A frictionless circular wire of unit radius lies in a horizontal plane. Two point particles of unit mass start simultaneously from A (θ=(π)/(2)) with identical uniform angular speeds in opposite directions and meet again at B (θ=-(π)/(2)). Which figure best represents the magnitude of the total linear momentum P of the system as a function of θ?

Solution & Explanation

Both particles have unit mass and equal angular speed ω on a unit-radius circle, so each has the same speed v = ωR. Start at A (θ = π/2) and run in opposite directions, meeting at B (θ = −π/2). At any common angular position the two are mirror images about the vertical AB line. Resolve velocities: the components perpendicular to AB are equal and opposite → they cancel. The components along AB add up. So the total momentum points along the diameter AB and its magnitude grows from the turning points toward the middle. At the start (A) and at the meeting point (B) the particles move oppositely along/around the same line so the along-AB components also vanish → P = 0 at both ends. Midway the components are fully aligned → P is maximum. Thus |P| versus θ is a single smooth dome: 0 at θ = π/2, a peak in between, back to 0 at θ = −π/2 — option C.