The Surprising Physics of Pulling a Bike With a Rubber Band
Do you need some practice with force diagrams for your introductory (or advanced) physics course? Here are three questions (and the last one is awesome) about the motion of a bike as it is pulled.
Case 1: Pulling the bike by the handle bars.
For this first case I will pull the bike at the top by the handle bars. I am going to pull in the direction the bike normally rides.
This one is probably pretty easy, but go ahead and see if you can figure out which way the bike will move when it is pulled. One quick note — since the bike is at rest, if it accelerates in some direction, it will also move in that direction. Really these questions are about acceleration, not moving.
Case 2: Pulling on the top of the front wheel.
Now I am going to take by giant rubber band (that is what I’m pulling with) and attach it to a spoke on the top of the front wheel. Just in case you can’t tell, I am drawing an arrow on top of the rubber band so it’s easier to see.
Which way will the bike accelerate (and thus move initially)?
Case 3: Pulling on the bottom of the front wheel.
This is the same as case 2 except that I am pulling on the bottom of the wheel.
Which way will the bike accelerate (move)?
Now you have three questions. You should have three answers. Discuss your answers with some people around you and convince them that your answer is correct. You might want to draw a force diagram to convince your neighbor that your answer is correct. If you like, you can also ask this question to random people on the street.
Here is a video version of these questions in case you want to show people with your phone.
Case 1: Answer.
I am going to go ahead with the answer to case 1. Really, this one should be easy (it wasn’t a trick).
The pulling force is to the left, the bike rolls to the left. No surprises here (hopefully).
Case 2: Answer.
With the rubber band on the top of the bike wheel, this is what happens.
The force pulls to the left. This makes the front wheel rotate in a counter clockwise direction. Both the rotation of the wheel and the force seem to suggest the bike should roll to the left. The bike rolls to the left. Everyone is happy.
Case 3: Answer.
This is the one you’ve been waiting for. Hopefully you had some excellent discussions with your neighbors and family — but not such a heated discussion that it caused a fist fight. You are right? Are they correct? Here is the answer.
If everyone answered this case correctly, it wouldn’t be that much fun. Still, it’s a great problem. But what is going on? Why does it move to the left? Well, I told you to draw a force diagram (free body diagram) so here’s what that would look like.
Yes, there is a frictional force between the front tire and the ground that pushes backwards on the wheel to cause it to rotate in the counter clockwise direction. However, this force has to be smaller than the force pulling the bike forward. The net force is pulling to the left so that the bike initially accelerates to the left. That might seem counter intuitive, but you need to practice thinking about the forces on the bike as it was just one system. For this system, the forces in the vertical direction have to add up to zero (because it doesn’t accelerate up or down) and the net force in the horizontal direction is to the left.
What if the bike did accelerate to the right? In that case, the frictional force would have to be greater than the external pulling force. This would be a little weird for a stationary object. It would be the same as if you pushed on a block to the left and it started accelerating to the right.
Could this bike accelerate to the left without having the wheel spin at all? Yes. If there was a very low coefficient of friction, the bike could just sort of slide along the ground without rolling. But it is still accelerating in the direction of the applied force.
So, did you get it this question wrong? That’s ok. It’s a difficult question — the best questions are tough. Finally, here is a video version of the answer and here is a similar question with a bit more details about rolling vs. sliding.