A friend of mine showed this problem he translated from a physics olympiad but I can't solve it:
Consider a perfectly symmetrical compass. It is constituted by two rigorously equal connecting rods, that join in a vertex the one that we call “pivot”. The opening angle is regulable. Imagine that we suspend the compass for the tip of one of the connecting rods, attaching it to a wire whose other end is attached at the ceiling.
a) Sketch the position of the bar at various angles of opening.
b) What is the opening angle of the compass, so that the pivot is the highest possible?
Hint: The center of mass of the bar, assuming that the rods have densities
uniform is the point where the bisector of the angle of the bar crosses the line
passing through the midpoint of both rods. The mass center has to be, for any angle of opening, in the vertical line of the point of suspension of the wire."
Sorry for the english in the translation. My question is how do you solve this geometrically (I know it seems like physics but the problem is all about geometry)
In the picture, the arms of the compasses, suspended from the point A, are AP and PB, with the pivot at P. The midpoints of AP and BP are L and M, and the centre of mass is at C, half way between L and M.
If the length of AP is a, then PL = PM = a/2, so L and M both lie on a circle of radius a/2 centred at P. The point C is half way from L to M, so lies on circle of radius a/4 centred at the point X half way between P and L.
The pivot point P will be highest when the angle between AP and the vertical line AC is greatest. That will happen when AC is tangent to the circle through P, C and L. Then ACX is a right angle, and the angle AXC is (because XC = a/4 and XA = 3a/4). But the angles AXC and APB are equal (because they are both equal to twice the angle LPC), so the answer to the problem is that the pivot is highest when the opening angle of the compasses is .
Linguistic note: In (British) English, a compass is a magnetic object that points north. The geometric object for drawing circles is usually called a pair of compasses.
Physical note: It follows from the geometry that the pivot is highest when the arm PB of the compasses is horizontal. There is probably a good physical reason for this, but I don't see it. If you could use physics to see why this is true, then you could deduce very quickly that the angle is , without having to do any geometry.