FAQ: What is the Significance of Weight in Rowing?

Prepared by Dr. A. Dudhia, Dept. Atmospheric Physics, Oxford University, an ARA
Silver Level coach and surprisingly slow sculler from St.Catherine's College BC

28th April, 1995

  1. Introduction
  2. Relationship between Power and Weight
  3. Power/Weight Ratios
  4. Relationship between Weight and Erg Speed
  5. Relationship between Weight and Boat Speed
  6. Speeds of different Boat Classes
  7. Effect of deadweight


1. Introduction

This Web FAQ page has been prompted by the large number of postings on
rec.sport.rowing relating to comparisons of erg scores from rowers of different
weight, how significant an advantage weight is when it comes to boat speed, and
what how much difference the weight of the cox makes to boat speed. If anyone
wants to query, argue or add to any of this, feel free to email me. But to
summarise the results regarding weight-advantage, this shows the theoretical
weight-dependence of various tests, in decreasing order of advantage towards the
heavier athlete.

Test                            Scaling   Notes
-----                           -------   -------
Erg Power (Anaerobic):          W^1       Anaer.Power prop. to muscle volume
Erg Power (Aerobic):            W^(2/3)   Aerobic Power prop.to surface areas
Erg Speed (Anaerobic):          W^(1/3)   Erg speed varies as cube root of Power
Erg Speed (Aerobic):            W^(2/9)   The famous 2/9, or 0.222, formula
Sculling Speed (Anaerobic):     W^(1/9)   Small weight advant.(ignore boat mass)
Erg Power/Weight (Anaerobic):   W^0       No weight advantage
Sculling Speed (Aerobic)        W^0       No weight advantage (ignore boat mass)
Erg Power/Weight (Aerobic):     W^(-1/3)  Lightweights have an advantage


2. Relationship between Power and Weight

In the following, the phrase `similar physique' means having the same build and
physiology but not necessarily same weight or height, i.e. the ratio of weights
of two people with `similar physique' will be the cube of the ratio of their
heights.

The body can produce Anaerobic Power Pa and Aerobic Power Po.

Anaerobic power depends on muscle bulk, i.e. for a given physique, will be
simply proportional to total Weight W:

(2.1)           Pa = a.W

where a is a constant.

Aerobic power is governed by the flow of oxygen across various membranes, so is
proportional to surface areas. Two people of similar physique will have surface
areas proportional to the square of their heights, or the 2/3 power of their
weights:

(2.2)           Po = b.W^(2/3)

where b is a constant.



3. Power/Weight ratios

From the previous section the Anaerobic Power/Weight ratio

(3.1)           Pa/W = a

should be a constant, i.e. over short distances on an erg, say 1000m or less,
two athletes of similar physique should have similar power/weight ratios.

On the other hand, the Aerobic Power/Weight ratio

(3.2)           Po/W = b.W^(-1/3)

is inversely proportional to the cube root of the weight, i.e. over long
distances on an erg, say 5000m or more, the lighter athlete should have the
larger power/weight ratio.



4. Relationship between Weight and Erg Speed

An ergometer, such as a Concept, fundamentally measures power P which is related
to speed V (=distance/time) via:

(4.1)          P = c.V^3

where c is a constant (nominally the same for all Concept machines, whatever
vent/gear setting is used). So, the time T to cover a given distance D (using
T=D/V) is related to power by:

(4.2)          T = d.P^(-1/3)

where d is a constant proportional to the distance D. Using the expressions for
aerobic power Pa and anaerobic power Po from section 2, the time-weight
relationship for anaerobic work (i.e. a few minutes or less) is given by

(4.3)          Ta = e.W^(-1/3)

where e is some constant proportional to the distance, and for aerobic work
(i.e. 20 minutes or longer)

(4.4)          To = f.W^(-2/9)

where f is some constant proportional to the distance. This second
expression is the origin of the (2/9) or 0.222 power that often appears in
formulae relating erg times to body weight. Note that it is only valid for
aerobic work.



5. Relationship between Weight and Boat Speed

Most of the resistance R (a Force) to a moving boat comes from surface drag
which is proportional to the wetted surface area A and the square of the
velocity V.

(5.1)         R = g.A.V^2

where g is some constant dependent upon hull shape. For a given submerged hull
shape, the surface area A varies as the (2/3) power of the volume enclosed,
which is proportional to the displacement, which is almost entirely the weight
of the crew (ignoring boat weight, cox, oars).

