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The Physics of Our Back Gate [Uncertain Principles]
It’s winter, and as usually happens in winter, I’m having a hard time opening the gate to our back yard. Why? It’s not the snow, it’s physics.
We have a standing policy that as much as possible, Emmy goes in and out through the back door for walks and small-animal-chasing in the backyard. This has occasionally been lifted, when a tree limb fell on our back gate, and when they were building the deck, but for the most part, when we had to go in and out through the front door, but she mostly knows not to go charging out the front door, which is the whole point.
Access to our fenced back yard is through a fairly wide gate consisting of most of one panel of the 6-foot stockade fence we have in the back (the front corner of the yard has 4′ fence, because of Town of Niskayuna requirements, but the bit where the gate is is allowed to be taller). This is, as you might imagine, kind of heavy, and we have a persistent problem with the gate sagging down to drag on the ground. This is particularly pronounced in the winter, when snow and frost heave raise the level of the path, and snow and ice weigh down the gate, so opening the gate now involves hauling it up a bit in order to be able to push it open about 1/3rd of the way until it hits the ground and won’t move any farther.
The root cause of this is classical physics, as is the likely solution. Which I will attempt to explain here, using this diagram:
Diagrams representing the two- and three-hinge configurations for hanging the gate to our back yard.
On the left in the figure, you see the current configuration of the gate: a large wooden panel supported by two hinges on one side. So, what are the forces that act, here? Well, this is a resolutely classical kind of system, so we have three forces that can act: the weight of the gate, due to the gravitational attraction of the Earth, represented by the green arrow labeled “Fgrav” in the figure, and two contact forces from the hinges.
Now, about two weeks into first-year Newtonian mechanics, any physics student should be able to say that since the gate isn’t moving, these must sum to zero:
Since gravity by definition pulls straight down, and only straight down, this means that the two hinge forces must have vertical components indicated by the purple arrows at the left.
That’s not terribly interesting or useful by itself, though. You need to get toward the end of the term before you learn what really matters here, namely the torque. The force of gravity acts on the center of mass of the gate, but the hinges act on the left edge. The distance between these means that while the net force on the gate is zero, the gate will have a tendency to rotate. And that’s the source of the problem– gravity wants to make the gate rotate in a clockwise manner, which brings the right edge down into the ground, and the hinges need to stop it.
To understand rotation, we need to think about torque, which combines force with the distance from the axis of rotation. The rotation point in this case is the lower hinge, more or less, in which case the torque due to gravity is given by:
where d/2 is half the width of the gate (assuming a uniform density rectangle, blah, blah, blah). This might seem too simple, because formally you would calculate the full distance from the hinge to the center of mass and then take a vector product, but there’s a convenient shortcut, namely using the full force and the perpendicular distance to the line along which the force acts; it saves doing a bunch of trigonometry.
Stopping this rotation must require an equal and opposite torque from the hinges. Now, if we’re taking the lower hinge as the rotation axis, it can’t exert any torque, which means everything has to come from the top hinge. And that, in turn, means that there must be a horizontal component to the top hinge force, acting to the left, whose magnitude we can find from setting the torques equal:
(where y is the distance between hinges, which is not quite the full height of the gate).
Of course, adding a leftward force on the top hinge would cause the whole gate to move to the left unless it’s balanced by a rightward force from something, which in this case must be the bottom hinge. So, to stop the rotation, we have the top hinge pulling left, and the bottom hinge pushing right. And the additional stress this puts on the hinges is what causes the dragging problem– when I look closely at the gate, the top hinge is clearly stretched somewhat to the right, and the bottom hinge is compressed to the left, exactly as you expect from physics. The hinges deform a little, and the gate rotates a little until the ground pushing up on the lower right corner provides the additional torque needed.
So, what to do about this? Well, the quick and easy solution is just to add a third hinge, as shown on the right (aligning the three hinges with the three horizontal rails on the stockade fense we’re using for a gate). You still have the same basic situation– gravitational and hinge forces acting some distance apart– but adding the third hinge will reduce the forces on the other two. The basic force equation thus becomes:
and the torque equation becomes:
As long as Fmid is greater than zero, this has to reduce the total force on the upper hinge, making it less likely to deform enough to drag the right edge of the gate on the ground.
Of course, hanging a big heavy gate with two hinges is kind of annoying, and adding a third hinge is going to be more annoying yet. Which is probably why the guys who fixed it the last time didn’t add the third hinge, despite my specifically suggesting that they should do that. But the physics of the situation is pretty clear, so once it warms up enough to melt the snow by the gate, I’ll recruit some neighbors to help with lifting it into place, and fix the damn thing myself.
Source: http://scienceblogs.com/principles/2015/03/12/the-physics-of-our-back-gate/
