Five separate force-couple systems act at the corners of a piece of sheet metal that has been bent into the shape shown. Determine which of these systems is equivalent to a force F = (10 lb)i and a couple of moment M = (15 lb·ft)j + (15 lb·ft)k located at the origin.
Identify the system having force equivalent
Answer to Problem 3.104P
System at corner D is the equivalent force-couple system.
Explanation of Solution
Take counterclockwise torques as positive quantities and clockwise torques as negative quantities.
Refer Figure P3.104.
There are five systems at corners A,D,I,F, and Among these, system at corner F cannot be equivalent. This is because that it does not have force in x-direction. All other systems have force
Write the expression to calculate the force part of force couple system.
Here,
Let check the couple part of force-couple system.
Write the expression to calculate the couple part of force couple system.
Here,
In the given system, at points A,D,G and I, moments in two directions are directly given. So the expression to find
Here,
Rewrite the equation for
Write the determinant form to calculate
Here,
Conclusion:
Substitute
Solve
Calculate
Substitute
Solve
Calculate
Substitute
Substitute
Solve
Calculate
From the above calculations, it is found that system at corner D has force equivalent
Therefore, System at corner D is the equivalent force-couple system.
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