1. Angular Momentum Balance

In the last lecture we had derived the torque due to traction forces as:

(1)

This chunk introduces the concept of torque resulting from traction forces. Torque is a measure of the force causing an object to rotate about an axis. In this case, it is described in the context of previous discussions where torques were calculated based on how traction forces act on the object. Understanding how traction affects torque is foundational in analyzing the dynamics of solid mechanics.

Consider a door handle: when you pull or push it, the force you exert creates a torque around the door's hinges, causing it to rotate open or closed. Similarly, traction forces exert torque in solid mechanics.

Similarly, we had derived the torque due to body force as:

T_body_force = o(∆V) (2)

Here, the torque from body forces is discussed. Body forces are the forces that act throughout the volume of a body, rather than at its surface (like gravity). This portion emphasizes that these forces contribute to the overall torque acting on the entire volume of the body, calculated as expressed in equation (2). Recognizing body forces is key to understanding how they influence the motion and equilibrium of a solid body.

Imagine a suspended balloon. The gravitational force pulling it downwards acts as a body force, affecting the entire volume of the balloon. Similarly, it contributes to the torque experienced by the balloon.

We need to now derive (H). In Figure 1, x is the center of the cuboid as considered previously and y is any arbitrary point in the cuboid. If we consider a particle of mass m moving with velocity v, its angular momentum is given by:

⟨r⟩ × mv (3)

In this chunk, the dynamics term is being defined. It involves deriving the angular momentum (H) concerning both the center of the cuboid and an arbitrary point within it, highlighting the role of velocity and mass. The angular momentum of a particle provides insights into how objects behave when subjected to forces. This relationship is crucial as it contributes to the overall angular momentum balance of the system.

Think about a spinning ice skater—a skater pulls in their arms to spin faster. The mass and speed of their limbs (akin to the particle described) influence their angular momentum and rotation speed.

To find the angular momentum of the mass m in the cuboid about its center, we need to integrate over all the particles that it contains.

As we are writing the equation in the center of mass frame...

This chunk describes integrating the angular momentum across all particles within the cuboid. It discusses the importance of considering every particle's contribution to the total angular momentum, emphasizing the need for balance in calculating the system's dynamics about the center of mass. The center of mass frame simplifies these calculations, allowing for clearer insights into the body’s overall behavior.

You can imagine a group of dancers holding hands and turning around a common center. Each dancer moves and contributes to the overall rotation, similar to how every particle affects the total angular momentum.

Just like linear momentum balance, angular momentum balance is also applied always to a fixed/identifiable mass. ... Taking the time derivative in control mass setting, we get (5)

This section emphasizes using a fixed or identifiable mass to maintain consistency when applying the angular momentum balance. The time derivative ties in the concept of how angular momentum changes over time under varying conditions. This mathematical treatment provides the framework for understanding dynamic responses as forces and motions change in the system.

Think of a spinning wheel: as you apply more force, the angular momentum changes based on how hard you push and for how long, demonstrating a clear application of time derivatives.

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

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 part combines all previously derived equations to establish the final angular momentum balance equation. The significance lies in its broad applicability, confirming that the equation holds true under various conditions, including motion and external forces. The deduction reinforces the equation's robustness in analyzing solid mechanics.

When baking, if you combine flour, water, and yeast, the final dough mix yields a consistent product regardless of the additional ingredients you might add (like flavoring). Similarly, regardless of forces acting on the system, we derive a consistent angular momentum balance.

Let us try to write the tensor equation (11) in component form in (e₁, e₂, e₃) coordinate system. ...

Thus, we get the following three scalar equations...

In this section, the final outcome of the angular momentum balance is conveyed through tensor representation in a specific coordinate system. This vector component breakdown facilitates ease in understanding how stresses and torques interface with the body's orientation. It provides a more practical framework for engineers and scientists when solving real-life problems.

Imagine a multi-directional compass that helps you find your path. Using a coordinate system is like using that compass—it helps you navigate through the complexities of angular momentum in a clear map-like manner.

There is also a simpler but approximate way to come to this final outcome which is given in many textbooks. ...

This is a simpler derivation but it involves a strong assumption of traction components not varying on cuboid’s faces.

This segment outlines an alternative derivation method for establishing the angular momentum balance, emphasizing the assumptions made for simplification. Acknowledging multiple derivation strategies expands understanding of physical concepts, although it's crucial to recognize potential limitations or oversights in assumptions as they can affect outcome accuracy.

Think of two routes to travel from one city to another—one may take a highway while the other is a shortcut through back roads. Each way can lead you to your destination, but they may have different timeframes and experiences that reflect their own unique assumptions.

--