8. Final Balance

We have got all the terms for Linear Momentum Balance. We can now plug in equations (22), (23) and (24) into the following balance law.

This chunk sets the stage for the final balance equation in the context of Linear Momentum Balance (LMB). It indicates that all necessary terms have been gathered from previous derivations and are now ready to be incorporated into a single balance law formula. This means the previous equations detailing forces, body forces, and momentum change are combined to describe the equilibrium of a system.

Think of it like gathering all the ingredients needed for a recipe before you start cooking. Just as you would combine various ingredients to create a dish, in physics, the various forces and momenta need to be combined to understand how a physical object behaves.

We also divide both sides by ∆V and take the limit ∆V→0. By doing this, the o(∆V) terms will drop out.

In this step, the equations are manipulated by dividing both sides by the volume of the cylindrical element, represented as ∆V. This process leads to taking the limit as the volume approaches zero, which is a common technique in calculus for simplifying equations. Here, the terms denoted as o(∆V) become negligible and are removed from the equation, making it easier to focus on significant terms.

Imagine trying to measure the drop of water from a faucet. If you increase the amount of water to measure the drop but then consider the drop when the water is minimal (i.e., very small), you only focus on the relevant parts of the measurement, ignoring practically insignificant dribbles.

We finally separate the terms in the final balance equation along different basis vectors to get the component form of the balance equation, i.e.,

In this part, the overall equation is broken down into its components based on the coordinate system being used—namely cylindrical coordinates. This means that instead of dealing with the equation as a whole, it is divided into parts that correspond to different directions or axes (r, θ, z) in cylindrical coordinates. This simplifies analysis and allows for clearer interpretations of the behaviors in each directional aspect.

Imagine trying to analyze a jigsaw puzzle. Instead of looking at the entire puzzle at once, it helps to break it down into sections—like corners, edges, or middle pieces—so you can work more effectively on fitting the right pieces together.

Equation in r-direction: (26) Equation in θ-direction: (27) Equation in z-direction: (28)

The final balance equations are presented in a component-wise format, indicating how the balance of linear momentum is expressed in each coordinate direction. Each equation captures forces and momentum changes specific to that direction, allowing for precise calculations in engineering applications. It also highlights differences from Cartesian coordinates by introducing extra terms relevant to the cylindrical system.

Consider how a smartphone detects orientation. Sensors provide data on movement in three directions (up/down, left/right, and forward/back). Each movement is processed separately but contributes to the total orientation of the device—similar to how these equations operate within their respective coordinate systems.

The first three terms in each of the equations are similar to what we get in Cartesian coordinate system. There is one extra term in each equation. The extra term for the θ-direction contains a factor of 2 because the two terms can be combined as one using the symmetry of stress matrix.

Here, it's explained that while there are similarities with Cartesian coordinates, cylindrical coordinates introduce additional factors into the equations. Specifically, the extra term in the θ-direction reflects the unique geometry of cylindrical systems, where the changes in radial position and angles must be both considered simultaneously due to the nature of stress distribution on cylindrical bodies.

Imagine using a circular track for training versus a straight line track. While both can measure speed (linear momentum) the circular track introduces additional factors—like the centripetal force—that require consideration, just as the cylindrical equations must accommodate.

We can conclude from this derivation that the equations for cylindrical coordinate system are quite different when compared with the equations in Cartesian coordinate system. So it was really important to work out this derivation explicitly.

The conclusion emphasizes the importance of understanding the differences between coordinate systems in mechanics. While both systems can describe motion and force balance, the nuanced differences—highlighted through this derivation—have significant implications for problem-solving in engineering and physics.

Consider learning to drive a car versus a motorcycle. While both are vehicles, their operation mechanics vary greatly due to their structure and dynamics. Understanding each system's specific requirements and behaviors is crucial for effective driving, just as it is vital for engineers and physicists to differentiate how cylindrical and Cartesian coordinates function.

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