Work by conservative and non-conservative forces
In my study of forces and energy, it’s crucial to distinguish between conservative and non-conservative forces and their roles in energy transformation. A conservative force is characterized by its independence from the path taken; the work it does depends solely on the starting and ending positions. Examples of conservative forces include gravitational force, electrostatic force, and spring force, all of which have an associated potential energy function.
For conservative forces, there exists a potential energy function U, and the force can be derived from it as:
\vec{F} = -\nabla U
This leads to two important properties: the work done by a conservative force in a closed loop is zero, and it is possible to define a potential energy associated with it. Non-conservative forces, on the other hand, depend on the path taken and often result in energy dissipation. Friction and air resistance are examples, converting some of the work done into thermal energy or other forms of irrecoverable energy.
To illustrate this, I analyzed the work done by a spring, a conservative force that obeys Hooke’s Law. Given a spring with stiffness k and an unstretched length L_0, any displacement from this natural length results in a restoring force:
F = -k \delta
where \delta = L - L_0 represents the displacement.
I then calculated the work done by the spring as it moves from one stretched position to another. By integrating the force over the displacement, I obtained the work done as:
W = -\frac{k}{2} \left( (L_2 - L_0)^2 - (L_1 - L_0)^2 \right)
This work done by the spring can also be expressed through the change in its potential energy, defined as:
U(L) = \frac{1}{2} k (L - L_0)^2
From this, it becomes clear that the work done by the spring force between two positions can be calculated as the negative change in potential energy.
For more insights into this topic, you can find the details here.