okay, this will be difficult to explain, and i don't have inventor here at home so i can't even try it, but give this a shot:
model your sphere as a revolution, making sure you projected the origin point of the part file onto the sketch so the sphere is definitely locked onto the centre of it's coordinate system.
whack another sketch on the xy plane, and slice graphics. use project geometry to project the outer "circle" of the sphere onto your new sketch.
draw a line from the centre of that circle to the perimeter, vertically downwards.
draw a line from the centre of the circle to the perimeter, on an angle down to the right. continue drawing from that point, horizontally to left, to join your vertical line somewhere.
you now have a right angled triangle in the lower right hand quadrant of your sketch, yes? if not, yell back and i'll post an image for ya on my site when i actually try this!
okay, the dimension of the horizontal line is related to the diameter of the hole you want to drop the ball into, to be precise, it's half (or equal to the hole's radius, if you prefer)... entering that dimension will lock the vertical position of that line.
use that line to create a workplane in your sphere part.
finish edit.
constrain.
simmer and repeat
as i mentioned, this is only in theory, i haven't tested it, so ask away if i either a) explained it poorly, or b) explained a method that doesn't work
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