Does the spring force do work on the spring itself?

[etotheipi]

Consider a spring with one end attached to a wall and the other to a free mass, which is then stretched so some potential energy U. After it has been released and has de-stretched, the change of elastic potential energy is -U which equates to the negative of the work done by the spring force on the system.

Here's my question: does the "system" in this case include the spring? Evidently work is done on the mass attached to the free end (which feels the full spring force -kx), but it is not so clear as to whether work is done by the spring on the spring itself.

For instance, if we were to split the spring into lots of tiny segments of width $dx$, the forces acting left and right on each segment should be equal in magnitude (tension is constant throughout the spring, assuming it is horizontal?) so no net work would be done on any piece of the spring.

However, it seems like common sense that if the spring is massive, some of the EPE will be converted into KE of the spring. If instead the spring is massless, though, it sort of makes sense that the only work done by the spring force is on the free mass.

 

[PeroK]

Consider a spring with one end attached to a wall and the other to a free mass, which is then stretched so some potential energy U. After it has been released and has de-stretched, the change of elastic potential energy is -U which equates to the negative of the work done by the spring force on the system.

Here's my question: does the "system" in this case include the spring? Evidently work is done on the mass attached to the free end (which feels the full spring force -kx), but it is not so clear as to whether work is done by the spring on the spring itself.

For instance, if we were to split the spring into lots of tiny segments of width $dx$, the forces acting left and right on each segment should be equal in magnitude (tension is constant throughout the spring, assuming it is horizontal?) so no net work would be done on any piece of the spring.

However, it seems like common sense that if the spring is massive, some of the EPE will be converted into KE of the spring. If instead the spring is massless, though, it sort of makes sense that the only work done by the spring force is on the free mass.

First, some of the PE must go into the movement of the spring itself. Normally this is simply neglected, as I think you understand.

Different parts of the sping accelerate at different rates during expansion and contraction and the tension in the string cannot be completely uniform.

 

[etotheipi]

First, some of the PE must go into the movement of the spring itself. Normally this is simply neglected, as I think you understand.

Different parts of the sping accelerate at different rates during expansion and contraction and the tension in the string cannot be completely uniform.

I'm slightly confused about one thing. When stretching the spring initially, we calculate the work done by the spring force as the integral of the spring force acting on the one object at the end (with no mention of the spring force acting on any parts of the spring), and this yields the increase in potential energy.

When it unstretches - sort of by symmetry - the decrease in potential energy must equal the work done by the spring force on the object at the end only.

When deriving the potential energy we don't seem to consider the work done by the spring force on each part within the spring whilst it is being stretched, so I don't see why we need to account for work done on individual bits of the spring during unstretching - that is, why should any PE go into the movement of the spring?

 

[Dale]

why should any PE go into the movement of the spring?

If it has mass and is moving then it has kinetic energy. Where else would that KE come from?

 

[etotheipi]

If it has mass and is moving then it has kinetic energy. Where else would that KE come from?

This is the confusing bit. When we derive the potential energy we think of the spring force as acting only at the ends of the spring.

For the spring to gain KE, the spring force must somehow do work on the "middle" of the spring - which is difficult for me to picture.

 

[Dale]

For the spring to gain KE, the spring force must somehow do work on the "middle" of the spring - which is difficult for me to picture.

As the spring accelerates the force available to accelerate a load is reduced. So it does less external work and instead converts some of its PE into its own KE

 

[etotheipi]

As the spring accelerates the force available to accelerate a load is reduced. So it does less external work and instead converts some of its PE into its own KE

That makes sense. So when deriving the equation for EPE we just consider the case where the spring is extending with no acceleration, so that all the tensions within the spring cancel and the only work done by the spring force is that on the free mass.

Whilst during rapid de-extension the tension within the spring decreases from left to right (since each subsequent chunk of spring needs to accelerate less mass), so the resultant work done on the mass is slightly lower?

 

[sophiecentaur]

This is the confusing bit. When we derive the potential energy we think of the spring force as acting only at the ends of the spring.

That's quite justifiable and it doesn't matter that it doesn't tell the whole story about the Forces and Energy within the spring. If the inner workings are of interest, think in terms of two springs connected in series and in this situation the spring near the anchor point would be exerting a force on the outer spring which would not be the same magnitude as the applied force when there is any acceleration involved (the outer spring would need a force ma to provide the acceleration a.

This can be taken further by considering elemental sections all along the original spring. The 'stationary' loop, right next to the anchor point would have zero KE and the furthest loop would have the maximum KE at any stage. The PE per unit length would be the same and the KE per unit length would vary over the whole spring so work done would vary over the spring length.

 

[Delta2]

Suppose we have a spring (that has mass M) attached to a body of mass m and compressed by $x_0$. Can someone propose a model and do calculations for how the initial energy of $\frac{1}{2}kx_0^2$ will split between kinetic energy of the mass of spring M and kinetic energy of the mass m?

 

[darth boozer]

Think about how a spring behaves without any extra mass attached, i.e. only consider the mass of the spring itself.