Particle Kinetics: Normal-Tangential, Polar, Cylindrical Coordinates; Solving Accelerating 3D System

LECTURE 05
Here, the equations of motion (EoM: F=ma) for particles in normal-tangential, polar, and cylindrical coordinate systems are stated, along with the kinematic equations for acceleration in each direction of each coordinate system. Formulas are derived to determine the angles between the tangent line to the path of motion and the radial direction and the z direction. An example is completed wherein a ball is pushed up a variable-radius helical ramp by a radial arm driven by a central shaft upon which a known torque is applied. The ramp then transitions into a planar parabolic shape. The contact forces applied to the ball and components of acceleration of the ball are found immediately before and after the transition from the helical portion to the parabolic portion. In the helical portion, the cylindrical equations of motion lead to a 3 equation-3 unknown system that is solved to find the force components and the angular acceleration. To properly set up the equations of motion, angles between the tangent line to the path and the radial direction and the z direction are computed. After the transition to the parabolic portion of the track, equations of motion are written in n-t coordinates. The tangential velocity is found based on the velocity at the end of the helical portion of the track, and the radius of curvature is found based on the equation defining the shape of the parabola. The velocity of the ball and the radius of curvature are used to find the normal component of acceleration, which is then used to find the contact force applied to the ball by the track. The tangential component of acceleration is computed based on the component of the ball's weight that aligns with the tangential direction.

Playlist for MEEN203 (Dynamics):
   • MEMT 203: Dynamics  

This lecture was recorded on June 14, 2018. All retainable rights are claimed by Michael Swanbom.

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