Kevin Zhou — Advanced Mechanics IV Problem 1 · Kevin Zhou #M8.1
Physics · mechanics
Problem
Problem 30. Using similar reasoning, design a machine that finds the point X that minimizes
(AX)2+(BX)2+(CX)2. What geometrical property can you conclude about this point?
Example 9
Show that the incenter of a triangle (i.e. the meeting point of the angle bisectors) exists.
Solution
Apply six forces at the vertices of a triangle as shown.
These forces clearly balance, and also produce no net torque on the triangle. Now combine
the forces applied at each vertex, yielding three forces that point along the angle bisectors.
By the principles of M2, the torques of these forces can only balance if their lines of action
meet at a point. Therefore the angle bisectors are concurrent, so the incenter exists.
Example 10
Let AB be a diameter of a circle, and let a mass be free to slide on the circle. The mass is
connected to two identical straight springs of zero rest length, which are in turn connected
to points A and B. At what points C can the mass be in static equilibrium?
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Solution
The potential energy of the system is proportional to (AC)2+(BC)2. Since ABC is a right
triangle, this is just equal to (AB)2 by the Pythagorean theorem. Since the potential energy
doesn’t depend on where the mass is, it can be at static equilibrium at any point on the
circle. Alternatively, you can show that the mass is in static equilibrium by force balance,
and use the reasoning in reverse to derive the Pythagorean theorem.