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 ?
D.
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 = ( i - j ) + (- i - j ) = -2 j . The magnitude of the total linear momentum is P = | P | = 2 (since 0 for [- 2 , 2 ]). The graph of P versus is a cosine curve which is zero at = 2 and maximum at = 0. This corresponds to the fourth graph. Answer:
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