# Why is it so much harder to ride a bike up a hill than to push it?

Title says it all. From a physics point of view, gravity is a conservative force and so the amount of work done should be the same, and yet one option is a lot more tiring than the other. It's especially odd if we consider friction, since rolling friction is less than static friction, which should imply that cycling up the hill does (slightly) less work. I understand that it's because of friction that if the terrain is flat, cycling is easier than pushing a bicycle.

Here're some hypotheses I've thought about:

1. If a person is standing on an inclined plane, he's probably stationary (because of friction). But a bicycle will roll downwards. This constitutes an extra force that a cyclist has to overcome.
2. Something to do with the human body, i.e. if we replaced the cyclist with a machine then it'll indeed do the same amount of work both ways.
3. Something to do with the human body, in the sense that if we had an Olympic-level cyclist then he'll have no problems cycling up the hill. In other words I only feel more tired because I'm unfit.
4. Tiredness correlates to power (physics "power" - work done / time taken). One cannot ride a bicycle slowly up the hill, since the bike won't stay upright; conversely if one is riding the bicycle faster than pushing speed, there's a difference in power.

I find #1 unconvincing since it seems to be double-counting the "roll backwards" force together with gravity. #4 sounds reasonable but on flat ground things reverse, which is contradictory. The other two are plausible. Does anyone know if either (or both / neither) are correct?

• The question asks why X is true, where X is either false or not clearly defined. It matters what gear you're in. In a low enough gear, X is false. – Ben Crowell Feb 6 '18 at 3:55

The bike's gears are set so that a large force over a shorter distance at the peddle gives a smaller force over a longer distance at the wheel. This is the opposite to most machines, eg. a screw-jack or winch where you put in a smaller force over a longer distance to move a much heavier load.

Normally when you are cruising on the level you want this - the force needed at the wheel to overcome drag and friction is low compared to the force you can easily push with on the peddle. But going uphill you reach the point where to move fast enough to stay upright you have to put a lot of force on the peddle.

Some extreme mountain bikes have low gear ratios <1, so that less force on the peddle than at the wheel - you have to spin to move at a crawling pace.

• However, shouldn't the work done be independent of the gear? This seems to imply that tiredness is correlated with the force exerted, not the total work done. – Allure Feb 6 '18 at 11:00
• Yes work is constant but power and force aren't. You need to lift the bike uphill at a certain speed = a certain power (work/time). If this is more energy (force * distance) than you can apply at the peddle it becomes impossible. – Martin Beckett Feb 6 '18 at 16:21
• Also, note that the bodies energy expenditure need not be constant for all velocities/powers. I would imagine that the body is more efficient at certain power "settings". – aidan.plenert.macdonald Feb 7 '18 at 1:46
• You can gear down to get more torque, but at some point the forward speed is reduced so much that balancing the bike requires the additional effort of having to use oscillating steering motions to remain upright. – docscience Feb 7 '18 at 2:06

Being a 65-year-old cyclist who lives at the top of a steep hill, I am well-familiar with this problem. When you are pushing a bike up a hill on foot, you are moving uphill very slowly and the power required to do this is less than that required to ride the bike (i.e., go fast enough to keep the bike upright) up the same hill.

On flat ground, things reverse, but for a different reason. Imagine you are trying to push your bike at a run and go as fast as you could while riding it. Even with maximum exertion, you simply can't do that because you can't move your legs back and forth fast enough, carry your own weight, and push the bike all at the same time. Because the bike lets you change gears as you go progressively faster, you do not have to whip your legs back and forth at some physically impossible speed to go fast on the bike- you just select a taller gear. In addition, you do not have to support your own weight while on the cycle, and you can take a break while riding by coasting momentarily.

• I'm not sure I understand your explanation for the reverse situation on flat ground. The person on the bike could cycle at a leisurely pace, which should be slower than the maximum speed the running person can go at. But the running person will tire a lot quicker. Alternatively the cyclist could go faster (= higher KE = higher work done), and again he would feel less tired (?). – Allure Feb 6 '18 at 11:05
• my point is it takes less work to travel on a bike on flat ground than it does to run at the same speed on flat ground. – niels nielsen Feb 6 '18 at 19:02

Friction is defined as

$F=μ*Ν$

Where $μ$ is the coefficient of friction, either static or kinetic, and N is the normal force.

When one walks uphill the leg muscles mainly carry the body weight efficiently, the foot making a minimal contact and the body weight is transferred up hill. Pushing a bike uses the hand muscles to transfer a force , the wheels of the bike making the minimal contact on the road surface , rolling resistance

My take is ,(as I do not even bike, much less change gears on a bike) that the spending of more energy (harder) is because of more friction. I.e the sum of frictional forces "weight on feet"+"rolling bike", is smaller than friction on "body +bicycle weights" on rolling bike. The reason: the weight will flatten the wheel contact on the tarmac , making the "rolling" friction larger, thus in total more work needs to be done.

• Even uphill I think a bike is more efficent than walking. The rolling resistance is very small compared to the energy wasted walking by lifting your body up and down on each step. – Martin Beckett Feb 6 '18 at 16:23
• @MartinBeckett well, at the extreme, for badly inflated tires, the extra 150 pounds would make a big difference.. An efficient walker keeps the center of mass of the body at a level, ( or a fixed angle if up an incline)I think, by synchronizing the lifting of thefeet, only in dancing one goes up and down. see this link.springer.com/article/10.1007/s11633-017-1088-5 – anna v Feb 6 '18 at 16:47
• But I always keep my bike tires fully inflated, but model my uphill (and up-Alp) gait on The Sound of Music ;-) – Martin Beckett Feb 7 '18 at 4:42

Your legs were made for running. That’s what they’ve gotta do, or one of these days the lions gonna get a bite of you.

Somewhat different sets of muscles are involved in biking uphill or climbing stairs versus walking or running. When biking, your hip extensor (gluteus maximus) does most of the work. In relaxed walking, your hip flexor (psoas) swings your thigh forward, but the gluteus is lazy. The knee extensor (quadriceps) and flexor (hamstrings) are important in all of these activities.

## protected by Qmechanic♦Feb 6 '18 at 23:33

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