# Efficiency of Bicycle Pedalling

Consider a bicycle with multiple gears. Suppose that you are in a starting position with someone holding your bike upright (so when you start there's no issue with clipping in etc). It's well-known (and easily testable) that if you start in a low gear, then you accelerate away faster than in a high gear. (Obviously, if you start in the lowest gear and have a climbing mtb, then you'll have to change gear which can slow you etc - to ignore this, suppose you're on a road bike and using a not too low gear etc.)

Now, if the bike (and your body) is (are) perfectly efficient, then applying the same amount of force over a given distance gives the same work done. However, this could be done at a different rate (power). (Does the human body output at a certain power or is it the work done?) However, this outcome clearly isn't realised, so there must be some inefficiencies. Here are a few that I can think of (mainly the first one then the next two):

1. Torque. Am I correct in thinking that changing gears is very similar to the situation, say, of undoing a nut and bolt with a spanner and changing the length of the spanner. (Consider also opening a door by pushing in different places (horizontally) on the door.)

2. The body is far more efficient using the optimal cadence than a very bad cadence. Eg, pushing really hard and slowly is inefficient: the extra effort doesn't translate into extra power; similarly, if the cadence is too high, then you cannot move your legs up and then back down fast enough to give the required speed.

3. A very high strain on the chain is inefficient ("uses up" a lot of the power).

4. Very fast/slow turning of the legs can cause loss of balance, so effort can be spent on trying to maintain this.

Is there something key that I'm missing, or are these just the main things?

Any insight would be appreciated! :) - thanks!

• "applying the same amount of force will give the same acceleration" --- I'm not sure this is true here, because different gear ratios mean that the same force applied to the pedal results in a different force applied by the bike to the ground. Jan 2, 2015 at 19:57
• That was what I wasn't sure about. The difference isn't force though, if completely efficient, surely? There may be a difference in torque. Consider for example if you push a door really near the frame: pushing really hard doesn't do much, while pushing not very hard at the far edge moves it easily. Jan 2, 2015 at 20:08
• Energy is conserved, so given output of a certain amount of energy the same amount of work should be done. But work is force times distance - doesn't matter how fast it's done (power). Thus, perhaps, the same amount of work is done, just over a larger period of time? Jan 2, 2015 at 20:09
• Ah just realised another key thing which I have added in (now number 1) in the list. Jan 2, 2015 at 20:23
• This question might be better answered in bicycles.stackexchange.com Jan 2, 2015 at 20:26

The question is quite complex, but there are several fairly simple things that can be observed:

First off, the standard bicycle, as a machine, is quite efficient. Very little energy is lost in the "drive train", with the vast majority of "lost" energy (mechanical energy input at the crank that is not converted to momentum) being expended as either air friction or friction between tire and roadway.

The human body, on the other hand, is often an incredibly inefficient machine. Not only are there simple concerns of "energy efficiency" -- how many calories of food, say, it takes to produce an erg of "work" -- but there are also major issues of "durability" and "endurance", both in the short term and long-term.

The average human body tends to have a "sweet spot" for cycling where the cadence is (depending on the individual and the circumstances) somewhere between maybe 60 and 90 RPM. Cycling within the "sweet" range for the individual produces a large amount of energy (though perhaps not the "peak" energy) and, more importantly, minimizes fatigue and optimizes endurance (as measured, say, in total energy produced in a given 24-hour period, including rest, eating, sleeping, etc).

In terms of gear ratio, in addition to determining cadence on relatively level ground, it also, of course, affects climbing. An individual is limited as to the total torque they can place on the bike crank arms, and hence what degree of incline they can climb at a given gear ratio. Lowering gear ratio (obviously) reduces the torque required to turn the crank arms and hence enables climbing a steeper incline. Here the "sweet spot" (for a relatively short climb) is below the level ground "sweet spot", but there still is one.

When considering cadence both on level ground and climbing it needs to be considered that muscles are more efficient when in "aerobic" mode -- burning "fuel" using oxygen supplied from the lungs via the bloodstream. Aerobic mode is perhaps twice as efficient as anaerobic mode (though don't quote me on that number), and, of major importance, it produces far fewer metabolic byproducts which can accumulate in the body and eventually become toxic. Although there are several factors that determine whether exercise is aerobic or anaerobic, a major one is, in fact, cadence, with lower cadences being more likely to be anaerobic.

