Problem 1: A ball of mass m attached to a string is being swung around in a horizontal circle of radius R centered at the origin, at angular velocity wo. (Neglect gravity throughout this problem.) At time t = 0, the ball is on the +x-axis and the string breaks. Newton's First Law tells us that the now-free ball will move at constant velocity in a straight line parallel to the y-axis. However, the angular velocity of the ball is no longer constant. (a) Find an expression for the angular velocity as a function of time, w(t) o(t), for this straight-line motion. Explicitly check your result for dimensional consistency and sensible limiting cases. (b) Assume instead that, after the string breaks, a purely angular force (i.e., zero radial component) acts on the ball in such a way as to keep its angular velocity constant at wo. Use the radial equation of motion in plane-polar coordinates to solve for the trajectory of the ball as a function of angle, p(p). Use Mathematica, Desmos, or some other
Problem 1: A ball of mass m attached to a string is being swung around in a horizontal circle of radius R centered at the origin, at angular velocity wo. (Neglect gravity throughout this problem.) At time t = 0, the ball is on the +x-axis and the string breaks. Newton's First Law tells us that the now-free ball will move at constant velocity in a straight line parallel to the y-axis. However, the angular velocity of the ball is no longer constant. (a) Find an expression for the angular velocity as a function of time, w(t) o(t), for this straight-line motion. Explicitly check your result for dimensional consistency and sensible limiting cases. (b) Assume instead that, after the string breaks, a purely angular force (i.e., zero radial component) acts on the ball in such a way as to keep its angular velocity constant at wo. Use the radial equation of motion in plane-polar coordinates to solve for the trajectory of the ball as a function of angle, p(p). Use Mathematica, Desmos, or some other
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Transcribed Image Text:Problem 1: A ball of mass m attached to a string is being swung around in a horizontal
circle of radius R centered at the origin, at angular velocity wo. (Neglect gravity throughout
this problem.) At time t = 0, the ball is on the +x-axis and the string breaks. Newton's
First Law tells us that the now-free ball will move at constant velocity in a straight line
parallel to the y-axis. However, the angular velocity of the ball is no longer constant.
(a) Find an expression for the angular velocity as a function of time, w(t) = o(t), for
this straight-line motion. Explicitly check your result for dimensional consistency and
sensible limiting cases.
(b) Assume instead that, after the string breaks, a purely angular force (i.e., zero radial
component) acts on the ball in such a way as to keep its angular velocity constant at wo.
Use the radial equation of motion in plane-polar coordinates to solve for the trajectory
of the ball as a function of angle, p(p). Use Mathematica, Desmos, or some other
program with plotting capabilities, to plot this trajectory for 0 ≤ < 4π. (Plotting p
versus is sufficient, but a polar plot is niftier.)
(c) For the situation in part (b), determine how the angular component of the force depends
on angle, Fo(o).
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