Representing a System by a Simpler Equivalent System System 1 in Fig. 4.44 consists of two forces and a couple acting on a pipe. Represent system 1 by (a) a single force acting at the origin O of the coordinate system and a single couple and (b) a single force.

Question

Representing a System by a Simpler Equivalent System

System 1 in Fig. 4.44 consists of two forces and a couple acting on a pipe. Represent system 1 by (a) a single force acting at the origin O of the coordinate system and a single couple and (b) a single force.

Strategy
(a) We can represent system 1 by a force F acting at the origin and a couple M (system 2 in Fig. a) and use the conditions for equivalence to determine F and M.

(b) Suppose that we place the force F with its point of application a distance D along the x axis (system 3 in Fig. b). The sums of the forces in systems 2 and 3 are equal. If we can choose the distance D so that the moment about O in system 3 equals M, system 3 will be equivalent to system 2 and therefore equivalent to system 1.

Accepted Answer

(a) The conditions for equivalence are

[latex](\sum{F} )_{2} = (\sum{F} )_{1} [/latex] :

F = 30j + (20i + 20j) = 20i + 50j (kN),

and

[latex] (\sum{M_{O}} )_{2} = (\sum{M_{O}} )_{1}[/latex] :

M= (30 kN)(3 m) + (20 kN)(5m) + 210 kN-m

= 400 kN-m.

(b) The sums of the forces in systems 2 and 3 are equal. Equating the sums of the moments about O,

[latex] (\sum{M_{O}} )_{3} = (\sum{M_{O}} )_{2}[/latex] :

(50 kN)D = 400 kN-m.

we find that system 3 is equivalent to system 2 if D = 8 m.

Discussion
To represent the system by a single force in (b), we needed to place the line of action of the force so that the force exerted a 400 kN-m counterclockwise moment about O. Placing the point of application of the force a distance D along the x axis was simply a convenient way to accomplish that.

Question 4.18: Representing a System by a Simpler Equivalent System System ...

Related Answered Questions

The radius R of the steering wheel in Fig. 4.50 is 200 mm. The distance from O to C is 1 m. The center C of the steering wheel lies in the x-y plane. The force F = sin α(10i + lOj — 5k) N. Deteimine the value of α at which the magnitude of the moment of F about the shaft OC of the steering ...

Verified Answer:

Determining Unknown Forces Two forces A and B and a 200 ft-lb couple act on the beam in Fig. 4.31. The sum of the forces is zero, and the sum of the moments about the left end of the beam is zero. What are the forces A and B? ...

Verified Answer:

Sum of the Moments Due to Two Couples Determine the sum of the moments exerted on the pipe in Fig. 4.32 by the two couples. ...

Verified Answer:

Distance Between the Lines of Action The force F in Fig.4.33 is -20i + 20j + 10k (lb). (a) What moment does the couple exert on the bracket? (b) What is the perpendicular distance D between the lines of action of the two forces? ...

Verified Answer:

Representing a Force by a Force and Couple System 1 in Fig. 4.43 consists of a force FA = 10i + 4j – 3k (lb) acting at A. Represent it by a force acting at B and a couple. ...

Verified Answer:

Determining Whether Systems are Equivalent Two systems of forces and moments are shown in Fig. 4.38, where FA = -10i + 10j – 15k (kN), FB = 30i + 5j + 10k (kN), M = -90i + 150j + 60k (kN-m), Fc = 10i – 5j + 5k (kN), FD = 10i + 20j – 10k (kN). Are they equivalent? ...

Verified Answer:

Determining Whether Systems are Equivaient Two systems of forces and moments act on the rectangular plate in Fig. 4.37. Are they equivalent? ...

Verified Answer:

Determining Whether Systems are Equivalent Three systems of forces and moments act on the beam in Fig. 4.36. Are they equivalent? ...

Verified Answer:

Representing a Force and Couple by a Wrench The system in Fig. 4.49 consists of the force and couple F = 3i + 6j + 2k (N), M = 12i + 4j + 6k (N-m). Represent it by a wrench, and determine where the line of action of the wrench’s force intersects the x-z plane. ...

Verified Answer:

Representing Parallel Forces by a Single Force System 1 in Fig. 4.46 consists of parallel forces. Suppose you want to represent it by a force F (system 2). What is F, and where does its line of action intersect the x-z plane? ...

Verified Answer: