Question

A mechanical device consists of two circular gears, one of radius 2 centered at (0, -2) and the other radius 1 centered at (0, 1). The gear of radius 2 rotates clockwise at unit angular velocity (1 radian per second), while the gear of radius 1 rotates counterclockwise without slipping at the contact point. The two gears each carry a small peg on their circumference, and these pegs are connected together by an elastic band. Initially, both pegs are at the origin. (a) Write parametric equations for the motion of the midpoint P of the elastic band. (Use vectors; begin by expressing the position vector in terms of simpler vectors, then express each of them in terms of (t)

Answer 1

der angular velocity Point P = (nt mis , Jot Ske). @ Let's consider angular" fo, -2) 4 (W) of Gear 1 . Isad/se. Since no slip cond" W,r, = Word * W x l = 1x2 W, = 2 rad/sec O, angular displacement after time t. of peg 1. 92: angular displacement after time t of peg 2. O, = wit, O2 = uz to x = r, sino, x2 = resine y, = 8,coso, , Y2 = 8 cose Point P ( rising + ogsind sesoso, - &coso, - 2 axp = rosino, + & sind . 2 yo = 7, cos o, -82COOL Xp = I sinat + asint Up = I cosat - acost 2