The coplanar force system in Fig. (a) consists of three forces and one couple. Determine the equivalent force-couple system with the force acting at point $O$ .

Question

The coplanar force system in Fig. (a) consists of three forces and one couple. Determine the equivalent force-couple system with the force acting at point [latex]O[/latex].

Accepted Answer

We will solve this problem with scalar algebra; however, the use of vector algebra would be almost as convenient.
The components of the resultant force [latex]R[/latex] are

[latex]R_{x}=\sum{F_{x}}\:\:\:\:\overset{ iny+}{\longrightarrow}\:\:\:\: R_{x}=50\cos 40^\circ +90-\frac{3}{5}(100)= 68.30\:N[/latex]
[latex]R_{y}=\sum{F_{y}}\:\:\:\:{ iny+}\big\uparrow \:\:\:\: R_{y}=50\sin 40^\circ -\frac{4}{5}(100)=-47.86\:N[/latex]

Thus the resultant force is

[latex]R=68.3i-47.9j\:N[/latex]

The magnitude of [latex]R[/latex] is

[latex]R= \sqrt{(68.30)^2+(-47.86)^2}=83.4\:N[/latex]

and the angle that [latex]R[/latex] makes with the [latex]x[/latex]-axis is

[latex]θ= an^{-1}\frac{47.86}{68.30}=35.0^\circ[/latex]

The force [latex]R[/latex] acting at point [latex]O[/latex] is shown in Fig. (b).

The magnitude of the resultant couple equals the magnitude of the total moment about point [latex]O[/latex] of the original force system. Referring to Fig. (a), we have

[latex]C^R=\sum {M_{O}} \:\:\:\:\underset{+}{\large \curvearrowleft}\:\:\:\: C^R=50\sin40^\circ (0.800)-50\cos40^\circ (0.500)-90(0.600)-40=-87.44\:N\cdot m[/latex]

Therefore,

[latex]C^R=87.4\: N\cdot m[/latex] clockwise

The resultant force-couple system is shown in Fig. (b).

Question 3.2: The coplanar force system in Fig. (a) consists of three forc...

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