Determine the resultant of the coplanar force system shown in Fig. (a) that acts on the arm of an excavator. Show your answer on a sketch of the coordinate system.

Question

Accepted Answer

For a coplanar force system, the resultant is either a force or a couple. If the resultant is a force, then Eqs. (3.5) provide three scalar equations: [latex]R_{x}=ΣF_{x}[/latex], [latex]R_{y}=ΣF_{y}[/latex], and [latex]ΣM_{O}=Rd[/latex].

We see that there are no unknown quantities in the original force system.

Therefore, if the resultant is a force, the three unknowns in this problem will be [latex]R_{x}[/latex], [latex]R_{y}[/latex], and [latex]d[/latex], which could be determined from the three scalar equations. Referring to Fig. (a), the three equations become

[latex]R_{x}=ΣF_{x}\:\:\:\:\overset{ iny+}{\longrightarrow}\:\:\:\:R_{x}=80\cos 40^\circ −50=11.284\:kN[/latex]

This equation is sufficient to tell us that the resultant is a force, not a couple; if the resultant were a couple, [latex]ΣF_{x}[/latex] would be zero.

[latex]R_{y}=ΣF_{y}\:\:\:\:{ iny+}\big\uparrow\:\:\:\:R_{y} =80\sin 40^\circ −60=−8.577\:kN[/latex]

[latex]ΣM_{O}=Rd\:\:\:\:\underset{+}{\large \curvearrowleft}\:\:\:\:ΣM_{O}=350−(80\cos 40^\circ)(3.3)−60(3)=−32.24\:kN\cdot m[/latex]

Therefore, the resultant [latex]R[/latex] is

[latex]θ= an^{−1}\frac{8.577}{11.284}=37.2^\circ[/latex]

Because [latex]ΣM_{O}[/latex] is negative (i.e., clockwise), the resultant [latex]R[/latex] must also provide a clockwise moment about [latex]O[/latex], as shown in Fig. (b). Therefore we obtain

[latex]ΣM_{O}=Rd\:\:\:\:\overset{\large\curvearrowright}{ iny+}\:\:\:\: 32.24=14.174d[/latex]

which gives

[latex]d=2.27\:m[/latex]

The final result is shown in Fig. (b).

Question 3.5: Determine the resultant of the coplanar force system shown i...

Related Answered Questions

The force system shown consists of the couple C and four forces. If the resultant of this system is a 500-lb · in. counterclockwise couple, determine P, Q, and C. ...

Verified Answer:

Verified Answer:

Determine the resultant of the line load acting on the beam shown in Fig. (a). ...

Verified Answer:

Determine the wrench that is equivalent to the force system described in Sample Problem 3.1. Find the coordinates of the point where the axis of the wrench crosses the xy-plane. ...

Verified Answer:

The parallel force system in Fig. (a) consists of the three forces shown and the 1250-N·m couple. (1) Determine the resultant, and show it on a sketch of the coordinate system. (2) Determine the resultant if the direction of the 100-N force is reversed. ...

Verified Answer:

The values of ΣFz, ΣMx, and ΣMy for three force systems that are parallel to the z-axis are as follows: Determine the resultant of each force system and show it on a sketch of the coordinate system. ...

Verified Answer:

The force R is the resultant of the other three concurrent forces shown. Determine P and R. ...

Verified Answer:

The values of Rx = ΣFx, Ry = ΣFy, and ΣMO for five force systems lying in the xy-plane are listed in the following table. Point O is the origin of the coordinate system, and positive moments are counterclockwise. Determine the resultant for each force system, and show it on a sketch of ...

Verified Answer:

The plate is acted on by four parallel forces, three of which are shown in Fig. (a). The fourth force P and its line of action are unknown. The resultant of this force system is the couple-vector C^R = −1100i + 1500j lb · ft. Determine P and its line of action. ...

Verified Answer:

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. ...

Verified Answer: