# The Physics of Power on a Bicycle

1. Mar 11, 2015

### 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??

2. Mar 11, 2015

### 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.

3. Mar 11, 2015

### A.T.

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.

Last edited: Mar 11, 2015
4. Mar 11, 2015

### A.T.

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.

5. Mar 11, 2015

### bike-nerd

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

6. Mar 11, 2015

### A.T.

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.

7. Mar 11, 2015

### 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.

8. Mar 11, 2015

### A.T.

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

9. Mar 12, 2015

### 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.

10. Mar 12, 2015

### 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?

Last edited: Mar 12, 2015
11. Mar 12, 2015

### A.T.

They obviously become more variable, just like the force.

What equilibrium?

12. Mar 12, 2015

### 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.

13. Mar 12, 2015

### A.T.

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

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

14. Mar 12, 2015

### 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?

15. Mar 12, 2015

### A.T.

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.

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

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.

Last edited: Mar 12, 2015
16. Mar 12, 2015

### 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?

17. Mar 12, 2015

### A.T.

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?

18. Mar 13, 2015

### 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?

Last edited: Mar 13, 2015
19. Mar 13, 2015

### A.T.

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

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

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.

20. Mar 13, 2015

### jimmy1010100

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.

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