A uniform rectangular plate has a mass of 5 kg and is held in position by three ropes as shown. Knowing that θ = 30°, determine, immediately after rope CF has been cut, (a) the acceleration of the plate, (b) the tension in ropes AD and BE.

Question

A uniform rectangular plate has a mass of 5 kg and is held in position by three ropes as shown. Knowing that [latex] heta= 30°[/latex], determine, immediately after rope CF has been cut, (a) the acceleration of the plate, (b) the tension in ropes AD and BE.

Accepted Answer

(a) Acceleration [latex]\quad +\ ⦫ 30^{\circ}\Sigma F=\Sigma F_{eff}:mg\sin30^{\circ}=m\bar{a}[/latex]

[latex]\bar{a}=0.5 \ g=4.905\mathrm{\ m}/\mathrm{s}^2\quad\bar{\mathbf{a}}=4.91\mathrm{\ m}/\mathrm{s}^2\ ⦫30^{\circ}\blacktriangleleft[/latex]

(b) Tension in ropes

[latex]+\circlearrowleft\Sigma M_{A}=\Sigma(M_{A})_{eff}:(T_B\cos30^{\circ})(0.3\mathrm{\ m})-mg(0.15\mathrm{\ m})=-m\bar{a}(\cos30^{\circ})(0.12\mathrm{\ m})-m\bar{a}(\sin30^{\circ})(0.15\mathrm{\ m})[/latex]

[latex]0.2598T_{B}-(5\mathrm{~kg})(9.81\mathrm{~m/s^{2}})(0.15\mathrm{~m})=-(5\mathrm{~kg})(4.905\mathrm{~m/s^{2}})(0.1039+0.075)[/latex]

[latex]0.2598T_{B}\,-\,7.35\,75=-4.388[/latex]

[latex]T_{B}=+11.43\ \mathrm{~N}\quad\quad T_{BE}=11.43\mathrm{\ N}\blacktriangleleft[/latex]

[latex]+\ ⦩10^{\circ}\Sigma F=\Sigma F_{eff}:T_A+11.43\mathrm{\ N}-mg\cos30^{\circ}=0[/latex]

[latex]T_{A}+11.43\mathrm{N}-(5\mathrm{\ kg})(9.81)\cos30^{\circ}=0[/latex]

[latex]T_{A}+11.43\mathrm{\ N}-42.48\mathrm{\ N}=0[/latex]

[latex]T_{A}=31.04\mathrm{\ N}\quad\quad T_{AD}=31.0\mathrm{\ N}\blacktriangleleft[/latex]

Question 16.14: A uniform rectangular plate has a mass of 5 kg and is held i...

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