The Physics of Power on a Bicycle

[bike-nerd]

Hello! I'm hoping someone here can answer a question for me...

I've observed that when pedaling a bicycle, if the rider moves from seated to standing position while maintaining the exact same cadence (pedal revolutions per minute) and resistance, power output (as measured on a power meter) often decreases.

My theory is that you add mass when you stand, and that alters the way power is produced - kind of like adding mass to one side of a fulcrum...

Anyone care to weigh in on how shifting to standing position might change the dynamics of power output when pedaling a bicycle??

 

[sk1105]

By power output do you mean the power of the rider's leg muscles? In which case standing up would cause a reduction because some work is now being done by gravity to maintain the same cadence.

 

[A.T.]

My theory is that you add mass when you stand, and that alters the way power is produced

When sitting, power is produced mainly by the leg muscles, which exhausts them faster than standing, where the work is distributed to the upper body.

 

[A.T.]

...because some work is now being done by gravity...

Gravity is not an energy source. It's still the muscles that have to lift the body against gravity in the first place, so it can push down the pedal with it's own weight. But you can employ a wide range of muscles of the upper body for this, which relieves the legs.

 

[bike-nerd]

By power I mean actual power - as in watts - generated by turning the pedals...

 

[A.T.]

By power I mean actual power - as in watts - generated by turning the pedals...

Given the same cadence pedal power depends mainly on the force applied to the pedal.

In standing position the rider usually just shifts his body over the upper pedal, and uses it's weight to push down the pedal. So the force is limited to about his weight.

In sitting position, a trained rider can potentially generate forces greater than his weight, by using his leg muscles, thus resulting in a greater power output. But it will exhaust his legs mush faster.

 

[Pierce610]

but isn't possible that moving from seated to standing position it generates in presence of gravity a torque, a moment of inertia and an angular momentum. For the principle of conservation of total angular momentum (wheels + standing rider) may be the power (the speed) had to decrease? I'm still thinking of the miracle of the cat coming from the space.

 

[A.T.]

but isn't possible that moving from seated to standing position it generates in presence of gravity a torque

The way I understand the OP, it's not about the transition from sit to stand, but continuous operation in either mode.

 

[rcgldr]

For competitive riders in a sprint situation, they stand up and pull up on the handlebars (alternating the amount of pull on left and right handlebars) while pressing down harder on the pedals for increased speed (cadence would be increased). You'll see the bike lean left and right while they do this. It takes more energy, but the goal in these situations is a relatively short burst of high speed.

Another reason for standing is to change the pattern of muscles involved while riding a bike to give some partial rest to the leg muscles. Non-compeitive bike riders may stand up on the pedals when pedaling up hill, mostly just standing up, then taking a rest while letting gravity drive a pedal downwards, so short bursts of muscle activity, rather than more continuous effort.

 

[bike-nerd]

My question is about the physics of what happens to power generation (P = F * V or P = torque * ω) when the rider stands if that is the only variable that changes. Let's forget it's a bicycle for a minute...

Assume you have a machine with a fixed center point and a pair a lever arm rotating around an axis.

Consider two different ways to apply force to the lever arm. Option 1 is relatively even distribution of force all the way around the the circular path of the lever arm (more similar to pedaling in seated position). Option 2 is to add mass to the lever arm at about the 1:00 position - changing the force application from a more even, circular distribution of force to a more sharp, downward application of force (more similar to pedaling in standing position).

When you change the force application as described above - what happens to torque and overall power generation if total revolutions per minute around the axis stay the same? (To maintain the same total revolutions per minute, the lever arm would need to pause or slow at some point as it continues to rotate around the axis to account for the sharp acceleration between 1:00 -> 6:00.)

To maintain constant power output, would you not need to counter-balance the mass added to the lever arm in order to maintain equilibrium?

 

[A.T.]

When you change the force distribution as described above - what happens to torque and overall power generation if both total revolutions per minute stay the same?

They obviously become more variable, just like the force.

To maintain constant power output, would you not need to counter-balance the mass added to the lever arm in order to maintain equilibrium?

What equilibrium?

