A small, 300-g collar D can slide on portion AB of a rod which is bent as shown. Knowing that \alpha=40° and that the rod rotates about the vertical AC at a constant rate of 5 rad/s, determine the value of r for which the collar will not slide on the rod if the effect of friction between the rod and the collar is neglected.
Question 12.58: A small, 300-g collar D can slide on portion AB of a rod whi...
First note v_D=r \dot{\theta}_{A B C}
+\uparrow \Sigma F_y=0: \quad N \sin 40^{\circ}-W=0or N=\frac{m g}{\sin 40^{\circ}}
\xrightarrow{+} \Sigma F_n=m a_n: \quad N \cos 40^{\circ}=m \frac{v_D^2}{r}or \frac{m g}{\sin 40^{\circ}} \cos 40^{\circ}=m \frac{\left(r \dot{\theta}_{A B C}\right)^2}{r}
or \begin{aligned} r & =\frac{g}{\dot{\theta}_{A B C}^2} \frac{1}{\tan 40^{\circ}} \\ & =\frac{9.81 \mathrm{~m} / \mathrm{s}^2}{(5 \mathrm{rad} / \mathrm{s})^2} \frac{1}{\tan 40^{\circ}} \\ & =0.468 \mathrm{~m} \end{aligned}
or r = 468 mm
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