Hobbits and Prime Ministers: The Physics of Doors

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...


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.


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

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One way around the hinge problem would be to mount the door on a center pivot like a butterfly valve. Of course, that would change putting the knob in the middle from merely inconvenient, to downright unworkable. Plus it would cut the door opening in half. But at least the weight of the door wouldn't tend to rip off the hinges.

By Tim Eisele (not verified) on 16 Nov 2012 #permalink

In the case of Number 10 the central knob makes some sense; the door can't be opened from the outside and is always opened by a policeman stationed inside, watching CCTV cameras to make sure he can let people straight in.

Assuming the doors are made of wood of equal density ρ and thickness a [1], the mass of the round door will be ρπr^2a. Rectangular doors are typically a bit more than twice as tall as they are high (the door to my office measures about 2 m tall by 0.9 m wide)--call it a mass of 2.2ρw^2a for the rectangular door. That gives a ratio of (3.3/π)(w/r). So the Prime Minister [2] comes out ahead by about 5% if w and r are equal. But if w and r are equal, that would be a huge hobbit door.

[1] In reality, I would expect the door at Number 10 to have some armor plating that a hobbit would not need. That means a higher density (or at least thickness) for the Prime Minister than for a hobbit.

[2] As Paul says above, the Prime Minister doesn't actually open his door. He has a security detail to do that for him.

By Eric Lund (not verified) on 16 Nov 2012 #permalink

With the round door and central knob arrangement, you could easily set up four or more bolts that extend radially into the frame, operated by a single twist of the knob, much like a naval bulkhead door. With heavy felt (assuming Hobbits lack natural rubber or polymers) weatherstripping, you'd have a nearly watertight seal.

Perhaps Hobbiton is subjected to frequent flooding?

We don't even really need the picture, once we've looked up the necessary reference material - Tolkein writes that hobbits are between two and four feet (0.61–1.22 m) tall; We can assume that like humans, hobbits build doors for the height of the upper end of norm, giving that round door a 1.22m diameter. Using my monitor and a piece of lego I had lying around, I estimate the door to be 1/12th as thick as it's diameter - .1 m. A cylinder with those dimensions encloses a volume of .11 cubic meters, Densities of woods range from 30-100 kg/m^3; Beechwood is a likely candidate (Native to New Zealand), and has a density of about 80 kg/m^3, giving us a weight of about 8.8 kg for the door of Bag End. Heavy, then, for a hobbit, but not unworkable.


By Thomas Hahn (not verified) on 16 Nov 2012 #permalink

These comments are all brilliant.

By Kate Nepveu (not verified) on 16 Nov 2012 #permalink

The reason you might want a knob in the center is if it connects to a crossbar extending the width of the door, something for the security conscious.

Thomas Hahn, you´ve got your wood densities wrong by a factor of ten: Hard wood like oak, or ironwood, approaches or even surpasses 1000 kg/m^3, some don´t even float in seawater.

By wereatheist (not verified) on 17 Nov 2012 #permalink

is 1/2 MR^2 the correct moment of inertia for the round door? For a thin disk rotating about its diameter, inertia = 1/4 MR^2, so I would expect the round door have a moment of inertia of 5/4 MR^2. I believe that would make angular velocity ratio 15R/4W, more than double 3R/2W. If I'm correct (and I have a cold, so I'm a bit foggy), the diameter of the round door would need to be less than half the width of a rectangular door to have the same torque at the outside edge.

I don't want people barging into my house and if putting the door knob in the center of the door makes it more difficult for them to enter then it is a good idea. Ten Downing Street has a policeman standing outside (not present in your picture) and when visitors turn up the door appear to magically open for them and close sharply after they enter.

The circular entrance is itself another feature designed to reduce the probability that more than one person will enter at a time since walking in left/right of center requires stepping over a raised edge and potentially stooping so as not to gang one's head.

Back to door design, what really jumped out at me was how the forces on the door hinge were concentrated over such a small distance and are much closer to the floor than most doors. The surrounding frame would need to be very robust and the hinges very solid to prevent the door rotating and rubbing on the floor. Perhaps the door rubbing against the floor is another security feature making it hard to enter from outside (the door would need to be lifted as well as pushed).

I don't think the frame as shown would be capable of handling the torque generated by such closely spaced hinges, if made of wood, without some serious distortion.

By Derek Jones (not verified) on 18 Nov 2012 #permalink

Regarding the last photo. There is no contradiction here. As is the case with Number 10 and many other doors in the UK, the knob is only for pushing and pulling -- there is no locking/unlocking function. In that still from the film, you can't actually see the far left of the door from the outside, where the keyhole should be!

IME, with a well hung door it doesn't cause a problem. Yes you need some more force but it's still not very much. I know this because I just checked by pushing a door closed[1] from the hinge side, (never mind the middle).

The problem comes about when you have doors with an autoclosing mechanism and you have to press against a spring or counterweight.

[1] Closed because that door isn't quite perfectly hung and tends to open on its own.

By Rick Pikul (not verified) on 19 Nov 2012 #permalink

Physics are nice, but the location of the door knob is really dictated by ergonomics. If you position the door knob on a round door in the traditional spot, you would be unable to open the door with the door knob more than about 20 degrees or so.
You cannot follow the door in if you're standing close to one side, unless you're a very athletic contortionist. And unless you're an orangutan, your arms will be too short to open the door all the way without stepping over the threshold.
Door knob in the middle allows both options.

Indeed, the round knobs in British doors of a certain period do not turn. They are for pulling and pushing only, and you open the door with a key.
Also, I think the "suboptimal knob location" at No. 10 is being resident at No. 10.

By Jeff Rubinoff (not verified) on 12 Dec 2012 #permalink

Mu's comment is spot on. Its the best spot from which to follow the door in.

The inner handle can be on the edge because you are pulling the door and not stepping thru that impractical threshold.