1.2. Body force contribution

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

T_body_force = o(∆V)

In this chunk, we introduce the concept of torque generated by body forces in a system. Define torque mathematically as the product of force and the distance from the point of rotation. Here, 'o(∆V)' signifies the torque generated by the body force over a small volume '∆V'. Understanding that body forces act throughout a volume is crucial since these forces might be due to gravity or electromagnetic forces. This means that the entire volume of the object is engaged in producing torque.

Imagine a large playground merry-go-round. If children push down on one side (acting like a body force throughout the unit), the entire structure begins to rotate. The torque created by their collective push around the center axis is similar to what we are discussing here—resulting from forces acting on a continuous volume.

Note again that the first integral is over fixed/identifiable mass whereas the second integral is over changing volume domain since as the identifiable mass moves in space, the volume contained by it changes with time. This volume happens to be our cuboid in the current time.

Here, we emphasize how torque calculations involve two types of integrals: one that relates to a constant mass and another pertaining to a mass that changes over time due to movement. In practice, this illustrates the distinction between static and dynamic systems. As an object moves, the volume it occupies—and thus its contribution to torque—may also evolve, which complicates the calculations.

Think about driving a car. When you accelerate in a straight line, the car's mass doesn't change; hence its resistance to changes in motion (its 'mass' under 'force') stays constant. However, if you were on a winding road, the curved path could represent a changing volume affecting how forces (or torques) are applied, demonstrating the dynamic nature of the system where speed and direction impact how these forces operate.

Thus, taking the time derivative in control mass setting, we get (Time derivative of velocity becomes acceleration).

In this part, we delve into the mathematical treatment of angular momentum when it changes over time due to acceleration. We derive that the time derivative relates directly to how quickly velocity changes, which is functionally understood as acceleration. This framework allows us to analyze more efficient ways to transfer momentum to calculate resulting torque and, consequently, angular momentum.

Consider pushing a shopping cart. At first, you might push it slowly (low acceleration). But as you push harder (high acceleration), the cart picks up speed more quickly, altering how you need to exert force on it to change its direction or stop. Just like the cart responds differently to forces based on how quickly you're changing speed, so do the dynamic components of torque due to body forces in physics.

Comparing this with the derivation of the torque due to body force term, we see that this is exactly similar to that term with just b replaced with ρa(y). So, upon integrating this dynamic term using Taylor’s expansion for acceleration, we would get a similar result as that of the body force contribution to torque.

This segment makes a critical link between the dynamic term derived from time derivative calculations and the previously discussed body force contribution. By equating both under similar forms (where 'b' transitions to avoid complexities introduced by forces), we demonstrate that analyzing the dynamic flow of momentum leads us back to the foundational principles of static torque derived from body forces, signifying a mathematical harmony in physics.

Imagine preparing a large cake. To make the icing spread smoothly, you can either add heavier cream (body force) or change how quickly you apply the frosting (dynamic force). The approach you'd take in either case leads to similar outcomes in cake smoothness—just as evaluating forces dynamically or statically leads to equivalent torque results in mechanics.

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