Consider the work done on the block by an external agent as the agent applies a force on the block and the block moves very slowly from xi = -x max to xf = 0.
W = Fs * dr = integral of (-kx) * dx from xi to xf
The Attempt at a Solution
The book illustrates F applied to the left, and Fs to the right.
"We can calculate this work by noting that at any value of the position, the applied force is equal in magnitude and opposite in direction to the spring force Fs if it moves at a very slow speed. Fapp = -Fs = -(-kx) = kx
Therefore, the work done by this applied force on the block-spring system is like the following integral as shown in the picture:
1. Was the illustration above a demonstration of the phenomena that if we carry the spring by the applied force very slowly, in theory and ideally, the applied force (which is pulling toward the right) is exactly the opposite of Fs?
But further more, the book continues:
"The work is equal to the negative of the work done by the spring force for this displacement." <--- I think this satisfy my comment above, yet
"The work is negative because the external agent must push inward on the spring to prevent it from expanding and this direction is opposite the direction of the displacement of the point of application of the force (F*r) as the block moves from -Xmax to zero."
I can understand everything except the phrase in bold. We applied the force outward (pulling it to the right, if I am correct).There the Fs due to the constant k, has tendcy to move backward, am I correct?
2. If I am, then why is this applied force pushing inward? And what exactly is this "expanding"?