# In the case of riding a bicycle, why can the system of the bicycle and rider be accelerated if no external net force?

According to Newton, if no external net force acting on a system then the system will keep its initial condition whether at rest or moving uniformly in a straight line.

Let's consider a boy riding a bicycle. He exerts a force to the bicycle and the bicycle reacts with the same magnitude but opposite direction to the boy. From an observer outside the system of boy-bicycle, the net external force acting on the system is always zero so there will be no acceleration. However, in our daily life, we have seen that accelerating a bicycle is possible. It seems to me inconsistent, doesn't it?

What is wrong with my argumentation above?

• You're missing a really big part of the system. If a bike rider was suspended in the air, or on a really smooth sheet of ice, you would be right. And the cyclist wouldn't move. – tpg2114 May 21 '14 at 19:48
• @tpg2114: Trust me, a cyclist on a really smooth sheet of ice does accelerate. On average, downwards. Briefly. BTDT. – Ilmari Karonen May 21 '14 at 21:08
• Newton's "equal and opposite reaction" is between the bicycle and the earth. The same amount of momentum that is added to the bike going one direction is added to the earth going the other. – Hot Licks May 22 '14 at 2:22
• If you understand this comic xkcd.com/1366 you will understand the answer to your question. – Dale Wilson May 22 '14 at 15:02
• @DaleWilson: It is identical to my imagination when I was young. Lift our body to a height by an helicopter and let the earth rotating, if we are above the destination than we land. – kiss my armpit May 22 '14 at 15:06

You're neglecting the interaction between the bicycle and the ground. If I start riding my bicycle to the east, the Earth rotates a tiny bit towards the west to compensate. Because I and my bicycle are small and the Earth is enormous, it's usually a good approximation to consider the Earth as fixed and immovable.

• So we just need to get everyone to start pedaling West to stop the Earth's rotation! – Cole Johnson May 22 '14 at 1:21
• Homework: estimate moment of inertia of $7\times10^9$ cyclists riding along the equator. Compare to moment of inertia of Earth. – rob May 22 '14 at 1:31
• what-if.xkcd.com/8 – Floris May 22 '14 at 2:07
• I hope you could tell I was being sarcastic. – Cole Johnson May 22 '14 at 2:37
• @Cruncher You are basically trying to assert that a moving bicycle has no kinetic energy, and that stopping it doesn't impart a force on the surface it rides on. This simply isn't true. There are a lot of places energy is lost, and the bicycle requires continuous energy input to keep moving due to these losses, but that doesn't mean that a moving bicycle won't impart force to the earth when it decelerates. Regardless, I think you are looking too deeply into the question. Assume a perfect bicycle with no internal losses, no air resistance, and excellent friction between the tires and earth. – Adam Davis May 22 '14 at 15:55

There are forces and torques in this system. Boy pushes on pedal and pedal pushes on boy. But for pedal to push on boy, bike frame pushes on pedal (at the axle of the pedals) and vice versa. Now there is a torque on the pedal which is transferred (through the chain) to the rear wheel. Rear wheel pushes on road, and road pushes on wheel (in the forward direction - because there is friction). Wheel pushes on frame. Frame pushes saddle. Saddle pushes boy. Boy moves with the whole bike.

I left out a few steps (like the fact that the torque is balanced by a different vertical force on rear wheel than on front wheel, or that angular momentum is conserved because of the force of the bicycle on the earth...) but I hope you get the idea.

Here is a little picture showing just some of these forces. In particular, the blue arrows correspond to forces from outside the "boy-bike" system: namely, the vertical force on the wheels (which stops the bike from disappearing into the earth) and the horizontal reaction force due to the force of the wheel on the earth (not shown) which in turn is a response to all the torques (described above - just some of them shown):

Source of bike cartoon

• Nice explanation! – Dave May 22 '14 at 17:04

Your assumption that there are no external forces is incorrect. There is the force of friction between the wheels and the ground. It is this force that is responsible for the acceleration. If you could remove this (as the other posters suggested) with ice or suspending the back wheel of the bike (as in my bike trainer), you would indeed go nowhere.

• Your first line summarizes the crux of the matter nicely. – Floris May 22 '14 at 9:06