The correct answer is b. the initial and final displacements of the spring.
To understand why, let's discuss the concept of work done by a spring. When a spring is stretched or compressed, it exerts a force that is directly proportional to its displacement from its equilibrium position. This force follows Hooke's Law, which states that the force is equal to the spring constant (k) multiplied by the displacement (x).
1. The work done by a spring is equal to the force applied by the spring multiplied by the distance over which the force is applied.
2. In the case of a spring, the force is not constant, as it varies with the displacement. Therefore, we need to consider the integral of the force over the displacement to calculate the work done.
3. The formula for the work done by a spring is given by W = (1/2)kx^2, where W represents the work, k is the spring constant, and x is the displacement.
4. As we can see from the formula, the work done depends on the square of the displacement, which eliminates options a, c, and d.
5. The formula also shows that the work done depends on the initial and final displacements of the spring, as represented by the variable x. Therefore, option b is correct.
6. The average displacement of the spring, as mentioned in option e, does not play a role in determining the work done by the spring. The work depends on the specific initial and final displacements.
The work done by a spring depends on the initial and final displacements of the spring (option b). The formula for calculating the work done by a spring is W = (1/2)kx^2, where k is the spring constant and x is the displacement. The inverse of the displacement, the square of the displacement, and the difference between the initial and final displacements are not factors in determining the work done by a spring.
Therefore, the correct answer is b. the initial and final displacements of the spring.