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Unformatted text preview: ME201 Advanced Dynamics (Fall 2007) Midterm Solution ( 100 pts ) Consider the system shown in the figure. Planar pendulum (no gravity) of mass m is free to rotate around the point O by 360 : The mass is connected to point O by a massless inextensible bar of length l . The mass is also attached to a massless linear spring of constant stiffness k and equilibrium length seq. The spring is attached to the slider B that is free to slide on the bar A: The spring always stays parallel to the (0; h ) axis. There is no friction. The system will be studied in cylindrical coordinates r-the radius measured from O and -the angle measured from the vertical axis passing through O in the counterclockwise direction, as shown in the figure. Figure 1: Phase space: m = 10 ,L = 1 ,h = 5 ,s = 0 . 1 ,k = 1 A. 10 pts Derive the Newtons equation of motion for the mass m for both the radial direction r and the tangential direction : Recall that the radial acceleration and tangential acceleration are given by a r = r- r 2 , a = r + 2 r Newtons equations of motion are: m a r = m r- r 2 =- T + k ( h- r cos - s eq )cos m a = m r + 2 r =- k ( h- r cos - s eq )sin Where T is tension in the rod. B. 10 pts Rewrite the equations as a system of first order ordinary differential equations....
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