I recently purchased a pull up bar to get chiseled in preparation for my photo shoot in the MindTribe Christmas calendar.
Tim, our director of Electrical Engineering, took a look at the contraption and concluded the load is borne solely on the door trim, a design choice of which he disapproved. This didn’t seem quite right to me, and it nagged enough to get me out of bed early Saturday morning to figure it out.
As an aside, whenever I think about the way my parents looked on the day I informed them I needed a 5th year at my very prestigious, very expensive university to complete my engineering degree I have no problem whatsoever getting out of bed early Saturday to do math. Thanks, Mom and Dad!
Here is a graphic showing how the Iron Gym hangs in a doorframe. The doorframe is rendered transparent to show the pink cross beam hanging on the internal door trim, while the yellow bar rests against the outside. You hang on to the straight section of the green “handles” to do pull ups.
Here is a cross section with the relevant loads and distances labeled:
When we sum the moments around 0 (the black dot) we see the following:
W*L3 – Rx*L2-Ry*L1 = 0
We also know Ry is the frictional force generated by the normal load Rx:
Ry = u*Rx
where u is the coefficient of friction for “rubber on wood”.
We also sum the forces in the Y direction to see:
Oy + Ry – W = 0
We substitute for Rx and solve the first equation for Ry:
Ry = W*L3 / (L1 + L2/u)
Plug in some rough numbers:
W = 1000N
u = 0.5
L1 = .20m
L2 = .20m
L3 = .25m
And observe that Ry is roughly 416N, or ~40% (see what I did there?) of the overall weight W. We may wish to vary the estimate for the frictional coefficient, or to include the moment from the force in the x-axis on the pink cross beam, but it turns out they don’t influence the answer very much. From this analysis we may conclude that although there is a lot of weight on the door trim, not all of it is borne there. A more useful analysis might calculate allowable loads on a residential door trim but that isn’t necessary to prove Tim wrong! I’ll stop here, then, and savor this rare moment.