12/12/11

In bodybuilding/strength training forums I regularly read this question "How much weight can I add to one side of a racked barbell before it tips over?" There are a lot of simplifying assumptions one can make to tackle this problem, and here is a general idea. Statics experts (of which I am not one) - please write in!

Let a racked barbell that weighs B lbs have its supports at x_{1} and x_{2}. Say there is m_{1} lbs at
x_{1} and m_{2} lbs at x_{2}. The center of mass of the unweighted barbell itself is at d/2, where d is the length of
the barbell.

Note that for a bar to not be tipping, it is not rotating about a support. Therefore the sum of the torque equals zero. Mathematically, for an arbitrary point X, this is

Note that if sum(torques) = 0, this implies that X = X_{cm} (center of mass), and

I've ignored the weight, B lbs, of the bar here. Adding that back in, we get

So when will the barbell tip over? Without loss of generality, assume we are just adding weight to m_{1}. The
bar will then tip when X_{cm} < x_{1}. That is, when the center of mass is outside of the support. The situation is:

Note something interesting here. Because each weight plate has a thickness, the new x_{1} point that we are loading with mass
is actually shifting to a number smaller than x_{1} as we add additional weight to the bar. So we're not just adding more weight,
but adding more weight further from the support.

If I substitute in specific numbers, these tend to match reality pretty well. For around 3 to 3+ plates it tends to tip for the setups I've used
if there are no plates on the other side (ie. m_{2} = 0). Here's a setup with some hypothetical numbers:

- B = 45 lbs
- bar's center of mass = d/2 = 42 inches
- starting m
_{1}= 45 lbs - m
_{2}= 0 lbs - starting x
_{1}= 24 inches - x
_{2}= 60 inches - assume x
_{1}shifts 3 inches to the left of the starting x_{1}with each plate added

Note that for 1 and 2 plates added to m_{1}, X_{cm} is still greater than 24 inches (the original x_{1}).
However, when a little over 3 plates are added, X_{cm} < 24, and therefore tipping occurs.

In practice, however, I'd recommend to not ever have the difference between m_{1} and m_{2} be large at all. Actually, there's really no need to use a barbell in the first place.

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