Problem 4: A bent rod OBA (0BA = 90) rotates with a constant counterclockwise angular velocity w about the vertical axis shown such that the rod OAB remains in a horizontal plane. A particle of mass m located at point A can freely slide on the rod AB without friction and is attached to pomt B by a spring with spring constant k and unstretched length [,. The particle will oscillate about an equilibrium position as it rotates in the horizontal plane. a) b) Find the change in length of the spring x, when the particle 1s at \"equilibrium (1.e., the particle 1s not oscillating). How 1s this equilibrium position x affected by the direction of spin of the mw?l, bentrod? Ans: x,, = 4 k-mw? Hint: At equilibrium particle A, is moving in a circle of radius \\/ 155 + (1o + x)? centered at O with constant angular velocity w. Determine the equation of motion of the particle as the particle oscillates. That 1s, find a differential equation involving terms x, x, X, m, k, l,, lopp and w, which when solved gives x as a function of time. Obtain the equilibrium position that was found in part a) starting from the equation of motion. Ans: (k mw?)x + mw?l, =0 Hint: Now that x is not constant anymore, particle A is not moving in a circle of fixed radius. We suggest to use a polar coordinate system with origin at O, 0, along OB, 1ig perpendicular and CCW to i, In this coordinate system, position vector of the particle: 14 = logu, (I, + x)ug. Now differentiate it to get , and dy. Note you'll have to differentiate the unit vectors U, and Ug because they are changing. This final expression of a will be used in Newton''s laws to find the equation of motion