Statics Exercise

Classical Mechanics Level 5

Two identical, uniformly massive rods are welded together at their end points to form an incomplete equilateral triangle (see above). The other end of each rod is connected to a flat horizontal surface by a pin support (not explicitly shown).

Assume that the system is in stasis. If the mass of each rod is $m$ and there is an ambient downward gravity field $g$ , the magnitude of the net reaction force at each pin support can be written as ${\sqrt{\frac{a}{b}} \,mg}.$ If $a$ and $b$ are coprime positive integers, determine $a+b$ .

Inspiration .

The answer is 25.

1 solution

Rohit Gupta
Feb 12, 2017

Let us draw the free body diagram of the left rod,

The force by the pin support is the resultant of the two forces $F_1$ and $F_2$ . From the vertical equilibrium of the rod, we can say that, $F_1 = mg$ . As the rod remains in equilibrium, the net torque about the pin support $O$ equals zero. Therefore, $mg \frac{L}{4} = F_2 \frac{\sqrt{3}L}{2} \Rightarrow F_2 = \frac{mg}{\sqrt{12}}$ The net force by the pin support is $\begin{aligned} F_{\text{net}} &= \sqrt{{F_1}^2 + {F_2}^2} \\ &= \sqrt{(mg)^2 + \frac{(mg)^2}{12} } = \sqrt{\frac{13}{12}}mg \\ & \Rightarrow a+b = \boxed{25}. \\ \end{aligned}$

So now, what if it was a "fixed support" instead of a pin support? Since the support could generate two forces AND a moment, there could be many possible values for F2 and the support could generate an extra moment to balance the torque. In such a case, I believe we would have an indeterminate system. Although it's easy to imagine constructing this very setup in real life. We could weld two rods together and cement the ends to the surface. And nature wouldn't have any trouble deciding what to do. So how do we resolve the paradox?

Steven Chase - 4 years, 3 months ago

I do not have a very concrete answer to resolve this paradox. However, I would like to bring up another paradox here.

Imagine a box placed on a rough table in contact with a vertical wall. We apply a force $F$ as shown, All we can say here is $N + f = F.$
But, what will be the actual value of $N$ and $f$ . I think it is also indeterminate.

Rohit Gupta - 4 years, 3 months ago

Based on some reading I've done, it sounds like these paradoxes arise because of our assumption of perfect rigidity in the objects. When material deformations are accounted for, extra constraints are imposed on the system that force unique solutions.

Steven Chase - 4 years, 3 months ago

Statics Exercise

The answer is 25.

1 solution

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