Exploring work and energy principle
In this blog post, I examine the principles of work and energy through practical examples involving gravitational forces, Coulomb friction, and spring resistance.
When a mass m moves vertically under the influence of gravity, the work W done by gravitational force can be expressed as:
W = -mg (y_2 - y_1)
where y_1 and y_2 represent the initial and final positions, respectively. This formula encapsulates how gravity performs work based on the vertical displacement of an object.
Coulomb friction introduces a resistive force \mathbf{f} = \mu \mathbf{N} proportional to the normal force \mathbf{N} and the coefficient of friction \mu. For a block moving over a distance d, the work W_f done by friction is:
W_f = -\mu \mathbf{N} d
This work is negative, reflecting the fact that friction opposes motion and dissipates energy.
In a round trip over the same path, frictional work is path-dependent, doubling the total work to -2 \mu \mathbf{N} d, unlike conservative forces.
Consider a block sliding along an inclined plane. This block has an initial kinetic energy and encounters friction and gravitational forces, as well as resistance from a spring. The work-energy principle allows us to calculate the maximum compression of the spring by equating the work done by external forces to the change in kinetic energy.
Given parameters, I compute each component’s contribution to the total work. The spring compression x_{ ext{max}} is calculated by setting the reference of spring compression to zero at rest and solving for the unknown using:
T_0 + W_g + W_f = -W_s
where W_s denotes the work by the spring force.
The final result demonstrates how gravitational, frictional, and spring forces interact, influencing the energy distribution within mechanical systems.
For more insights into this topic, you can find the details here