A frictionless circular wire of unit radius is fixed on the horizontal plane. Two point particles of unit mass start moving simultaneously from point A ( = 2 ) with identical uniform angular speeds in opposite directions, and meet again at point B ( =- 2 ) . During this time, which of the following figures schematically represent the magnitude of the total linear momentum P of the system, as a function of ?

Step-by-step solution

Let the uniform angular speed of the particles be . The position vector of the particle on the right at an angle is given by r ₁ = i + j . Since it moves from = 2 to = - 2 , its angular velocity is d dt = - . The velocity of the first particle is v ₁ = d r ₁ dt = (- i + j ) d dt = i - j . By symmetry, the position vector of the particle on the left is r ₂ = - i + j . Its velocity is v ₂ = d r ₂ dt = ( i + j ) d dt = - i - j . The total linear momentum of the system is P = m v ₁ + m v ₂ . Given m = 1 , we have P = (