How Can Oak Tree Branches Be So Long?

In this a part of the world, we’ve oak trees. Technically they’re known as stay oaks – however I do not get it. Of direction they’re alive. I become at a football sport and that is the tree I constantly appearance at.

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Look how a ways the ones limbs enlarge horizontally. That department is set 12 meters lengthy. Why is that this amazing? Have you ever attempted to preserve an eight foot 2 x four board horizontally through retaining one end? Pretty tough. 

How approximately I calculate the forces had to preserve that department in place? I will do a easy version after which perhaps later I could make it greater complicated. Suppose I update that limb with one immediately uniform limb that looks as if this:

In this alternative limb, I am going to mention it’s miles a cylinder this is nine meters lengthy and 30 cm in diameter. 

Let me expect that this limb is attached at  factors to the tree (the white dots). So, for this limb to live there, the subsequent should be actual:

The first  equations say that the whole pressure should be 0. The final one says that the torque approximately any factor should be 0 (on the grounds that it’s miles in rotational equilibrium approximately any factor). First for the forces.

 There is the gravitational pressure. This pulls on all elements of the limb, however I can constitute this as one pressure pulling at the limb on the middle of mass (lengthy ago, I stated I could explicitly display this – however I have not yet). 

Then there are  different forces. Let me fake like there are  pins that preserve the limb to the tree. Each of those pins can exert a pressure withinside the vertical and horizontal route I will name those F1-y F2-y etc…in which the pinnacle pin can be 1. That is five forces.

So, already I even have a few constraints. The horizontal additives of the forces from the 2 pins should be identical and contrary. 

The vertical additives of the forces from the pins must upload as much as the load of the log. Now for the torque, I am going to feature up the torques approximately the decrease pin. 

Let me draw a distorted view of the log in order that the vital distances may be seen.

What is torque? Torque is sort of a rotational pressure. Here is an example, what in case you try and open a door through pushing close to the hinges? It is a great deal more difficult than pushing close to the handle, right? When rotating approximately a few axis, the torque is:

Here F is the carried out pressure, r is the space from the factor in which the pressure is carried out to the axis. 

Theta is the attitude among F and r. I will name torques that could make a rotation counterclockwise positive (really, torque is a vector). So, what’s the a number of torques approximately axis O (that passes thru factor O)? First, there are a few forces which have 0 torque. Both of the vertical pin forces have both theta = zero or r = zero in order that the torque is 0. 

The equal is actual for the horizontal pressure on the lowest pin. This simply go away  forces which have non-0 torques:

Now that I even have the horizontal pressure at the pinnacle pin, the lowest pin has the equal value (however withinside the contrary route). I do not have an expression for the 2 vertical pin forces. Let me simply say that every has a pressure identical to have the load of the limb.

How approximately a few values? First, I want the mass of the limb. If that is a cylinder of wood (with a density rho) then the mass is:

So the importance of the 2 horizontal forces at the pins could be:

If I use my values from above and an estimation of the density of wood, I get:

Wow. Oh, I recognise I made a few estimations however even 50,000 Newtons could be huge. Impressive, maximum impressive. I salute you effective tree.