# Cantilever equilibrium problem

• UMath1
In summary, the forces in equilibrium must be equal, but the normal force produced by the fulcrum is far greater than the force of gravity at center of mass. The problem is that for an object to be in mechanical equilibrium, the forces must also be equal. So how can both of these be true? The other issue I am having is that you are allowed to find torques using any axis. If that is true and I set the axis of rotation to be the point of contact with the fulcrum, then there will be a net non zero torque. So how is this situation possible?So at the attachment of a cantilever beam there is a torque as well as a net force. Without that additional torque, as you
Since a point by definition has virtually no area, the pressure would be infinite, right?

Yes. Exactly.

Now, how realistic do you think that is? Do you think that you can really have an infinite pressure at a single point?

No, the force must be spread across a small area. So how exactly would that work in the case of the cantilever?

UMath1 said:
No, the force must be spread across a small area.
Yes. So the model of a point force is just a limiting approximation. It is not realistic, but if the pressure is relatively high over a relatively small area, then it can be a useful simplifying assumption.

Do you now understand the idea that a point force is a limit of a large pressure over a small area, and it is just a useful approximation?

Assuming you understand the above, now consider what would happen if the pressure were not uniform. Suppose that it varies linearly from one side to the other. Do you see that the average of the pressure still gives you the applied force as before, but the linear variation also gives you an applied torque?

I think I understand. But wouldn't the location the torque and applied force coincide?

UMath1 said:
I think I understand. But wouldn't the location the torque and applied force coincide?
Yes, that is the idea. We are modeling the force as a point force and the torque as a point torque.

So how would you do that in this case. The weld force and normal force must be treated as one point force producing one point torque?

Yes.

For clarity, you have a cantilever force and a cantilever torque. If you choose an axis through the attachment point then the torque for the cantilever force is 0 so only the cantilever torque (bending moment) remains.

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Not sure wht I did wrong.

Do the torques about the joint so that the lever arm of ##F_{joint}## is 0. Then you can easily calculate the point torque applied at the joint, call it ##\tau_{joint}## in order to satisfy the torque equilibrium condition. Note that ##\tau_{joint}## is a completely separate independent degree of freedom and is not related to ##F_{joint}##

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Is the location of Tjoint the same as Fjoint, though? Does it have a zero moment arm?

UMath1 said:
Is the location of Tjoint the same as Fjoint, though?
Yes.
UMath1 said:
Does it have a zero moment arm?
It is not a force, it is a torque (technically a bending moment). It doesn't have a moment arm, the concept is irrelevant for it.

So then in this case, based on how I set it up, Tjoint would be 195 N*m, correct?

And what is a bending moment?

UMath1 said:
And what is a bending moment?
Tension breaks a beam by pulling it apart. Compression breaks a beam by crushing or buckling. A shear moment breaks a beam by slicing it. A bending moment breaks a beam by snapping it. And torsion breaks a beam by twisting it.

The math is in the right ballpark. I can't check exactly since you didn't label the distances.

So is bending moment a force or a torque? Tension is a force if I understand correctly.

Neither. They are all stresses. The units for stress are the same as the units for pressure, but in solids they can point in directions other than perpendicular to the surface.

So Tjoint is not a torque but a stress? How does it cancel out the other torques then?

If you integrate a stress over an area you can get a torque or a force.

This is related to what we discussed earlier about the point force and point torque approximations.

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