A small, 200-g collar D can slide on portion AB of a rod which is bent as shown. Knowing that the rod rotates about the vertical AC at a constant rate and that α = 30° and r = 600 mm, determine the range of values of the speed ν for which the collar will not slide on the rod if the coefficient of

Question

A small, 200-g collar D can slide on portion AB of a rod which is bent as shown. Knowing that the rod rotates about the vertical AC at a constant rate and that α = 30° and r = 600 mm, determine the range of values of the speed v for which the collar will not slide on the rod if the coefficient of static friction between the rod and the collar is 0.30.

Accepted Answer

Case 1: [latex]\underline{ u= u_{\min }}[/latex], impending motion downward

[latex]\begin{aligned}+ earrow F_{x}=m a_{x}: \quad N-W \sin 30^{\circ} & =m \frac{ u^{2}}{r} \cos 30^{\circ} \ ext{or}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad N & =m\left\lgroup g \sin 30^{\circ}+\frac{ u^{2}}{r} \cos 30^{\circ} ight group \+ warrow \Sigma F_{y}=m a_{y}: \quad F-W \cos 30^{\circ} & =-m \frac{ u^{2}}{r} \sin 30^{\circ} \ ext{or}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad F & =m\left\lgroup g \cos 30^{\circ}-\frac{ u^{2}}{r} \sin 30^{\circ} ight group\end{aligned}[/latex]

Now [latex]\quad\quad\quad\quad F=\mu_{s} N[/latex]

Then [latex]\quad m\left\lgroup g \cos 30^{\circ}-\frac{ u^{2}}{r} \sin 30^{\circ} ight group=\mu_{s} imes m\left\lgroup g \sin 30^{\circ}+\frac{ u^{2}}{r} \cos 30^{\circ} ight group[/latex]

or

[latex]\begin{aligned} u^{2} & =g r \frac{1-\mu_{s} an 30^{\circ}}{\mu_{s}+ an 30^{\circ}} \& =\left(9.81 \mathrm{~m} / \mathrm{s}^{2} ight)(0.6 \mathrm{~m}) \frac{1-0.3 an 30^{\circ}}{0.3+ an 30^{\circ}}\end{aligned}[/latex]

or [latex]\quad\quad\quad u_{\min }=2.36 \mathrm{~m} / \mathrm{s}[/latex]

Case 2: [latex]\quad \underline{ u= u_{\max }}[/latex], impending motion upward

[latex]\begin{aligned}+ earrow \Sigma F_{x}=m a_{x}: \quad N-W \sin 30^{\circ} & =m \frac{ u^{2}}{r} \cos 30^{\circ} \ ext{or}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad N & =m\left\lgroup g \sin 30^{\circ}+\frac{ u^{2}}{r} \cos 30^{\circ} ight group \+\searrow \Sigma F_{y}=m a_{y}: \quad F+W \cos 30^{\circ} & =m \frac{ u^{2}}{r} \sin 30^{\circ} \ ext{or}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad F & =m\left\lgroup-g \cos 30^{\circ}+\frac{ u^{2}}{r} \sin 30^{\circ} ight group\end{aligned}[/latex]

Now [latex]\quad\quad\quad\quad F=\mu_{s} N[/latex]

Then [latex]\quad m\left\lgroup-g \cos 30^{\circ}+\frac{ u^{2}}{r} \sin 30^{\circ} ight group =\mu_{s} imes m\left\lgroup g \sin 30^{\circ}+\frac{ u^{2}}{r} \cos 30^{\circ} ight group[/latex]

or

[latex]\begin{aligned} u^{2} & =g r \frac{1+\mu_{s} an 30^{\circ}}{ an 30^{\circ}-\mu_{s}} \& =\left(9.81 \mathrm{~m} / \mathrm{s}^{2} ight)(0.6 \mathrm{~m}) \frac{1+0.3 an 30^{\circ}}{ an 30^{\circ}-0.3}\end{aligned}[/latex]

or [latex]\quad\quad\quad u_{\max }=4.99 \mathrm{~m} / \mathrm{s}[/latex]

For the collar not to slide [latex]\quad\quad 2.36\mathrm{\ m}/\mathrm{s} \lt u \lt 4.99\mathrm{\ m}/\mathrm{s}\blacktriangleleft[/latex]

Question 12.59: A small, 200-g collar D can slide on portion AB of a rod whi...

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