3.2. Body force contribution

We also need the body force contribution. As body force is defined as force per unit volume, we need to integrate over the volume of the cuboid to get the total force due to body force which we denote by Fb.

This chunk introduces the concept of body force, which is a fundamental idea in mechanics. Body force refers to forces that act throughout the volume of an object rather than on its surface. To find the total body force acting on a solid object, we calculate the body force per unit volume and then integrate it over the entire volume of the cuboid. This gives us the total force due to body forces, denoted as Fb.

Consider a sponge soaked in water. The water inside the sponge exerts a force on the sponge material from all directions. Similarly, in a solid object under gravity, the weight of the material above acts as a body force acting on the volumes beneath, pulling them downwards.

The body force, denoted by b, can also vary within the volume of the cuboid in general. So, it will be a function of the position vector of the point of interest in the cuboid.

In this chunk, it's explained that the body force is not constant throughout the cuboid but can vary depending on the position within the volume. This means we must consider how body forces change at different points inside the cuboid. To express this mathematically, we use a position vector that defines the location of a point within the cuboid. As you move through the cuboid, the body force can change based on that point's position, which we denote as a function of the position vector.

Think of a high-rise building under the influence of wind. The force exerted by the wind on different floors (where some are higher than others) varies. Each floor feels a different amount of wind pressure depending on its height and location, similar to how the body force varies within the volume of the cuboid.

The total force would be: ... The higher order terms in the Taylor’s expansion when integrated will give terms of order less than the volume of the cuboid. In mathematics, ‘o’ is used to denote a smaller order term.

Here, we cover the integration of the body force across the volume of the cuboid. By integrating, we are summing up all the incremental body forces throughout the cuboid's volume to find the total force. The mention of Taylor’s expansion signifies that we approximate certain functions that describe body force. Higher order terms (those that become less significant as our volume shrinks) are dropped because they contribute negligibly to the total force in the limit.

Imagine measuring the total weight of apples in a basket, where some apples may be far heavier than others. If you were to use an uneven scale that only gives precise readings for certain weights, the 'smaller' weights become insignificant, and you focus only on the bulk weight—this is akin to ignoring smaller terms in our calculations.

Thus, the body force contribution to the torque has all terms of the order less than the volume of the cuboid. We will continue from this point in the next lecture.

In this final chunk, we conclude that the total body force contribution is composed of terms that are of lower order than the actual volume of the cuboid. This simplification is crucial for further calculations involving torques and angular momentum in the subsequent discussions in the lecture, setting the stage for future analyses of how these forces interact within the system.

Think of a small pebble dropped into a large pond. The ripples (representing smaller effects) diminish rapidly in size as you move away from the source of disturbance (the pebble). In our calculations, we focus on the significant disturbances (larger effects) and disregard the insignificant ones.

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