Finding difficulty in understanding spring potential energy?

when we stretch a spring it stores a potential energy due to its elastic properties. if i assume a block connected to the spring (as potential energy is defined as external work done against internal conservative force) then at some displacement both block and spring are applying equal and opposite forces on each other which are conservative. so does this means both will store a potential energy and if yes how will both be represented and if no what i am missing in my concept.
It will be very helpful if u explain it both ways mathematically and with an example.
thank you

Newtonian mechanics is usually taught using idealized concepts like "point particles" and "rigid bodies" which do not exist in reality.

Also, Newton did not have any clear concept of "energy" at all so the original historical descriptions of the theory made no use of energy concepts.

"External work done against internal conservative force" is a rather vague description of what is going on. Things become much clearer conceptually in continuum mechanics, where the internal strain energy of a body whose volume is $$V$$ is simply $$\frac{1}{2}\int_V \sigma \cdot \epsilon\,dV$$ where $$\sigma$$ and $$\epsilon$$ are the stress and strain tensors.

With that formulation, you can find the internal strain energy without having to consider the process which created it. Of course the work-energy theorem still holds, to relate the change in internal energy to the external work done on the system.

The above equation implies that the internal strain energy in a rigid body, where $$\epsilon = 0$$, is always zero. If you consider a rigid block attached to a flexible spring, there is no energy stored in the block, unless it is moving and therefore has kinetic energy.

• so for example there is a massless spring attached to a block from one end and to the wall from another. and initially at unstretched position the block has speed (u) and i have to calculate the extension when the block stops. so if i take block and spring as a system and try to apply work energy theorem both block and spring has done work on each other internally which tend to cancel each other so how does the block stops? Jun 23 '21 at 1:49

An example that might be easier to think of: gravitational potential energy. If we have two masses, there is a gravitational force between them. As the objects move closer or farther apart, the potential energy of the system changes. Even though mass 1 exerts a force on mass 2 while, by Newton's third law, mass 2 also exerts an equal but opposite force on mass 1, there is still only one potential energy: the potential energy of the system. If you were to consider both forces in the N3L pair, then you would be double-counting the energy.

Similarly, in your spring example the potential energy is a property of the system. You don't count both forces in the N3L pair.

• so for example there is a massless spring attached to a block from one end and to the wall from another. and initially at unstretched position the block has speed (u) and i have to calculate the extension when the block stops. so if i take block and spring as a system and try to apply work energy theorem both block and spring has done work on each other internally which tend to cancel each other so how does the block stops? Jun 23 '21 at 0:59
• @Agastha if the system is the spring and the block then the spring force isn't doing work on the system. For more than one body or considering internal forces you have to generalize the work-energy theorem. If you want to apply the work-energy theorem more simply, just look at the block: the work done on it by the spring force will be equal to its change in kinetic energy. Jun 23 '21 at 1:03
• i quite didnt understand why the spring is not doing any work on the system and in what way i have to generalise work energy theorem for using in system. Can u please elaborate. Jun 23 '21 at 1:22
• @Agastha Internal forces can't do work on the system they are in. But it's all subjective based on how you define the system. The spring force is always doing work: and it's that work that changes the kinetic energy of the block. Jun 23 '21 at 5:05
• But internal forces can do work that can change the kinetic energy (ex work done by kinetic friction between two blocks)and in work energy theorem we consider work of all the forces. Jun 23 '21 at 6:18