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I am having trouble with this moments question: enter image description here

A force F balances a 6m long, 100kg uniform beam at A. The beam will be shortened in length by 2.5m at B. Determine the new position of support A that balances the shortened beam.

Now, my initial thinking was just to calculate what A and B were with the shortened beam (3.5m instead of 6m). So I tried the following (I am assuming clockwise is positive and anticlockwise is negative. Obviously if it were the other way round, you'd get the same answer.):

0 = -f + 2.5b

f = 2.5b

Sub f back into f:

0 = -(2.5b)+2.5b

0 = 0

So it equals 0, which leads me nowhere. But even if it did, how do calculate the new position of a support? Any help would be appreciated.

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I cannot decipher your first equation. I think it is meaningless.

The first step is to determine the force F. That needs to be determined using the weight and center of gravity of the 100 kg beam.

After you know the force F, you can determine where the support A needs to be positioned to balance the shorter beam of weight X and a new center of gravity.

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A simple solution approach is as a two step problem:

  1. Determine the magnitude of Force F that keeps the original 6m beam balanced.
  2. With Force F known, determine the shifted position of Support A that will maintain static equilibrium of the shortened 3.5 beam.

Before we jump into writing equations, another good step is to draw a free-body diagram of the beam and the applied forces. This will make it far easier to develop meaningful equations. For example, your proposed equation omits the self-weight of the 100 kg beam and drawing a FBD will help prevent errors like that.

free body diagram

By using variables to represent the beam length and support position, we can use the same FBD to write equations for both beam configurations.

First, in the original system, we know $x_1$, $x_2$, and $L$. We then solve for Force F by summing moments about Point A.

Second, in the shortened length system, we know Force F and Shortened Length L, and solve for the modified $x_1$ by once again summing moments about Point A.

Because this looks like a homework assignment, I'm not going to write out the equations but conceptually this is the approach I'd use.


Supplemental Thought: A nice zeroeth step is to consider what your intuition tells you. Imagine balancing a bar on your fingertip with your finger positioned like Support A (or, heck, go find a ruler and do an experiment!). Think about how you need to apply Force F to keep the bar stable. Now, imagine suddenly cutting off a chunk of the right end of the bar. What will happen to the system if Force F is kept constant? How might you shift your finger to keep that from happening? Taking a minute to do a thought experiment both develops intuition and serves as a gut-check on your final calculated answer.

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