1900 at the Olympics in Paris

François Brandt and Roelof Klein had qualified in the coxed pairs, but had lost a race to a French pair. Thinking it over with an unspecified amount of alcohol they came to the conclusion they couldn't change their rowing performance, but perhaps Hermanus Brockmann was overweight. Brockmann was their coxswain. In the early days the cox was largely responsible for steering and sometimes setting the rhythm of the strokes, but they were frequently seen as dead weight. Brockmann weighed sixty kilograms. The pair spotted a young boy in the crowd and ran a trial. He only weighed thirty-three kilos and they had to attach another five kilos of lead to their shell to make the rudder set properly in the water.

The flying Dutchmen leapt to an early lead but couldn't keep their initial pace. Halfway into the race the French started their move, but at the finish the Dutch were still ahead - but by less than a meter. A dramatic race by any measure, but more curious is the fact that the cox disappeared into the crowd never to be awarded his gold medal.

That brings me to the trigger for the post. *American Experience* in PBS recently aired a piece on the 1936 US men's Olympic rowing team - one of the most amazing bits of magic ever seen at the Olympics. You can watch it here.

How important is that dead weight? In theory a cox aims the boat in the straightest possible line, gives pacing and coaching, and removes the effects of uneven thrust as much as possible. But he or she also adds to the load. We can easily make an estimate.

In doing very quick calculations to test ideas dimensional analysis is frequently invoked. The kid destined to be a natural scientist would look at the units and note that gallons is a volume or a mass. Assuming you are using the same liquid in your comparisons - at least liquids of similar densities - it doesn't matter ... take your pick. Let's pick volume .. the dimension of volume is a length cubed - L^{3}. Miles are easy - just a length. So miles per gallon is dimensionally L/L^{3} or 1/L^{2} - an inverse area. It could be inverse square meters, inverse square feet, or ... hey .. inverse acres has a nice rural feel to it.

The question is how much does it cost for a racing skull to carry a cox? Rather than a boat with two rowers let's consider the more general case with N rowers. A four man boat has twice the power of a two man, but its weight is about twice as much. Does having twice the engine and almost twice the weight help or hinder?

I've received some complaints about being a tad too technical. I'll indent and highlight the reasoning behind the estimate. You can skip it if you like. It turns out to be about two or three minutes of work at the blackboard and a good example of how physicists play. You make simple estimates, often little more than dimensional analyses, to build a simple model. That can give you some insight into the problem and further questions. It is a very quick method for going through dozens of ideas when you're in conjecture and hypothesis land. Here it gives a quick and dirty answer to the question.

The power the rowers deliver mostly is used to overcome the drag on the hull. It is

p = f_{d} * v

where p is power, f_{d} is the force of drag, and v is the boat's velocity relative to the water

The force of drag is reasonably complex and would require some careful measurements that depend on the shape of the hull and some other factors. The important thing to realize it is proportional to the wetted area - something that goes as a length squared - and the square of the velocity.^{ } So

f_{d} ∝ L^{2} * v^{2}

the power needed is

p_{d} ∝ L^{2} v^{2} * v or L^{2}v^{3 }

The power to overcome the drag is supplied by N rowers (assuming all are about equally strong). A racing shell is pretty minimal, but its volume needs to increase to float with the heavier load. The volume goes as L^{3} and also is proportional to N, so L ∝ N^{1/3}. Getting tricky we substitute

p_{d} ∝ L^{2}v^{3} ∝ N^{2/3}v^{3}

getting to the end! .... We know power supplied is proportional to N so

N ∝ v^{3} N^{2/3}

solving for velocity we see

v ∝ N^{1/9}

The speed of the boat only increases slightly with an increasing number of rowers.

For those of you who have rejoined us it only increases at the one ninth power of the rowers or N^{0.111}. Try plugging some numbers into your calculator and see! The time is just to the inverse of the speed over similar distances so t ∝N^{-1/9}. If you look at world record times over similar distances this holds for one to four person coxless skulls reasonably well and is a good enough model for this purpose.

You can play with some numbers yourself - assuming the cox is just along for the ride the impact of having one decreases with crew number. Since races are tight you don't want to race coxed vs uncoxed boats and you don't want a heavy cox. Hopefully they add more value and perhaps you can get a sense of quantifying that value in real vs expected times.

