
Solving for the Coefficient of Kinetic Friction using Work-Energy Theorem
Figure 1: Air-track glider with spring and friction
Given:
Mass of the glider (m)
Initial velocity of the glider (v)
Spring constant (k)
Acceleration due to gravity (g)
Distance traveled by the glider (d)
To find:
Coefficient of kinetic friction (μ_k)
Approach:
Apply the Work-Energy Theorem:
The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. In this case, the net work is done by the spring force and the friction force, and the kinetic energy change is from initial to zero (momentarily at rest).

Calculate the Work done by the Spring:
The spring force is conservative, so the work done is equal to the change in potential energy of the spring.
(Since the spring is initially unstretched, the initial potential energy is 0)
Calculate the Work done by Friction:
The friction force is non-conservative, so the work done is equal to the negative of the energy dissipated by friction.
(Negative sign indicates energy loss)
Substitute the work expressions and ΔKE:
(Initial kinetic energy)

Solve for μ_k:
Rearrange the equation to isolate μ_k:

Therefore, the coefficient of kinetic friction (μ_k) can be expressed as:

This equation shows that the coefficient of kinetic friction depends on the spring constant, the distance traveled, the initial velocity, the mass of the glider, the acceleration due to gravity, and the distance traveled.