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Assume a double-sided lever where the left arm has a length of a and the right arm has a length of b + c. Please refer to this link for a picture.

a, b, c and F are known and F1, F2 and F3 are not. How do I find the F1, F2 and F3 forces?

EDIT:

The lever is weightless and it is in balance. However, it is also in motion and F is a force acting upon the lever as a whole - it's supposed to be some kind of sum of the F1, F2 and F3 forces - I'm just not certain if I could assume that F = F1 + F2 + F3. Anyway, the picture could be misleading - it is not acting upon the lever in addition to the other forces.

It feels kind of intuitive to me that I've got a sufficient amount of data. I just don't seem to be able to write down all the possible equations. In other words - how do you describe a double-sided lever with multiple forces applied using math?

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1 Answer

up vote 1 down vote accepted

You haven't given us enough information to solve the problem. Is there any motion? Do you know the weight of the board? Anything else?

Typically, lever problems have two types of equations that can be useful. The first is just the standard $\sum_i F_i = m\, a$. Note that this is the sum of all the forces. For example, in your picture, the sum of all forces is $F + F_1 + F_2 + F_3$. If this is nonzero, the center of mass will accelerate. In particular, if you really do have $F = F_1 + F_2 + F_3$, and if $F$ is nonzero, then that sum will be nonzero, and you will get acceleration.

The second type of equation is basically the same thing, except with torque: $\sum_i \tau_i = I\, \alpha$, where the $\tau_i$ are your torques, $I$ is the moment of inertia, and $\alpha$ is the angular acceleration. Your equation $F_1 a = F_2 b + F_3 (b+c)$ can be rearranged as $F_1 a - F_2 b - F_3 (b+c) = 0$, which says that the total torque about the fulcrum is zero. (Since $F$ is applied at the fulcrum, it does not apply any torque.) You can sometimes get more than one equation out of this if you move your origin around. For example, if you choose one end of the lever as the origin, $F$ could come into it.

From what you've told us, you have three unknowns (the $F_i$), and one or two equations. That still leaves at least one unknown. So you'll have to find something more in the physics to get another equation, so that you can solve for all three unknowns.

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I updated the description. –  Andrew Retainer May 21 '13 at 19:56
    
I've updated my answer to outline the general method for solving lever problems, but you really still haven't given enough information. Either you're not telling us relevant details or the question is ill posed. Just count up the number of equations and the number of unknowns, and you'll see that there's at least one missing equation. –  Mike May 21 '13 at 20:10
    
I think the three required equations could be: F as the sum of the Fi forces, F1a = F2b + F3(b + c) as long as it's correct and some relation between the Fi forces, making use of a, b and c. And, as I mentioned before, maybe I could substitute the F2 and F3 forces in order to find the value of the F1 force? Sorry for the lack of clarity. –  Andrew Retainer May 21 '13 at 20:22
    
Okay, first, as I've said, F as the sum of the Fi forces means that the total force on the system is just 2F, so if this is not zero, the system will accelerate, which means that it will fly off the fulcrum. I kind of doubt that this is what you want. –  Mike May 21 '13 at 20:31
    
Second, F1a = F2b + F3(b + c) is the torque-balance equation, which is right if and only if the lever is not accelerating. This is just something you have to tell us – whether it is or not. You still haven't told us, which is why this question is vague and unanswered. –  Mike May 21 '13 at 20:32
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