 

[bike-nerd]

I'm no physicist - so forgive me if I get any of this wrong...

I obviously know it becomes more variable - but my question is HOW? In what way does it change? Is overall power (watts) generated by adding mass to the lever on one side, and un-weighting the mass on the other side of the circle higher or lower?

By equilibrium, I'm thinking along the lines of what would happen if you had, instead of a rotating lever arm, a lever arm balanced on a fulcrum. Imagine a board balanced on a fulcrum. You want to move that board up and down at a constant rate of speed with a constant input of power. If you add mass to one side, wouldn't you need to add mass to the other side to balance it out.

 

[A.T.]

I obviously know it becomes more variable - but my question is HOW?

Look up the definitions of torque and power, and how they relate to force.

You want to move that board up and down at a constant rate of speed with a constant input of power. If you add mass to one side, wouldn't you need to add mass to the other side to balance it out.

Yes, but the assumption of constant power is unrealistic for pedaling in standing.

 

[bike-nerd]

LOL - I have looked them up, but still don't quite get it... sorry - as I said - not a physicist!

So - what would need to change in order to maintain constant power if you were to add mass to the lever as described?

I'm actually debating this issue with someone else (neither of us are physicists...) His argument is that there is a certain amount of power required to move pedals around the axis at given revolutions per minute, and that that power requirement does not change regardless of whether the rider is seated or standing. My argument is that when you add MASS to the pedals by standing, you change the power requirement.

Which of us is right?

 

[A.T.]

So - what would need to change in order to maintain constant power if you were to add mass to the lever as described?

You can't, because weight on the pedals doesn't produce a constant torque throughout the cycle. In particular, when the lever is vertical, the weight produces no torque.

His argument is that there is a certain amount of power required to move pedals around the axis at given revolutions per minute, and that that power requirement does not change regardless of whether the rider is seated or standing.

Assuming for example you want to keep some constant bike speed, and ignoring changes in drag from changed stance, that is correct.

My argument is that when you add MASS to the pedals by standing, you change the power requirement.

That doesn't make any sense. The power requirement comes from the resistance (like drag) that the bike has to overcome. How the torque on the pedals is generated has nothing to do with that requirement.

 

[bike-nerd]

I'm not sure I'm expressing the question I have very well... I'll try again... Let's assume we're talking abut a stationary bicycle.

In seated position, the rider's body mass is not, for the most part, applied directly to the pedal. Force is applied in a more circular fashion throughout the pedal stroke. When the rider stands, that changes. A large portion of the rider's body mass is applied directly downward on one pedal. For the most part, the opposite pedal is "unweighted" at that point.

Again, thinking of it not as a bicycle, but as a lever (or pair of lever arms) rotating around an axis...

When a rider shifts to standing position, it's like adding mass the the lever at about the 1:00 position, unweighting at about 6:00 or 7:00, and adding mass again at 1:00 every revolution.

Since Power = Force x Velocity, and Force = Mass x Acceleration, doesn't adding mass change the amount of power that is required to turn the lever arms?

 

[A.T.]

Since Power = Force x Velocity, and Force = Mass x Acceleration, doesn't adding mass change the amount of power that is required to turn the lever arms?

I think you and your friend are talking about two different "power requirements":
- He talks about the power that needs to be applied to the bike to keep it going.
- You talk about the amount of power the body needs to generate, in order to apply a certain amount of power to the pedals.

Since in the standing position you are accelerating a larger proportion of your body, there is more energy going into that acceleration, and you have to generate more power in order to keep the power at the pedals the same. Is that what you mean?

 

[bike-nerd]

Thinking out loud here...

So - there is a certain amount of power that is required to keep a bicycle moving at a constant rate of speed. The real question here is - where does that power come from, and does that change in seated vs standing position?

Here's my thinking - when seated, the power required to move the bicycle comes entirely from the the rider turning the pedals - moving the lever around the rotational axis. When standing, you add mass to the lever, and I think that at that point - gravitational force contributes some amount of power to the system. Now you have TWO sources of force (and power) - the force of the rider physically pushing on the pedals and gravitational force. So in order to maintain a constant speed, the amount of force (and power) contributed by the rider pushing on the pedals would be decreased. Does that make sense?

