The Requirements of Equilibrium — Question 3

Question 3

A two-force member transmits force only along its endpoints

A weightless member is acted on only at endpoints $A=(0,0)$ and $B=(2,1)\,\mathrm m$, with no end couples. The force at $B$ has magnitude $10\,\mathrm N$ and points generally from $A$ toward $B$.

1. Find both endpoint forces required for equilibrium.

2. Could the force at $B$ instead be purely vertical with nonzero magnitude?

Question 3 - Solution

A two-force member transmits force only along its endpoints

(a) Derivation and calculation

Force balance requires $\vec F_A=-\vec F_B$. About $A$, $$\vec r_{AB}\times\vec F_B=0,\qquad \vec r_{AB}=(2,1)\ \mathrm m.$$ The stated direction selects $$\vec F_B=10\frac{2\hat x+\hat y}{\sqrt5} =\boxed{(4\sqrt5\hat x+2\sqrt5\hat y)\ \mathrm N},$$ $$\boxed{\vec F_A=-\vec F_B}.$$

(b) Result and interpretation

A nonzero purely vertical force at $B$ would give $$\tau_A=x_BF_{By}-y_BF_{Bx}=2F_{By}\ne0.$$ The force at $A$ has no moment about $A$, so it cannot cancel this torque. A third force or an end couple would be needed.

Physical check

The calculated force has magnitude $\sqrt{80+20}=10$ N and zero moment because $2F_{By}-F_{Bx}=0$.