A conveyor system is fitted with vertical panels, and a 15-in. rod AB weighing 5 lb is lodged between two panels as shown. Knowing that the acceleration of the system is 3 ft/s² to the left, determine (a) the force exerted on the rod at C, (b) the reaction at B.

Question

A conveyor system is fitted with vertical panels, and a 15-in. rod AB weighing 5 lb is lodged between two panels as shown. Knowing that the acceleration of the system is 3 ft/s² to the left, determine (a) the force exerted on the rod at C, (b) the reaction at B.

Accepted Answer

Geometry:

[latex]\begin{aligned}& {\overline{{C B}}}=d={\frac{10\;\mathrm{in}.}{\cos20^{\circ}}}=10.642{\mathrm{~in}}.\& b={\frac{1}{2}}(15\mathrm{~in.})\sin20^{\circ}=2.5652\mathrm{~in.} \& c=\frac{1}{2}(15\mathrm{~in.)cos}\,20^{\circ}=7.0477\mathrm{~in}.\end{aligned}[/latex]

Mass: [latex]\quad m={\frac{W}{g}}={\frac{5\ \mathrm{lb}}{32.2\ \mathrm{ft}/s^{2}}}=0.15528\ \mathrm{slug}[/latex]

Kinetics: [latex]\quad m{\overline{{\mathbf{a}}}}=(0.15528{\mathrm{~slug}})(3{\mathrm{~ft}}/{\mathrm{s}}^{2})=0.4658{\mathrm{~lb}}\longleftarrow [/latex]

(a) Force at C.

[latex]\begin{aligned}& +\circlearrowright \Sigma M_B=\Sigma(M_{_B})_{\mathrm{eff}}\,;\quad C d-W b=-m\overline{{{a}}}c\& C={\frac{W b}{d}}-{\frac{m\overline{{a}}c}{d}}={\frac{(5\ \mathrm{lb})(2.5652\ \mathrm{in}.)}{10.642\ \mathrm{in}.}}-{\frac{(0.46588\ \mathrm{lb})(7.0477\ \mathrm{in}.)}{10.642\ \mathrm{in}.}}\& C=0.89669\ \mathrm{lb} & {\bf C}=0.897\ \mathrm{lb}\ ⦨20^{\circ}\blacktriangleleft\end{aligned}[/latex]

(b)Reaction at B.

[latex]\begin{aligned}& +\uparrow \Sigma F_{y}=\Sigma(F_{y})_{\mathrm{cff}}\,;~~~B_{y}-W+C\sin20^{\circ}=0\& \quad\quad\quad\quad\quad\quad B_{y}=5\mathrm{~lb-(0.89669\mathrm{\ lb})\,sin}\,20^{\circ}=4.6933\mathrm{\ lb}\& \stackrel{+}{\longleftarrow}\Sigma F_{x}=\Sigma(F_{x})_{\mathrm{eff}}\colon\ \ B_{x}-C\cos20^{\circ}=m a\& \quad\quad\quad\quad\quad\quad B_{x}=(0.89669\mathrm{\ lb})\cos20^{\circ}+0.46588=1.3085\mathrm{\ lb} \ \boldsymbol{B}=1.3085 ext{ lb}←+4,6933 ext{ lb}↑& \quad\quad \mathbf{B}=4.87\ \mathrm{lb}\ ⦩74.4^{\circ}\blacktriangleleft\end{aligned}[/latex]

Question 16.2: A conveyor system is fitted with vertical panels, and a 15-i...

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