 

[A.T.]

So - there is a certain amount of power that is required to keep a bicycle moving at a constant rate of speed.

Yes, because of losses, from air resistance, rolling resistance friction in bearings etc.

and does that change in seated vs standing position?

The power requirement changes, if the losses change. Air resistance for example might be different,

Here's my thinking - when seated, the power required to move the bicycle comes entirely from the the rider turning the pedals - moving the lever around the rotational axis. When standing, you add mass to the lever, and I think that at that point - gravitational force contributes some amount of power to the system. Now you have TWO sources of force (and power) - the force of the rider physically pushing on the pedals and gravitational force. So in order to maintain a constant speed, the amount of force (and power) contributed by the rider pushing on the pedals would be decreased. Does that make sense?

No. Gravity is not a energy source. All the energy that goes into the pedals must have been generated by the rider at some point.

 

[jimmy1010100]

I've observed that when pedaling a bicycle, if the rider moves from seated to standing position while maintaining the exact same cadence (pedal revolutions per minute) and resistance, power output (as measured on a power meter) often decreases.

Do you have a strain gauge power meter or one that estimates from speed (and thus gives you the wrong answer)? If one's power really is decreasing it's because of personal reasons rather than physics which I'll explain below. For others including myself, we stand to increase power. However, the dynamics of pedaling while standing is slightly different as one loses a contact/balance point (i.e., the saddle) and as A.T. mentioned, engages upper body muscles which fatigue sooner. But for short periods such as attacking a climb or starting a sprint, this is actually desired, as rcgldr mentioned. If you watch professional bike racing you'll often see people standing for more power. This is more common among lighter riders as they have less mass to accelerate up off their saddles. Unless the upper body is a huge drain, human power should always increase while standing because standing increases the blood flow to the legs (by un-pinching the femoral artery) so one can deliver more energy to the working muscles.

But there are trade-offs to standing: energy lost to accelerating off the saddle, holding up one's body weight (as the body does internal work when standing due to dynamic balancing), as A.T. mentioned, energy lost to upper body muscles (engaged for pulling and balance), and most importantly, the energy lost to aerodynamic drag from increasing frontal area.

My theory is that you add mass when you stand, and that alters the way power is produced - kind of like adding mass to one side of a fulcrum…

...I'm actually debating this issue with someone else (neither of us are physicists...) His argument is that there is a certain amount of power required to move pedals around the axis at given revolutions per minute, and that that power requirement does not change regardless of whether the rider is seated or standing. My argument is that when you add MASS to the pedals by standing, you change the power requirement.
Which of us is right?

Sorry, your theory is wrong. Your friend is correct. Adding mass would change moment of inertia, but it would have an minuscule effect on the power to move the bike. And you're not really adding mass you're just moving it around. What happens when you stand is you actually throw the bike slightly backwards then later pull it back forwards due to conservation of momentum. This is in fact a common cause of crashes when drafting as people throw their back wheel into the front wheel of the rider behind them. Also, if one increases torque by leveraging their weight differently it will only slow their cadence (in RPM or pedal velocity) as ##\tau=power/cadence##, and the power delivered to the cranks will remain the same as it's determined by physiology rather than mechanics. The linkage has a negligible effect on the amount of energy one can put into the pedals because there is barely any loss through the cranks.

You are correct that ##power=Fv##. The power ##P## to move a bike at a steady pace can be summed roughly as the power to overcome each of the three major forces as
$$P=K + \underbrace{c_r cos\theta \,v}_{friction} + \underbrace{mg sin\theta \,v}_{gravity} + \underbrace{\frac{1}2 \rho c_dA \,v^3}_{aero drag}$$ where ##K## subsumes power to overcome bearing drag, chain friction, acceleration and other small effects because they are so small compared to the three emphasized. ##\theta## is the arctan of the hill grade. ##\rho## is air density. ##A## is area orthogonal to wind which is frontal area in this case. The other constants are the coefficient of rolling resistance (due to internal friction of tire deformation) and the coefficient of aerodynamic drag (due to shape of body and bike that affects airflow). And ##v## is the speed.

