4 Force System Resultants Free Online FlashCards

What does an equivalent force-couple system preserve?

An equivalent force-couple system preserves the original system’s total force and total moment, and therefore its external effect on a rigid body.

How do you calculate a system’s resultant force?

The resultant force is the vector sum of all forces: $\mathbf{R}=\sum_i \mathbf{F}_i$.

How do you calculate the resultant moment about $O$?

About point $O$, sum each force’s moment and every applied couple: $\mathbf{M}_O=\sum_i(\mathbf{r}_i\times\mathbf{F}_i)+\sum_j\mathbf{C}_j$.

How does an applied couple contribute to a resultant moment?

Add an applied couple’s moment directly to the resultant moment; it is a free vector and contributes the same moment about any reference point.

What is the planar moment formula for a force at $(x,y)$?

With counterclockwise moments positive, a force at $(x,y)$ with components $(F_x,F_y)$ contributes $M_O=xF_y-yF_x$.

How do you transfer the resultant moment from $O$ to $P$?

If $\mathbf{r}_{OP}$ points from $O$ to $P$, then $\mathbf{M}_P=\mathbf{M}_O-(\mathbf{r}_{OP}\times\mathbf{R})$.

What happens to the resultant force when the reference point changes?

The resultant force remains $\mathbf{R}$. Changing the reference point changes the moment assigned to the equivalent system, not the physical loading it represents.

What is the resultant force in the worked example?

The forces sum to $\mathbf{R}=(4,-3)\,\text{kN}$: $4\,\text{kN}$ in the $+x$ direction and $3\,\text{kN}$ downward.

What is the resultant moment in the worked example?

Taking counterclockwise as positive, the total is $M_O=-15\,\text{kN}\cdot\text{m}$, or a $15\,\text{kN}\cdot\text{m}$ clockwise moment.

How can a planar force-couple system with nonzero resultant force be simplified?

In a planar system with $\mathbf{R}\ne\mathbf{0}$, the force’s line of action can generally be shifted so the force alone creates the same moment.

What does a system with zero resultant force but nonzero moment reduce to?

If $\mathbf{R}=\mathbf{0}$ but $\mathbf{M}_O\ne\mathbf{0}$, the system reduces to a pure couple.

What does a zero moment at a point imply when the resultant force is nonzero?

If $\mathbf{R}\ne\mathbf{0}$ and the resultant moment is zero at a point, the resultant force acts through that point.