Question 3.SP.7: Replace the couple and force shown by an equivalent single f...
Replace the couple and force shown by an equivalent single force applied to the lever. Determine the distance from the shaft to the point of application of this equivalent force.
STRATEGY: First, replace the given force and couple by an equivalent force-couple system at O. By moving the force of this force-couple system a distance that creates the same moment as the couple, you can then replace the system with one equivalent force.
MODELING and ANALYSIS: To replace the given force and couple, move the force F = −(400 N)j to O, and at the same time, add a couple of moment \pmb{M}_O that is equal to the moment about O of the force in its original position (Fig. 1). Thus,
\pmb{M}_O=\overrightarrow{OB} × F =[(0.150 m)i +(0.260 m)j]×(−400 N)j
= −(60 N⋅m) k
When you add this new couple to the couple of moment −(24 N⋅m) k formed by the two 200-N forces, you obtain a couple of moment −(84 N⋅m) k (Fig. 2). You can replace this last couple by applying F at a point C chosen in such a way that
−(84 N⋅m) k= \overrightarrow{OC} × F
= [(OC)cos 60°i +(OC)sin 60°j]×(−400 N)j
= −(OC)cos 60°(400 N) k
The result is
(OC)cos 60° = 0.210 m = 210 mm
OC = 420 mm
REFLECT and THINK: Because the effect of a couple does not depend on its location, you can move the couple of moment −(24 N⋅m) k to B, obtaining a force-couple system at B (Fig. 3). Now you can eliminate this couple by applying F at a point C chosen in such a way that
−(24 N⋅m) k= \overrightarrow{BC} × F
= −(BC)cos 60°(400 N) k
The conclusion is
(BC)cos 60° = 0.060 m = 60 mm BC = 120 mm
OC = OB + BC = 300 mm + 120 mm
OC = 420 mm
