The retractable shelf shown is supported by two identical linkage-and-spring systems; only one of the systems is shown. A 20-kg machine is placed on the shelf so that half of its weight is supported by the system shown. If the springs are removed and the system is released from rest, determine (a)

Question

The retractable shelf shown is supported by two identical linkage-and-spring systems; only one of the systems is shown. A 20-kg machine is placed on the shelf so that half of its weight is supported by the system shown. If the springs are removed and the system is released from rest, determine (a) the acceleration of the machine, (b) the tension in link AB. Neglect the weight of the shelf and links.

Accepted Answer

The links AB and CE keep the line AC on the shelf parallel to the fixed line BE. Thus the shelf moves in curvilinear translation.

[latex]\alpha=0\qquad\qquad{\mathbf{a}}_{G}{\mathrm{~is~perpendicular~to~}}{\overline{{A B}}}[/latex]

Since link AB is a massless two-force member, the force at A is along link AB.

Since link CDE is massless and the spring DF is removed, the force at C is directed along the link CDE.

Consider the kinetics of the shelf.

Force perpendicular to link AB.[latex]\quad \ ⦨30^{\circ}[/latex]

[latex]m g\cos30^{\circ}=m{\overline{{a}}}[/latex]

[latex]\overline{{{a}}}=g\cos30^{\circ}=(9.81\;\mathrm{m/s})\cos30^{\circ}=8.4957\;\mathrm{m/s}^{2}[/latex]

[latex]\overline{{{\mathbf{a}}}}=8.50~\mathrm{m}/{\mathbf{s}}^{2}\ ⦪60^{\circ}\blacktriangleleft[/latex]

(a) Acceleration of the machine:

[latex]\overset{\curvearrowright }{+} \Sigma M_{C}=\Sigma(M_{C})_{\mathrm{eff}}\colon~(F_{A}\cos30^{\circ})(0.080)+(F_{A}\sin30^{\circ})(0.100)-m g(0.150)[/latex]

[latex]=(m a\sin30^{\circ})(0.160)+(m a\cos30^{\circ})(0.150)[/latex]

[latex]0.11928F_{A}-0.150\,m g=-0.04990\,m g\cos30^{\circ}[/latex]

(b) Tension in link AB. [latex]\quad F_{A}=89525\,m g[/latex]

Taking mg to be half the weight of the machine,

[latex]m g={\frac{1}{2}}(20\,\mathrm{kg})(9.81\,\mathrm{m/s}^{2})=98.1\,\mathrm{N}[/latex]

[latex]F_{A}=(0.89522)(98.1\mathrm{~N})[/latex]

[latex]F=87.8\ N\blacktriangleleft[/latex]

Question 16.13: The retractable shelf shown is supported by two identical li...

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