Suppose there is a (uniform rectangular) plank of mass $m$ and total length $L$ that is on top of a building with some piece overhanging or jutting out past the building. Assume it isn't past the point where it tips over, so assume it stays in place. See the image below.
Let's say the overhanging piece is of length $L_{1}$ and the piece on top of the building is $L_{2}$. There are two forces acting on this plank: gravity and the normal force.
Gravity acts on the entire plank uniformly, so the resultant of the gravitational force will be $F = mg$ downward at the center of the plank, i.e. at $x = L/2$ in the images below.
Now my question is, where does the resultant of the normal force go? Should it be $N$ upwards at $x = L/2$ or at $x = L_{2}/2$? Moreover, what is the magnitude of $N$?
Remember that the plank doesn't tip over, so both the net torque (relative to, say, the pivot at $x = L_{2}$) and the net force have to be zero.
- If the normal force acts at $x = L_{2}/2$, then since the net torque has to be zero, we must have $N < mg$. But if $N < mg$ then the overall net force on the plank body is nonzero, so it must move.
- If the normal force acts at $x = L/2$, then we can set $N = mg$ to get zero net torque and zero net force. However, this doesn't quite make sense to me, because the normal force is only acting on the $L_{2}$ portion of the plank. If it's acting uniformly on the $L_{2}$ piece, shouldn't the resultant act on the middle of that piece? Moreover, if only a fraction of the plank is in contact with the top of the building, shouldn't the overall normal force be less than the usual $mg$?
Which placement and magnitude of the resultant normal force is correct? And why?
Best Answer
Adding to Chemomechanics' answer, the resulting forces is dependent also on the plank's geometry and material. Here's the behaviour of a particular material and geometry of a plank.
Fine, it's actually a piece of paper. But a piece of paper is great because the deformation is highly exaggerated and we can see it with our eyes. A real plank would have similar but less obvious deformations.
As you can see, there's a point force on the ledge and some distributed force at the right end. There's obviously none in the middle because the plank is curved away from the surface.
If the plank is very long, then far enough to the right you'd expect it to behave as if there was no ledge and it would lie completely flat on the surface and have more or less uniformly distributed normal force equal to its weight.