Lotta good answers on here.
There are definitely ways bridge length affects your stroke and the shot.
In a nutshell, I think the basic tradeoff is:
Shorter = More accurate, less room for the tip to deviate from the intended contact point.
But there's less room to get the cue up to the necessary speed... so if you have to strain
(aka lunge, jab, poke, stab) at the ball, you increase the chance of your own muscles
steering the stick off course.
Longer = easier to put a full stroke on the ball, less effort to move the cue ball, so the above issue
with muscle strain goes away. But more room for the tip to steer off course. I find the further
back I pull my arm, the more it wants to swing outward, away from my body.
So long bridges can introduce a steering movement (aka a hitch) in the stroke, and may
lead to extra unwanted spin on the cue ball.
But as for your original comment, "the angle of your back arm when it contacts the cue ball"...
You mean the angle of the stick when the tip makes contact?
Are you referring to the horizontal or vertical angle? With a short or long bridge, your stick
should be very level. So vertically, bridge length doesn't really matter.
You shouldn't be angling downward into the ball either way.
In terms of left and right spin, it definitely matters. Over time players will find themselves subconsciously
adjusting bridge length when applying heavy sidespin, as they adjust for deflection.
Good stuff CreeDo, i add, long bridge allow you to see your cue moving back and fourth and detect if it is in line or not, allows you to gauge speed better since your arm is taking bigger swing, and you are actually doing a dry run of the shot; just remember to pause as close to CB as possible to avoid unwanted spin and follow through. Efren does warm up before pause, and two or three warm ups after pause no pause then; tried to copy it helps gauge speed but did not help much with accuracy! i can see hitting CB while arm is warming up is good but who knows..