A small, 300-g collar D can slide on portion AB of a rod which is bent as shown. Knowing that α = 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

Question

A small, 300-g collar D can slide on portion AB of a rod which is bent as shown. Knowing that [latex]\alpha=40°[/latex] 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.

Accepted Answer

First note [latex]\quad u_{D}=r \dot{ heta}_{A B C}[/latex]

[latex]+\uparrow \Sigma F_{y}=0: \quad N \sin 40^{\circ}-W=0[/latex]

or [latex]\quad\quad\quad\quad N=\frac{m g}{\sin 40^{\circ}}[/latex]

[latex]\overset{+}{\longrightarrow } \Sigma F_{n}=m a_{n}: \quad N \cos 40^{\circ}=m \frac{ u_{D}^{2}}{r}[/latex]

or [latex]\quad\frac{m g}{\sin 40^{\circ}} \cos 40^{\circ}=m \frac{\left(r \dot{ heta}_{A B C} ight)^{2}}{r}[/latex]

or

[latex]\begin{aligned}r & =\frac{g}{\dot{ heta}_{A B C}^{2}} \frac{1}{ an 40^{\circ}} \& =\frac{9.81 \mathrm{~m} / \mathrm{s}^{2}}{(5 \mathrm{\ rad} / \mathrm{s})^{2}} \frac{1}{ an 40^{\circ}} \& =0.468 \mathrm{~m}\end{aligned}[/latex]

or [latex]\quad\quad\quad\quad r=468 \mathrm{~mm}\blacktriangleleft[/latex]

Question 12.58: A small, 300-g collar D can slide on portion AB of a rod whi...

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