Over at Tor.com, Kate has begun a chapter-by-chapter re-read of The Hobbit , and has some thoughts on Chapter 1. It's full of interesting commentary about characters and literary technique, but let's get right to the important bit: Physics!

Kate mentions in passing in the post that the Hobbit style round door with a knob in the middle seems a suboptimal design choice, however pretty it may look once Peter Jackson's set designers get done with it. This draws a couple of comments noting that the doorknob-in-the-middle thing is an English affectation, and pointing to the Prime Minister's residence at Number Ten Downing St. as an example:

Number Ten Downing St. Note the suboptimal knob location.

(Image from this blog post.)

My comment to Kate when she mentioned this was "See, that's why we had a Revolution-- to get out from under people dumb enough to do that."

Why does this matter at all? Well, because when you push open a door, what you're really doing is trying to make a solid object rotate about an axis defined by the hinges. For making something rotate, what matters is not just the force, but the torque you exert, which is the product of the force you exert and the distance from the rotation axis to the point where you exert that force.

To most effectively open a door, then, you want to push on it as far away from the hinges as you can manage. Which is one reason why doorknobs are usually located on the opposite side of the dor from the hinges, and also why it's really annoying when office buildings have those glass doors where you can't easily tell the location of the hinge as you approach. If you guess wrong, you end up trying to open the door by pushing on the near edge, and end up looking like an idiot as you walk smack into the glass. I use this in class as an example of how physics turns up in unexpected places in the design of everyday objects.

So, having established that putting the knob in the middle is kind of dumb from a physics perspective, how can we wring some more physics out of this. Well, Kate raised an interesting point: it might actually be the case that the knob-in-the-center for a round door is less dumb than for an ordinary rectangular door. A round door is, after all, exactly as wide as it is tall, which makes for dramatic framing of arriving dwarves:

Fili and Kili arrive at Bag End.

(Image from this collection of PR stills)

It also gives you a longer lever arm-- halfway across a round door is farther from the hinge than halfway across a narrower rectangular door. Then again, a round door probably has more surface area than a rectangular door of the same height, meaning it contains more material, and would thus be harder to move. So, which is it?

The relevant quantity for our purposes is the change in the angular speed of the door (ω, the rate at which it's rotating) for a given applied force. This is determined from the Angular Momentum Principle which relates torque to change in angular momentum:

$latex \Delta L = \tau \Delta t$

Where τ is the torque exerted, Δ t is the time the torque is applied, and L is the symbol for angular momentum, for some obscure reason. We can write this in terms of force and distance and the "moment of inertia" I, which is the analogue of mass for a rotating system:

$latex rF\Delta t = I \Delta \omega $

So, what's this "moment of inertia" thing, I? Well, it depends on the mass of the object and also the distribution of that mass relative to the axis of rotation. You can find tables of the moments of inertia for common objects in lots of references, such as Hyperphysics and Wikipedia. These are usually listed for rotations about the center of mass, but you can use the parallel axis theorem to figure out the moment of inertia for doors of different shapes.

For a circular door of radius R and mass M, then, we have:

$latex \frac{R}{2}F \Delta t = \frac{1}{2}MR^{2} \Delta \omega $

Which gives us a change in rotational velocity:

$latex \Delta \omega = \frac{F \Delta t}{MR} $

The corresponding calculation for a rectangular door of width W is:

$latex \frac{W}{2}F \Delta t = \frac{1}{3}MW^{2} \Delta \omega $

Which gives us a change in rotational velocity:

$latex \Delta \omega = \frac{3}{2} \frac{F \Delta t}{MW} $

The ratio of these two tells you which would be more efficient, assuming you used the same force for the same amount of time trying to open them:

$latex \frac{\Delta \omega_{rect}}{\Delta \omega_{circ}} = \frac{3}{2} \frac{M_{rect}}{M_{circ}} \frac{R}{W}$

If the two doors have the same mass, you would need the round door to have a radius no more than 2/3rds the width of the rectangular door to break even. Any bigger than that, and the rectangle with a knob in the middle will open more easily than the round door with a knob in the middle.

So, it turns out that British prime ministers have life a little easier than hobbits, at least when it comes to opening their front doors. But then both of them are working twice as hard as they would need to if they had the good physics sense to put the knob on the opposite side from the hinges...

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A couple of other issues relating to weird doors with knobs in the middle:

1) In the comments to the Tor post, it's pointed out (comment #55) that putting hinges on a round door is kind of a hassle. It took a little while to find an image where you could see the hinges on the Bag End set door, but I eventually Googled up this collection of observations about the Fellowship of the Ring movie, which includes this image:

Both sides of the front door to Bag End.

This not only shows how the door is hung, but also that Jackson's set designers have better sense than hobbits, and put the interior knob on the outer edge, where it belongs.

2) Of course, physics isn't the only reason to put the knob at the edge of the door-- it also allows you to easily integrate the latching mechanism to hold the door closed with the knob. If you want a centered knob, you have to either have the latch controlled elsewhere, or have some really long connections between the knob in the middle and the edge of the door.

3) For extra credit, use one of the images above to estimate the size and mass of the door to Bag End, and determine whether it would, in fact, be more difficult to open than a reasonable estimate of the size and mass of the door to Number Ten Downing St.. Show all your work for full credit.

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(Peter Jackson picture at the top of this post from this page.)