You can see the biggest term is aerodynamic drag which is cubic with speed whereas the other terms are linear. So beware of standing or sitting up too much into the wind. For a more detailed and precise model see [1] and [2] below. For further reading, Fajans has looked at some interesting bike physics in [3].

Now the question really is why does someone in particular produce less power standing on the pedals? The answer is one or more of the following factors: too much body weight, poor position/weight distribution/balance, poor handlebar setup, poor front geometry/mechanics of the bike itself, or simply not being used to it.

[1] Martin et al., Journal of Applied Biomechanics 1998, 14, 276-291
[2] http://www.analyticcycling.com
[3] http://socrates.berkeley.edu/~fajans/Teaching/bicycles.html

 

[bike-nerd]

Thanks all for your analysis!

I am talking about a strain gauge power meter on a stationary bike - so we can ignore factors like aerodynamic drag for the purpose of this discussion... In particular, when riders shift from seated to standing WITHOUT ADDING RESISTANCE, especially if resistance is on the lighter side to begin with, they often observe a reduction in power - or at least the power meter REPORTS lower power. He contends that this is particular to a specific type of power meter - I contend that it's just what happens (without sufficient resistance added).

I have a couple of follow up questions/observations...

1) I still contend that standing does add mass at the pedal, or at least it changes effective pedal force & torque, which (if I understand correctly) would have an affect on power. It's kind of like the difference between turning a wrench by hand (or perhaps pushing the handle of a wrench with your foot from a seated position would be a better analogy) and standing on the wrench handle to loosen a particularly stubborn nut or bolt.

2) Gravity is not a source of energy - but it is a FORCE - is it not? So when you stand, you have two separate "forces" at play - the rider's muscular force rotating the pedals, and the gravitational force of the rider's body weight pushing down on the pedals.

Imagine a bicycle crank, or other fixed rotating object with a lever arm attached. If you want to physically rotate that lever arm by hand, YOU would need to contribute power by pushing on the lever. If, on the other hand, you placed a weight on the end of the lever arm, YOU would not need to contribute any power (or at least much LESS power) to move the lever. With lighter resistance on a bicycle, I contend that a similar thing happens when the rider stands on the pedals.

 

[A.T.]

1) I still contend that standing does add mass at the pedal, or at least it changes effective pedal force & torque, which (if I understand correctly) would have an affect on power.

You have to distinguish between:
- instantaneous power at some point during a cycle (that varies in both cases, but probably more in standing)
- average power over a cycle (usually what the power meter shows)
Note that average power can drop, even if peak instantaneous power increases (and vice versa).

It's kind of like the difference between turning a wrench by hand (or perhaps pushing the handle of a wrench with your foot from a seated position would be a better analogy) and standing on the wrench handle to loosen a particularly stubborn nut or bolt.

That's about peak force/torque, and has little to do with average power over a cycle that your power meter measures.

Imagine a bicycle crank, or other fixed rotating object with a lever arm attached. If you want to physically rotate that lever arm by hand, YOU would need to contribute power by pushing on the lever. If, on the other hand, you placed a weight on the end of the lever arm, YOU would not need to contribute any power (or at least much LESS power) to move the lever.

To lift the weight onto the lever arm you have to provide energy, which is then used to push it down. The instantaneous power might be less at some point, because it was more at another point. The average power you would have to provide over a whole cycle is not less.

Both of your examples are flawed in the same way: They do not represent a cyclic process that always returns to the same state. The do not correspond to the average power over a cycle measurement, because there is no whole cycle. You just pick the part of the cycle where stored potential energy is transmitted to the pedals, but are ignoring the part where it is stored by lifting the body.

 

[bike-nerd]

Thanks - that all makes perfect sense.

You have to distinguish between:
- instantaneous power at some point during a cycle (that varies in both cases, but probably more in standing)
- average power over a cycle (usually what the power meter shows)
Note that average power can drop, even if peak instantaneous power increases (and vice versa).

