3. Angular Momentum Balance

Weknowfromfirst yearmechanicscoursethat arigid bodyhastoalso obeythebalance ofangular momentumapart from theNewton’ssecondlaw. Forarigid body,netexternaltorque on it equals the rate of change of its angular momentum. We’ll use T to denote Torque and 𝑳 to represent angular momentum. Let ’O’ denoteafixedpoint and ’cm’ thecenter of mass. We can do thebalance ofangle momentum eitherabout a fixed pointorabout thecenter ofmass.

This chunk introduces the concept of angular momentum balance, stating that rigid bodies must follow specific laws of motion just as they do with linear momentum. Angular momentum, denoted by 𝑳, is linked to torque (denoted by T), which is the rotational equivalent of force. The balance equation indicates that the total external torque acting on a body equals the rate of change of its angular momentum. The reasoning behind this is intuitive; just as pushing a door causes it to swing, the net torque applied to a body determines how quickly and in what direction it will rotate.

Imagine a person spinning a basketball on their finger. The speed and direction of the spin depend on how much force (torque) they apply with their fingers. If they apply more force, the spin accelerates, showing that the relationship between the force applied and how the ballet ball's spin changes aligns with the principles of angular momentum.

Let us begin with torque due to traction on the plane with normal along e (denoted by T1 ). For an arbitrary point y on this face, the arm of the force will be y−x and force on a small area around y would be t1dA. Hence, we have again used Taylor’s expansion here about the center of the cuboid.

This chunk focuses on calculating the torque produced by traction (forces acting on the surface) on the cuboid. Here, we consider how much torque is generated at a specific face of the cuboid, with the force acting at a distance from the center of mass. By using Taylor expansion, we approximate the torque contributions by systematically exploring how these forces change across small areas (dA) on the cuboid's surfaces. This analysis ultimately leads to a formula for total torque acting on that specific face, which is critical for the overall angular momentum balance.

Think of opening a door. If you push the door at the edge (far from the hinges), it swings open easily (more torque). If you push it near the hinges (close to the pivot point), it’s much harder to move (less torque). This is similar to how traction forces create torque, and the position where the force is applied affects the overall ability to rotate the object.

Adding torques on the one and −e faces, we get T1 + T−1 = e × σe ∆V + o(∆V). Doing similar analysis for the remaining four faces, we obtain total torque contributions.

In this chunk, the analysis extends to include all faces of the cuboid, showing how to calculate the total torque acting on the cuboid. By summing up the contributions from the traction forces on opposite faces, we derive an expression that accounts for how these forces generate torque relative to the center of mass. The use of mathematical operations like addition and cross products (e × σe) illustrates the vector nature of these torque contributions, thereby tying together all the components into a complete equation.

Consider a seesaw where the torque on either side changes based on how far from the middle the forces act. When children sit further from the pivot point, they generate more torque, balancing against smaller forces on the opposite side. The principle applied here is consistent across different faces of the cuboid, reinforcing how individual contributions can add up to a more significant system behavior.

Now, consider torque due to body force about the centroid. Now, y will represent a general point in the volume of the cuboid. The volumetric integration together with Taylor’s expansion lead to an expression that simplifies based on the centroid's position.

In this chunk, the focus shifts to the contributions of body forces (like gravity) acting throughout the volume of the cuboid. By utilizing Taylor's expansion to simplify calculations, we derive an expression that shows how these forces contribute to the overall torque about the center of mass. The position of the centroid is crucial here, as it helps to establish a reference point from which the torque is calculated, ultimately leading to an understanding of how distributed forces affect the angular momentum balance.

Imagine an elevator lifting a heavy object. The gravitational force acts every part of the object, and its center pulls down directly. To maintain the elevator's balance, you need to assess the 'weight distribution' (or body force) when designing the elevator’s mechanics, much like how we consider the torques on the cuboid.

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