Final answer:
The work done by friction along a circular path can be derived using the concept of dot product where the work is the product of the force vector and displacement vector, simplified for friction which is tangential to the path.
Step-by-step explanation:
The work done by friction along a circular path can indeed be derived using the dot product. The work (W) is the integral of the dot product of the force (→F) and the displacement (→Δs).
Simplistically, if the force is tangential to the path, the angle between the force vector and the displacement vector is 0 degrees, meaning the dot product equates to the product of the magnitudes of the two vectors.
For a particle moving along a circular path under the influence of friction, we have →W = →F␣ →Δs.
Since the force due to friction is always tangential to the path and thus parallel to the displacement, their dot product simplifies to the magnitude of the force times the magnitude of the displacement (W = Fd cos 0), where cos 0 is 1 because the force and displacement vectors are aligned.
To include rotational quantities, we consider work in terms of torque and arc length, with net work done expressed as net W = (net F)As, where As represents the arc length traveled by the point of application of the force.