The observation I'm talking about is actually what appears as a brief drop in power (as reported on the power meter) when the rider stands. It usually comes back up (at least to some degree) after a few pedal strokes, unless resistance is insufficient and the rider "bottoms out" at the bottom of the pedal stroke. In which case rpms may remain constant, but rather traveling in a smooth, continuous circle, the pedal stroke is more choppy - if that makes sense...

Since the strain gauge of this particular power meter is located on the crank arm, it may be more sensitive to those instantaneous changes in power...

 

[zanick]

Think of standing up on the pedals vs sitting and pedaling, as the difference between running and walking. standing provides more burst energy and force, then constant reciprocating forces, which tends to be much more constrictive on circulation. standing pedaling, applies more force for a shorter duration, but gives the riders legs an aerobic /cardiovascular break between strokes. a rider can ALWAYS applie more power by standing vs sitting. by standing up during the, some of the dead time where you can't efficiently apply force, you store up kinetic energy as potential energy and release it and add it to the forces applied when the rider presses down on the peal. by standing and moving up and down, you are able to apply a force greater over a greater distance over slightly less time, vs sitting and pedaling. even the forces applied on the up stroke, have more force over distance (more work). standing optimizes the force angles of the leg, well beyond sitting and pulling up on the pedals on those strokes. ever see the bike sprinters sit down and pedal at the end of the race? never... there is your answer.

 

[bike-nerd]

I completely get all of that. But that's not really what I'm asking about.

In order to put out the massive amount of power riders put out in say, a final sprint - they need to be in a big gear so they have enough resistance to apply that force to. It's impossible to sprint - impossible to put out that kind of power - in a small gear. My question is about what happens to power output when you stand and apply that greater downward force WITHOUT changing gearing/resistance - i.e., - when you increase downward force without a increasing the opposing force? Often in that scenario you see recorded power (on the power meter) drop. If you shift gears/add resistance (increase the opposing force), it comes back up.

 

[jbriggs444]

My question is about what happens to power output when you stand and apply that greater downward force WITHOUT changing gearing/resistance - i.e., - when you increase downward force without a increasing the opposing force?

That cannot be done. Newton's third law applies. The force of the rider's foot on the pedals is always exactly equal to the force of the pedals on the rider's foot.

However... For a fixed gearing, as the force exerted on the pedal increases, the speed of the pedal will increase. As the speed of the pedal increases, the cadence increases. The amount of force that a rider can apply may drop as a result.

Like throwing a feather, will be dissipating effort moving your hand instead of moving the feather.

 

[A.T.]

The observation I'm talking about is actually what appears as a brief drop in power (as reported on the power meter) when the rider stands. It usually comes back up (at least to some degree) after a few pedal strokes,

A transient effect like that is most likely due to something physiological, like the body adapting to the new mode of movement.

 

[sophiecentaur]

Now you have TWO sources of force (and power) - the force of the rider physically pushing on the pedals and gravitational force.

I know it's tiresome for a non-Physicist to be told to use the right terminology but it is really critical in the understanding of these things. You can't afford to confuse Force with Power.
You can alter the forces involved on the bike frame, pedals and the rider but there is only one source of Power / Energy and that is the rider's muscles. Gravity doesn't give you power, You can use gravity to store energy, temporarily, in order to be more effective in some operations. For instance, you can lift an axe up high, storing gravitational potential energy, and then bring it down with a bash - expending all the input energy in a fraction of a second as the axe splits the wood. No extra energy is available - it's just redistributed in time. With a bike, the situation is not like using an axe because you don't require short bursts of high power during each turn of the pedals. In fact, it would be better, in many ways, to have a steady source of energy - as with a motor. Unfortunately, our bodies don't work that way and the way cyclists use the pedals is a sort of work around to get the best overall result (hence the selection of the best gear for each situation). 'Good' cyclists tend to have learned different styles from your average guy on a bike, who can be very inefficient in the way they pedal, just because they are knackered and clutching at straws. I always know I'm on the way out when I start standing on the pedals rather than just dropping a gear.