(5.2)         R = h.W^(2/3).V^2

At steady speed V, the resistive power (R.V) equals the motive power.

Using the derived weight-dependence of Power from section 2, the speed Va for
anaerobic distances, using R.Va = Pa, is given by:

(5.3)         Va = q.W^(1/9)

where q is some constant. Thus, over short distances, heavier scullers have a
(small, only the ninth root of weight) advantage over light scullers.

The speed Vo over aerobic distances, using R.Vo=Po, is given by:

(5.4)         Vo = r

where r is a constant. This suggests that over longer distances, e.g. Head
races, light crews are as fast as heavy crews. This isn't quite true, since the
weight of the boat has been ignored, but lightweight scullers certainly feature
more regularly at the top of the Tideway Scullers Head than in the open 1x
finals of Nat.Champs.

 

6. Speeds of different Boat Classes

Most racing boats have essentially the same hull shape e.g. the submerged volume
of an eight has approximately twice the dimensions of length, width and depth as
that of a scull since it has to displace 8 times (=2x2x2) as much water. We can
then use the same resistance formula Eq.(5.2) for different boats, replacing the
weight of the individual W with the total weight of the crew N.W

(6.1)         R = h.(N.W)^(2/3).V^2

To simplify things, assume each oarsman weighs the same and absorb W into the
constant h:

(6.2)         R = k.N^(2/3).V^2

where k is a constant proportional to the (2/3) power of the average weight. The
total power generated will also be N.P, where P is the power generated by an
individual, so equating resistive power R.V with motive power N.P, the basic
boat speed is given by

(6.3)         V = m.N^(1/9)

where m is a constant proportional to the cube root of the average power of an
individual. Since 2^9 = 1.08, this implies an 8% difference in speed between
different boat classes (singles,doubles/pairs,quads/fours,eights).

To compare the winning times from the Men's Lightweight events in the 1995 World
Champs:


Event:      1x          2x          4x
Time:      6:53  <9%>  6:18  <9%>  5:47

Event:      2-          4-          8+
Time:      6:34  <11%> 5:54  <7%>  5:31

This also suggests that a Men's Lwt 8x would take 5:21


7. Effect of Deadweight on Boat Speed

Now the really tricky one: how much difference does that extra kilo of
deadweight make to your speed? First of all, to point out the obvious, it
doesn't really matter where the extra kilo is: on the cox, on the boat or on one
of the oarsman, it slows you down just the same.

To take a simple model of a hull, imagine a submerged portion with a
constant semicircular cross section, radius X and length Y. The displaced mass
of water, equal to the weight of crew, cox, boat and oars, W, is given by:

(7.1)        W = d.pi.X^2.Y/2

where d is the density of water, and the total surface area A is given by:

(7.2)        A = pi.X.Y

If a small weight dW is added, the boat sinks a by an extra depth dZ (assume the
sides of the hull are vertical at the water line) until the equivalent
additional weight of water is displaced:

(7.3)        dW = 2.d.X.Y.dZ

and the extra wetted surface area dA is given by:

(7.4)        dA = 2.Y.dZ

Putting these equations together, we get:

(7.5)        dA/A = (1/2).dW/W

Note that a simpler assumption, that surface area increases as the (2/3) power
of weight, is really only true for comparing different boat classes, or perhaps
single scullers. However, this would lead to a factor (2/3) rather than (1/2) in
the above equation, which would not significantly affect the answer.

From section 5, a boat's speed V is determined by the balance between the motive
power P and the resistive power R.V, where the resistance itself depends on
wetted surface area A and the square of the speed (Eq.(5.1)), so

(7.6)          P = g.A.V^3

Rearranging, and assuming constant power (i.e. same crew rowing the boat and
that the extra weight dW is deadweight),

(7.7)          V = p.A^(-1/3)

where p is a constant proportional to the cube root of the total power.Using
some basic calculus, the change of speed with surface area is given by:

(7.8)         dV/V = -(1/3).dA/A

and using Eq.(7.5), the relationship between speed and deadweight is given by:

(7.9)         dV/V = -(1/6).dW/W

Which tells you that the percentage loss of speed is one sixth the
percentage increase in weight.

An example: assume an VIII, total weight 800 kg (=8x80kg rowers, 50kg cox, 100kg
boat, 10kg oars). An extra 10 kg (=22 lbs) represents 1/80=1.25% increase in
weight. So the boat moves 1.25/6=0.2% slower. Over a 6 minut race (eg 2000m)
this corresponds to 0.6 sec, or 4m (about 1/5th of a boat-length )