Is restoring force a particular type of force?

Question

I have a question about the restoring force in elastic band or rope which confusing me for a long time.

As I was told in high school physics, for an elastic band (or spring), if Hooke's law holds, we have $F = k\Delta x$. What's confusing me is: should F be the total force acting on the object or the restoring force only? Or I ask this way: is there anything called "restoring force" existing independently, just like gravity, friction or tension?

To my understanding, restoring force should be the total force which is pointing to the equilibrium point. For example, if we consider a bungee cord, we should always count the tension of the cord as well as the gravity so the restoring force at any time should be the total force of tension and gravity; hence, when we apply Hooke's law, we should always have $F$ being the total force, not just the tension. Is that correct?

This is pretty confusing to me because there use many terms in the book. Sometimes they said it is the tension in Hooke's law, sometimes they say the restoring force and sometimes the total force....

Answer 1

"Restoring" forces refer primarily to forces that try to return a system to equilibrium. So a spring has a restoring force of $F = -k\Delta x$. This means that if you choose the origin as being $x = 0$, then compressing the spring would correspond to a negative $x$ (displacing the spring to the left), and stretching the spring would correspond to a positive $x$ (stretching the spring to the right). In that sense, by extending the spring, a positive $\Delta x$ creates a negative force ($-1 \times \Delta x$) that acts to restore the spring to equilibrium (pulling back on the spring extension) and by compressing the spring, you would have a negative $\Delta x$ ($-1 \times \Delta x$), which creates a positive force that restores the spring equilibrium.

So Hooke's Law is actually $F=-k \Delta x$

Hope that helps.

Answer 2

The restoring force is defined as a force which acts to bring the system back into equilibrium. Total force is just that: total force. For example, if a mass is hung vertically on a spring, then the restoring force is given by Hooke's law: $F_{rest}=kx$, acting upwards. The gravitational force is $F_{grav}=-mg$, acting downwards. The total force is:

$$F_{tot}=F_{rest}+F_{grav}=kx-mg$$

Answer 3

Good question! What you probably haven't been told is that forces can sometimes go by multiple names. There can be a name that describes what produces the force, but also a name that describes how the force is acting, or something else. The term "restoring force" falls in this latter category: it's a name that describes which way the force points, i.e. toward the equilibrium point. The same force might also have another name which describes what produces it. For example, it's quite possible that the tension of a string, or the elastic force of a spring, or gravity, is the restoring force.

In fact, it's possible that multiple forces contribute to the restoring force. "Restoring force" just means the total force that acts toward the equilibrium position. In the case of the bungee cord, the restoring force is the sum of gravity and the elastic force of the cord.

Another case where you might have seen the same thing is with the term "centripetal force," which refers to whatever forces are pointing toward the center of an object's circular motion. Sometimes gravity is the centripetal force, sometimes it's tension, sometimes it's a normal force, etc., or even a combination of different forces.