1.4. Final balance

So, now we can substitute equations (1), (2), and (7) in (8)

In this step, we take equations from previous sections (1, 2, and 7) and plug them into equation (8). This is a crucial step in our calculation, as it allows us to combine previous findings into a single expression which takes into account all the effects on angular momentum balance.

Think of it like making a recipe. You have several ingredients (equations) from previous steps, and now you are combining them to create a final dish (the final balance equation). Just as you need the right amounts of each ingredient to make a successful meal, you need the right equations to derive the correct balance.

As this equation is valid for a cuboid of any size, we can shrink it to its centroid (x). So, first we divide by ∆V on both sides and then take the limit: ∆V→0

This chunk describes how to generalize our findings. By stating the equation is valid for a cuboid of any size, we can simplify it further. We focus on the centroid of the cuboid, which represents an average center point. Dividing by ∆V and taking the limit approaches a situation where the volume is infinitesimally small, which helps in refining our calculations to find specific behavior of the system.

Consider it like zooming in on a map. Initially, the map shows a whole city, but when you zoom in, you focus on a specific neighborhood (the centroid). As you continue to zoom in, details become clearer, just like how we refine our equations by focusing on smaller scales.

We finally get → Angular Momentum Balance (AMB)

After substituting the equations and taking the limit, we arrive at the final expression known as Angular Momentum Balance (AMB). This equation summarizes all the previous analyses and gives us a formula to evaluate the angular momentum in relation to the forces acting on the body. Its formulation captures the essence of dynamic interactions within the system.

Imagine constructing a model. At first, you have many pieces (equations and concepts). But as you assemble them thoughtfully, you create a final model (the AMB equation) that represents a complete picture of how things work together. This model allows you to analyze how the system behaves under different conditions.

This equation holds even if a body force is acting or if the body is accelerating as these terms vanished in the derivation itself.

This statement emphasizes that the Angular Momentum Balance equation is robust; it remains valid regardless of additional forces acting on the body or whether the body itself is in motion. This illustrates the power of the balance equations in mechanics where they can simplify complex scenarios and help us analyze the motion without additional complications.

Consider a car moving on a road. Whether the car is going uphill (experiencing a body force) or changing speed (accelerating), the principles governing its motion still apply. In the same way, the AMB holds true under different conditions, providing a reliable framework for understanding angular momentum.

--