13-111. A 0.2-kg spool slides down along a smooth rod. If

Chapter 13, Problem 13-111

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A 0.2-kg spool slides down along a smooth rod. If the rod has a constant angular rate of rotation \(\dot{\theta}=2 \mathrm{\ rad} / \mathrm{s}\) in the vertical plane, show that the equations of motion for the spool arc \(\ddot{r}-4 r-9.81 \sin \theta=0 \text { and } 0.8 \dot{r}+N_{s}-1.962 \cos \theta=0 \text {, where } N_{s}\) is the magnitude of the normal force of the rod on the spool. Using the methods of differential equations, it can be shown that the solution of the first of these equations is \(r=C_1e^{-2t}+C_2e^{2t}-(9.81/8)\sin2t\text{. If }r,\ r\text{, and }\theta\) are zero when t = 0, evaluate the constants \(C_{1} \text { and } C_{2}\) determine r at the instant \(\theta=\pi / 4 \mathrm{\ rad}\).