The original question was what is the impact of the cox - the little guy or gal with the rudder and the loud voice? I did a similar analysis increasing the size of the boat, and thus its drag, by going to N + 1 crew members, but assuming only N are contributing to the power. I won't go through the reasoning as it takes much more time to type than to think about it, but it is very similar. The predicted time with a cox, t_{cox} is:

t_{cox} ∝ (N + 1)^{2/9}/N^{1/3}

To get something that is easy to think about look at the ratio of cox and coxless times and apply a bit of rearranging

t_{cox}/t ∝ ((N + 1)/N)^{2/9}

(N + 1) is always bigger than N, so the cox always slows things down ... not by a lot and the effect decreases with increasing N. A two man boat should be about nine percent slower if the weight of the cox is similar to the rowers. But it isn't. The Dutch used a kid who was about a third the weight of each rower, so a reasonable guess would be about (2.33/2)^{2/9 }or something over two percent slower - probably several percent faster than an equal muscled crew with a cox that weighed as much as poor Hermanus Brockmann.

These days coxswains have much greater value. They steer the boat and usually run the strategy. Recently I was thinking about four man (or women) boats - there are coxed and coxless events. Thinking about the pattern of the rowers led to a slightly veiled surprise.

The pattern of the rowers is called the rig of the the boat. It looks very symmetric - right, left, right, left for the four and twice that for the eight (note that left, right, left, right is the same by symmetry). Assuming the crew are evenly matched you would expect the shell to move straight ahead without the intervention of the cox. If this was true the uncoxed four should be a bit faster with the cox only offering on-bard coaching. Of course crew members aren't evenly matched so the cox has to work the rudder.

A bit of thinking about the forces on the oarlocks leads to a slightly hidden asymmetry.

High school physics tells us we can decompose forces into vector components - when examining the flight of a ball you can break the motion into horizontal and vertical components allowing you to isolate the effect of gravity. On a shell there is an oscillating force at each of the oarlocks. When the oar is in the water there is a strong force on the oarlock in the forward direction of the boat's motion. Assume each crew member is equally strong with the same form and perfect timing. In a four man boat we get four times this force to move the boat ahead. Dandy - that's what you usually think about when you calculate the potential speed of a racing shell.

There is another force at the oarlock perpendicular to the forward force. At the first half of the stroke it is directed towards the shell and in the second half it is directed away. The force drops to zero at the midway position of the oar - a sinusoidal force. It is smaller than the forward force and at first glance appears to be balanced by the fact that every oar has an opposing oar on the other side of the shell. Two oscillating forces that are mirror images of each other at any moment. They should cancel each other so we might be tempted to move on.

Here's the nut of the question. The forces act over a distance. We're not concerned with a force on the left balancing a force on the right, but rather each oar produces a torque on the rudder. Now things get interesting.

The torque is just a force on a long arm. Imagine the sideways force is F at any moment in time. If you are distance L from the rudder, the torque at the rudder is F*L (foot-pounds, newton-meters...). Let's give our model a distance r from the rudder - essentially the cox - to the first oarsman. Let the oarsmen be separate by the same distance - call it d. The total torque at any given moment (note I'm just looking at instantaneous torque rather than worrying about a messy integral) for a four man shell with the rig shown would be:

T = rF - (r + d)F + (r +2d)F - (r - 3d)F

this = -2Fd certainly not zero over most of the stroke! This wiggle-waggle torque, remember F is oscillating in magnitude over ^{+}/_{-} its maximum value, either goes into a wasteful motion of the shell or the cox can counteract it. Both outcomes waste energy.

So you wonder about other patterns. Consider these rigs. The first is what we just examined and the second is the standard eight which alternates between ^{+}/_{-} 4d as you might expect (check my arithmetic).

The third rig is interesting. Performing a similar calculation I get:

T = rF - (r + d)F - (r + 2d)F + (r + 3d)F

Now we get zero. By a clever arrangement we can balance out the torque at every instant. It turns out this has been tried and successfully, although there isn't any mention they understood what was going on.

Moving to the eight- man gives a trivial wiggle-waggleless rig - 4a is just a couple of the rigs used in example 3. For fun I found three more and showed that is all that exists. 4b gives zero - you may want to check the arithmetic again. I found two more in short order and won't give the solution for those of you who find pleasure in figuring things out. Ping me if you want my solution.

In the real world the forces aren't equal and timing and form are less than perfect - mostly that is. There is this notion of swing in rowing. It happens rarely and is the result of near perfect timing and application of force. Even in world class crews it isn't that common although you may see it in one or two of the races in Rio. An eight-man boat is producing something over five kilowatts most of the time and can burst much higher. Individual differences on the order of a few watts can destroy swing. It is a dramatic thing to see and crew members who have been there cite it as something amazing - a very special kind of flow. You can see it in the motion of the boat and in the expressions of the crew. Quite literally a poem of motion.

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Recipe Corner

From an athlete friend:-)

**Peanut Butter Smoothies**

**Ingredients**

° 2 sliced bananas - frozen

° 3/4 cup milk or soy milk

° 3 tbl peanut butter

° 2 tbl unsweetened cocoa powder

° 1/4 tsp vanilla extract

**Technique**

° surely you jest - blend until smooth and server

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