In this section we will use integration and differentiation to determine the change in Kinetic Energy. You may already know the basic equation for Kinetic Energy but it is important to understand the concepts and how to derive the Formulae.
We will start with this simple diagram, which shows a tangential gradient of value we do not pocess. There is a particle of a certain mass, that is applying a force against a stationary wall along the tangential gradient, and preventing the particle from hitting the wall is a spring of value K. Friction is not neglected.
The initial work done in this system is known as non-conservative, and the calculation is shown. The force applied to the particle, in the direction (angular, not normal) minus friction, all integrated by the distance the spring compresses or extends. In this case its a compression.
Having obtained the Formula for the non-conservative function, we now are missing just two functions that we require, both are conservative functions (Work done under gravity, and work done by the spring).
The spring processes a stiffness of K, and is compressed a distance delta. Therefore the Force is K*(Delta). This makes K=0.5*F*(Delta) also equal to area, as shown in the diagram below.
The final function is work done by the spring, the calculation is shown below, but this only stands should the spring be at what is known as free length.
A spring in the case of this problem can exist in three states, free length, initial length and final length.
A pictorial representation is shown below, highlighting the different states.
Now we have all the required functions, the total work done can be calculated by addition of the non-conservative and conservative portions.
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