 

[bike-nerd]

That cannot be done. Newton's third law applies. The force of the rider's foot on the pedals is always exactly equal to the force of the pedals on the rider's foot.

However... For a fixed gearing, as the force exerted on the pedal increases, the speed of the pedal will increase. As the speed of the pedal increases, the cadence increases. The amount of force that a rider can apply may drop as a result.

Like throwing a feather, will be dissipating effort moving your hand instead of moving the feather.

See - THAT'S what I'm getting at. That "dissipating effort" when you stomp on the pedal WITHOUT adding resistance.

 

[A.T.]

See - THAT'S what I'm getting at. That "dissipating effort" when you stomp on the pedal WITHOUT adding resistance.

Speaking of resistance, how is it generated exactly, and what dependency on speed and changes in speed does it have? How does the resistance generated by the machine compare to the resistance on an actual bike?

 

[bike-nerd]

I know it's tiresome for a non-Physicist to be told to use the right terminology but it is really critical in the understanding of these things. You can't afford to confuse Force with Power.

But isn't "FORCE" part of the power equation?

You can alter the forces involved on the bike frame, pedals and the rider but there is only one source of Power / Energy and that is the rider's muscles. Gravity doesn't give you power, You can use gravity to store energy, temporarily, in order to be more effective in some operations. For instance, you can lift an axe up high, storing gravitational potential energy, and then bring it down with a bash - expending all the input energy in a fraction of a second as the axe splits the wood. No extra energy is available - it's just redistributed in time.

In some ways - standing, shifting your weight over the pedal, and pressing down hard is a little like that axe falling. The input energy is expended in a shorter time.

In your example - if you could measure the torque of the axe falling - would it be higher or lower than if you took an axe of the same weight and pressed it down against some type of reciprocating force - like a see-saw, for example?

With a bike, the situation is not like using an axe because you don't require short bursts of high power during each turn of the pedals. In fact, it would be better, in many ways, to have a steady source of energy - as with a motor. Unfortunately, our bodies don't work that way and the way cyclists use the pedals is a sort of work around to get the best overall result (hence the selection of the best gear for each situation). 'Good' cyclists tend to have learned different styles from your average guy on a bike, who can be very inefficient in the way they pedal, just because they are knackered and clutching at straws. I always know I'm on the way out when I start standing on the pedals rather than just dropping a gear.

Again, I know that steady power output is more efficient for cyclists. I know about gear selection for terrain and riding style (seated vs standing). That's not my question. My question is more academic than performance related.

The question is - When a rider stands, shifts their weight over the pedals, and expends force quickly without providing additional reciprocating force to push against (shifting gears) - what happens to torque and/or power?

 

[A.T.]

The input energy is expended in a shorter time.

Okay.

and expends force quickly

There you go again, confusing force and energy.

without providing additional reciprocating force to push against

What?

 

[bike-nerd]

Speaking of resistance, how is it generated exactly, and what dependency on speed and changes in speed does it have? How does the resistance generated by the machine compare to the resistance on an actual bike?

I'm not exactly sure I'm answering the question you're asking - but on a bike, "resistance" is changed by shifting gears. To maintain the same cadence when you add "more resistance" (shift to a bigger gear), you need to apply more force to the pedal (apologies if I'm using the wrong terminology - but I think you get the idea). The result is an increase in speed (assuming all other factors - aerodynamic drag, etc., are equal) because as the gearing increases, each revolution of the pedals translates to further rotation of the wheels.

 

[A.T.]

To maintain the same cadence when you add "more resistance" (shift to a bigger gear), you need to apply more force to the pedal

On an real bike, If you shift to a bigger gear, the cadence doesn't stay the same but inevitably drops intially. To bring cadence to the old value you have to accelerate the bike.

Is this also the case for the simulated resistance on the bike machine? Is there a flywheel or something?

 

[bike-nerd]

Yes - there's a fly wheel and a resistance unit - either a brake pad or magnetic resistance unit.

It occurs to me that perhaps the question I should have been asking all along is what happens to torque & angular velocity, since that's what the power meter is measuring.