it acts on the wall (the action of one part of the system on another) at any given moment is equal to the elastic force arising in the spring. The projection of this force onto the axis of oscillation is:
$$F(t) = k \cdot x(t) = kA \cos(\omega t + \varphi_0)$$

As can be seen from the equation, the force $F(t)$ with which the spring acts on the wall varies harmonically with time.