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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....

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4 Answers 4

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"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.

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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:

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

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Thanks. So based on your explanation, Hooke's law basically applies to the force bringing the object back to equilibrium but not to the total force. So can I say that for an elastic cord, the restoring force is tension. What about the spring? Can I say that the restoring force for spring is due to the compression or stretch of the spring so the potential energy change, which leading to the restoring force? –  user1285419 Feb 24 '13 at 5:55
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That's right. The tension on an elastic cord is a restoring force. Similarly, the restoring force for a spring is given by Hooke's law, whether it is stretched or compressed, because it always acts to bring the system back into equilibrium. –  elfmotat Feb 24 '13 at 6:00
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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.

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Thanks David, let me modify my question in this way. Use davecoulter's example, if I initially compress the spring and then left it goes, we know that the energy store in the spring will bring the spring back to it original position, let says the Hooke's law holds so $F=k\Delta x$, based on your explanation, F is the restoring force. But what happen if while the sping is being restored natually, I add additional force $F_1$ along the same direction of $F$ by some way, so in this sense, will the restoring force changed to $F+F_1 = k\Delta x$? –  user1285419 Feb 24 '13 at 6:05
    
What confusing me is my instructor told us that k is the property of the system (cord, band or spring) so it gives me a sense that restoring force is coming from the system but not from external environment. So in my above example, it looks to me that even we add the additional force $F_1$, the restoring force is still $F$ not $F+F_1$. –  user1285419 Feb 24 '13 at 6:11
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Different people may define "restoring force" in different ways, so perhaps what we're saying here doesn't apply in your class. But consider this: the restoring force should be zero at the equilibrium point, right? If that's the case, then you have to include both forces, $F + F_1$, to get a force that is zero at the equilibrium point. A constant external force $F_1$ will change the equilibrium to a point where $F$ by itself is not zero. That implies that $F + F_1$ is the restoring force. It's not necessarily the case that forces from the external environment aren't included. –  David Z Feb 24 '13 at 6:16
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restoring force is refer to the system which bring force back to it normal position with out the effect of distance which the force existed to it Constance force. F1+F2= total force. F= kx

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