Force on +r and −r planes

Today, we will start discussing the forces acting on the +r and −r planes of a cylindrical element. Can anyone recall what we discussed about cylindrical coordinates in previous lectures?

We learned that cylindrical coordinates are useful for dealing with problems involving circular symmetry.

Exactly! Now, we will apply this understanding to compute the forces. Recall that the assumption we will use is that the traction is constant across the plane. Does anyone know why this assumption is helpful?

It simplifies the calculations because we do not have to analyze variations in traction.

Correct! Using the constant traction assumption allows us to accurately compute total forces efficiently. We can express the total force on the +r plane as the traction multiplied by the plane area. This is a critical step in our analysis.

Let's calculate the forces now. The area of the +r plane can be found as the product of its height and curved edge. Who can remind us of the formula for the area A?

Is it A = r∆θ∆z for the z planes?

Close! For the +r plane, we need to consider the curvature. The area is actually calculated as the curved edge multiplied by the height. Here, it will be related to ∆r instead.

So we multiply the traction at the center with the area to find the total force?

Exactly! And once we calculate the forces on both +r and −r planes, we can sum them up to understand the net effect and how it influences momentum balance.

We also briefly mentioned Taylor's expansion earlier. Can someone explain its role in our derivations?

It expands functions to account for changes in variables, right?

That's correct! In this context, it helps us approximate the variation in traction as we observe the cylindrical element in different planes.

But we don't actually need to perform the full calculation because our assumption simplifies the process?

Exactly! The Taylor's expansion confirms that smaller order terms drop out when averaged over the volume. Thus, we can effectively use our simplified models without compromising accuracy.

Now let's bring everything together. Using the formulas we discussed, how would you combine the forces from the +r and −r planes?

We add them since they both act on the same cylindrical element!

Correct! The final expression gives us an insight into how forces act within the cylindrical coordinate system. Understanding these forces helps lay the groundwork for the Linear Momentum Balance moving forward.

So this will influence our next topics about momentum balances too?

Absolutely, and this is why mastering these concepts is crucial for tackling subsequent subjects.