I am having quite difficult time to understand the meaning of total potential energy on spring.
All references I read say that the total potential energy on particle attached to spring is:
total potential energy (U)=elastic potential energy (from spring) + potential energy from external force
The work done on particle due to system of forces is equal the difference in potential energy between point A and point B.
If we chose point A to be the datum with 0 potential energy, then the total work done on the particle to move it from point A to point B is equal to the potential energy of the particle at point B.
U (Total potential energy) = W (Total work) = Ws (word done be spring force) + We (work done by external force)
In this picture, it is well know that the spring force is a function of the displacement x so the work done by spring force on the red particle is
dws=-Fs.dx=k.x dx
ws=-1/2kx^2
Now the problem is in the work done on the particle by the external force F. References I read say that the work done by the external force is
we=F.x
because the force is independent of displacement x.
My question is as follows:
At the initial stat, there is no force F and no spring force Fs. Once we apply the external force one the particle the work start to be exerted on the particle.
we normally do that gradually and the external force F should be larger than the spring force Fs in order to move the particle to the left.
Through distance x, external force should be larger than the spring force and then at some point it should be adjusted to make the total force on the particle equal zero at final position (with zero velocity).
so I thing the work done be external force should be some thing like this:
we=Integrat(F(x,t)dx)
Why the external force is considered independent of the displacement x and then the work done by external force is calculated as:
we=F.x
Best Answer
Are you wondering why your two equations $W=\frac{1}{2}kx^2$ and $W=Fx$ don't match? That's because the latter equation, $W=Fx$, is only true for a constant force. The more general expression is $W=\int \vec F \cdot d \vec x$, similar to what you wrote in differential form $dW=F\,dx$.
The expression with $\frac{1}{2}$ in it is correct. The expression with just $Fx$ is only correct for constant forces, which spring forces are not.