A small mass is free to move on a frictionless horizontal surface. Let die horizontal surface be a circle of radius R = 1 m, and let it rotate counterclockwise about a vertical axis with a constant angular velocity ω =1 rad/s. Let the coordinates of die mass be described by a rectangular (x, y) coordinate system centered on die axis of the rotating system and rotating with it. (a) Find die equations of motion of die mass in terms of the rotating (x, y) coordinate system, (b) If the initial position of the mass is (−R, 0), what initial y-component of the velocity (relative to the rotating frame) is necessary if die mass is to be projected across a diameter of die circular surface, from the perspective of a fixed, inertial observer? (c) Find an expression in terms of integers n = 1, 2, 3 . . . for die initial x-component of die velocities diat results in die mass traversing a diameter of the circular surface in a fixed inertial frame of reference and landing at the same point (−R, 0) from which it was projected in the rotating frame (let n = 1 represent the fastest initial velocity), (d) Plot these trajectories from the perspective of die rotating frame of reference for the five largest initial x-components of the velocities (n = 1 ... 5). (e) Describe die resultant trajectory as seen from the rotating frame of reference as die x-component of the velocity approaches zero (n → ∞).
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