Answer to Question #100332 in Mechanics | Relativity for EIRRA EIRWANI

Question #100332
Q5: Determine, by resolution of forces, the resultant of
the following three coplanar forces acting at a point:
12 kN acting at 32° to the horizontal; 25 kN acting at
170° to the horizontal; 40 kN acting at 240° to the
horizontal
Q6: Determine by resolution of forces the resultant of
the following three coplanar forces acting at a point:
300N acting at 20° to the horizontal; 500N acting at
165° to the horizontal; 600N acting at 250° to the
horizontal.
1
Expert's answer
2019-12-17T09:54:16-0500

Since we deal with forces having both x- and y-components, we can find the resultant by both axles first and then find the force in "polar" form, i.e. its magnitude and angle above horizontal.


Q5:


Ox:Fx=12 cos32+25 cos170+40 cos240==34 kN.Oy:Fy=12 sin32+25 sin170+40 sin240==24 kN.Ox: F_x=12\text{ cos}32^\circ+25\text{ cos}170^\circ+40\text{ cos}240^\circ=\\ =-34\text{ kN}.\\ Oy: F_y=12\text{ sin}32^\circ+25\text{ sin}170^\circ+40\text{ sin}240^\circ=\\ =-24\text{ kN}.

The magnitude:


F=Fx2+Fy2=42 kN,F=\sqrt{F_x^2+F_y^2}=42\text{ kN},

acting at


θ=180+atanFyFx=180+55=235.\theta=180^\circ+\text{atan}\frac{F_y}{F_x}=180^\circ+55^\circ=235^\circ.

Q6 can be solved by analogy:


Ox:Fx==300 cos20+500 cos165+600 cos250==406 kN.Oy:Fy==300 sin20+500 sin165+600 sin250==332 kN.Ox: F_x=\\ =300\text{ cos}20^\circ+500\text{ cos}165^\circ+600\text{ cos}250^\circ=\\ =-406\text{ kN}.\\ Oy: F_y=\\=300\text{ sin}20^\circ+500\text{ sin}165^\circ+600\text{ sin}250^\circ=\\\\ =-332\text{ kN}.

F=Fx2+Fy2==(406)2+(332)2=524 kN,F=\sqrt{F_x^2+F_y^2}=\\=\sqrt{(-406)^2+(-332)^2}=524\text{ kN},

θ=141.\theta=-141^\circ.


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