Integrating to find moments?

In summary, the conversation discusses using calculus in unfamiliar situations, specifically in relation to finding the total force and moment of a wall that is depressed into the ground and experiences varying wind speeds. The conversation also touches on the principles of integrating with respect to a variable twice and clarifies that the height h=0 can be interchangeable depending on the context.
  • #1
I am trying to get more familiar with using calculus in unfamiliar situations, although I am stuck when thinking about moments. I am considering a wall that is depressed 0.7m into the ground and sticks out above ground by 2.0m (and has a width of w metres) and I am assuming that wind speed varies linearly with height about ground so that the wind speed at the ground is zero and at 2m is a max speed of V_0, so that V=V_0 h/2.

Now assuming the wind is stopped by the wall, the pressure of the wind at a given height is [itex]P=\frac{Force_{on small area,dA}}{dA}=\frac{mass per second_{hitting area,dA} v_{wind speed}}{dA}=\frac{\rho_{air}vdA v}{dA}=\rho v^2 = \rho v_0^2 h^2/4[/itex]

Now to find the total force of the wind, I would integrate the pressure with respect to the area, and since the width of the wall w is constant, this turns out to be
[itex]\int_{0}^{2}Pwdh =\rho w v_0^2/4 \int_{0}^{2}h^2dh[/itex]

But then for the moment about the lowest edge of the wall wedged into the ground, it would be [itex]\int_{0.7}^{2.7}Fdh[/itex] but the force is an integral of h because it is pressure x w x dh, so how do I deal with integrating with respect to h twice? Could someone please explain the principles behind this as well because I am really trying to improve my skills in applying calculus...

Note: I realize that this sounds like a homework question, but it is not. It is just something I was thinking about and trying to fiddle with myself. If it would be better suited to the homework thread though, I would be happy to move it, it's just that my main question is about applied calculus methods rather than just finding an answer to a question :) - latest science and technology news stories on
  • #2
I also think I may have made a mistake because in my calculation for the pressure at a given height, I assumed that the height h=0 occurs at ground level, and I used this resulT of pressure in my caulculation for the moment, however in my calculation for the moment I assumed that the point 0.7m below ground was zero, so then I think it pressure equation no longer applies. Is there a way I can go about this?

1. What is the purpose of integrating to find moments?

Integrating to find moments is a mathematical technique used to calculate the internal forces and external loads acting on a body or structure. This information is important for analyzing the stability and strength of the body.

2. What are the steps involved in integrating to find moments?

The first step is to determine the function that represents the distributed load or force acting on the body. Then, the function is integrated over the length or area of the body to find the internal forces and moments. This process is repeated for each direction and the results are combined to find the total internal forces and moments.

3. How is integration used to find moments in 2D and 3D structures?

In 2D structures, integration is used to find the internal forces and moments acting on the x and y directions. In 3D structures, integration is used to find the internal forces and moments acting on the x, y, and z directions. The same steps are followed for both cases, but the integration is done in multiple dimensions.

4. What are some common applications of integrating to find moments?

Integrating to find moments is commonly used in structural and mechanical engineering for analyzing the strength and stability of buildings, bridges, beams, and other structures. It is also used in physics and materials science to study the stress and strain in different materials under different conditions.

5. Are there any limitations or assumptions when using integration to find moments?

Yes, integrating to find moments assumes that the body or structure is in static equilibrium and that the material being used is linearly elastic. This means that the internal forces and moments are balanced and the material follows Hooke's law. Additionally, this technique may not be accurate for complex or non-uniformly distributed loads, in which case numerical methods may be used instead.

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