# Restoring forces and oscillating systems

My book states, 'Restoring forces give the system it's potential energy.' And it also states, 'Inertia due to mass in mechanical system gives the system it's kinetic energy.' I don't get what is all this supposed to mean. This was all in regards to oscillating systems and I don't get how do these forces give these energies.

• "Inertia" isn't a force, and the restoring forces change both the KE and PE of the oscillating system. Maybe you should find a better book. – alephzero Feb 10 at 17:19

My book states, 'Restoring forces give the system it's potential energy.' And it also states, 'Inertia due to mass in mechanical system gives the system it's kinetic energy.' I don't get what is all this supposed to mean. This was all in regards to oscillating systems and I don't get how do these forces give these energies.

Ignore about the SHM problem first. If a body is acted upon by a single force from a spring, that body will pick up velocity (as it accelerates - Newton's second law). That is the kinetic energy it gains. If this were some elementary school problem, you would not have needed to know the potential energy connection to all of this.

What you must realise is that the high amount of compression in the spring (it does not like being compressed all that because it is not actually stable) and if you let the system go from rest, a force acts on the body causing it to gain kinetic energy (as work is done over the body by that spring). But by conservation of energy, we know that if the body has kinetic energy, it should have had a source. In this case, the only source is the spring itself. So, we say that the spring has potential energy that it converted into the kinetic energy of the body.

"Restoring forces", in SHM context, is this action done by the spring after the body starts moving. This is the force acting on a body in a direction and magnitude in order to get back to its natural length.

"Inertia due to mass in system gives the mass it's kinetic energy" is also true. If you don't have mass, you are not really talking about any kinetic energy (assuming the spring is massless). But remember that the same restoring force is responsible for the body's motion (from Newton's first law - things accelerate and change direction only because of a force acting).

Consider the following mass-spring system:

The bob has mass $$m$$ and the spring's Hookean spring constant is $$k$$. $$y$$ is the axis of motion.

A Hookean spring provides a restoring force:

$$F=-ky$$

where $$y$$ is the deformation of the spring.

By deforming the spring, mechanical work $$W$$ was performed on it:

$$W=-\int_0^yFdy=-\int_0^ykydy=-\frac12ky^2$$

According to the work-energy theorem, this is also the change in potential energy $$\Delta U$$ of the bob.

Now the bob has gained velocity $$v_y=\dot{y}$$, because the restoring force has provided acceleration ($$\ddot{y}$$), so it has obtained kinetic energy $$\Delta K$$:

$$\Delta K=\frac12 m\dot{y}^2$$

Apply conservation of energy:

$$\Delta K=\Delta U$$

So:

$$\frac12m\dot{y}^2=-\frac12ky^2$$

Take the derivative with respect to time, $$\frac{\mathrm{d}}{\mathrm{dt}}$$, of both sides and rework slightly to get:

$$m\ddot{y}+ky=0$$

This is the equation of motion of a harmonic oscillator and has the solution:

$$y(t)=A\cos(\omega t + \phi)$$

With:

$$\omega=\sqrt{\frac{k}{m}}$$

The constants $$A$$ and $$\phi$$ are determined from the initial conditions.