• Excellent answer, thank you. I agree that the bike is very efficient. Slip-streaming/drafting is so good! =P Although, that doesn't necessarily show that the drive chain is efficient (but it is!). Jan 2, 2015 at 23:09
• Thanks for the biological/nutritional insight also. I, along with my fellow skinny cyclists, enjoy spinning at 90-100 RPM, whereas some more hench cyclists go down to 60 or less. I find that lowering the cadence (so putting in more effort more slowly) means that I can't last as long. I've heard that this causes a build up of lactic acid in the legs. (I believe that) this is very difficult to get rid of, and only really goes with rest; as such, once it's basically going to be there until the end of the ride, so need to be careful when "red-lining". Does this seem correct to you? :) Jan 2, 2015 at 23:13
• @SmileySam - That's about right. There's been amazingly little scientific work studying optimal cadence (not counting, perhaps, the work by the TdF teams who keep their stuff very secret). 20-odd years ago one of the cycling magazines reported on experiments that showed that a cadence above 80 (for the half-dozen cyclists in the study) was not as optimal as lower (even though the cyclists believed the higher cadence was better). But, again, there's surprising little "science". Jan 2, 2015 at 23:19
• I know that Specialized do quite a lot of stuff - even have their own wind tunnel. But as you said, very secretive. I think it also depends an enormous amount on your build. As I said above, I like a high cadence - I cycle with someone who's up to 120 RPM and is as skinny, if not more, than me - but others will like lower if they have bigger legs. Jan 2, 2015 at 23:21
• (My advice to inexperienced cyclists is to always maintain a cadence that is at least equal to your breathing rate, and, ideally, about twice your breathing rate. This works remarkably well for a variety of circumstances, from a slow "amble" to flat-out racing.) Jan 2, 2015 at 23:22

The power inserted into a mechanical system equates to the power dissipated by the system plus the rate of gain of energy that stays in the system. As a formula it would look like:

$$P_{in} = P_{out} + \dot E_{internal}$$

$P_{in}$ is the power the cyclist is introducing into the system by moving his legs.

$P_{out}$ is the energy being lost from the system. This is what you're looking for here in your question.

$\dot E_{internal}$ is the rate of gain of internal energy, and, for a bike, mainly deals with rates of gain of kinetic and gravitational potential energy. Elastic energy might also be including, in the metal frame of the bike, but I doubt it will have too much of an impact over time.

Some of the sources of $P_{out}$ that I can think of:

When you pedal, you will be combating a frictional torque due to the rubbing together of the mechanical parts. The work done against this frictional torque leads to some heat loss.

When you are moving, you will be working against air resistance. The air particles collide with the moving bicycle, and the collision of the air transfers some of the energy away from the bicycle. The faster the bicycle, the faster air resistance dissipates energy. This also occurs on a smaller scale when the pedals go around very fast: the pedals collide with the air particles when rotating, and hence lose energy.

The tyres on the road do a good job at not slipping, but when they do (which I would suspect must happen regularly, if not constantly to some extent), the tyres are doing work against the friction from the road. This results in heat dissipation.

And, as an intentional design of the bicycle, the brakes do a wonderful job at contributing to $P_{out}$, because, when you use your brakes, you really want to drain as much of the kinetic energy (stored as part of $E_{internal}$) as possible. This works by constantly applying a strong frictional force against each wheel, dissipating a lot of heat energy.

That is what I could think up of for now, but I'm sure there are additional means of losing energy from the system that I missed.

• I would argue that frictional torque due to rubbing together of mechanical parts is fairly minimal. I have a fairly high-end racing bike ('05 model, but was about £1.8k when new - I got it at the start of summer for £600 =D!). Jan 2, 2015 at 23:16
• Tyres do play a part. But I use ~100psi and very thin tyres, so it's not a big deal. (You can see that the above two are not a big deal since, when in a good draft, you can coast along the flat for a long time.) But, from a pure physics point of view*, the tyres don't need to be slipping in order to be working against friction. [*As mainly a pure mathematician, "pure physics" is not a phrase I like to use... ;)!] Jan 2, 2015 at 23:18
• I do 100% agree that air resistance is a big thing - the biggest by far. (I love drafting someone downhill - they're pedalling hard and I'm just cruising!) Jan 2, 2015 at 23:18
• That's a fair point. In most cases, the frictional torque is really not going to be that significant, but I thought I'd add it for completeness sake. It would only be significant for bicycles with rusted or poorly fitting mechanisms. Jan 2, 2015 at 23:24
• @SmileySam - Actually, as seen here, the amount of energy stored in the wheels is not nearly as substantial as it is often claimed. It basically only increases the apparent mass of the bike by the weight of the tires and rims being included a second time. Jan 2, 2015 at 23:26