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Here’s a deceptively hard free body diagram problem: draw a free body diagram for a motorized cart moving at constant velocity. Even though these constant velocity buggies are staples of the first weeks of many physics classes, I’ve come to realize the free body diagrams for these buggies are more complicated than you might expect.

Because the cart is moving at a constant velocity, the net force has to be zero, but I’ve always wondered if there are any horizontal forces acting on the cart.

It could be that there’s a forward frictional force (we could call this traction) exerted by the table on the rotating rubber wheel, but then I struggled to think about what force would act in the backward direction to make the forces balanced. The car is moving to slowly for drag to be significant (though the drag force balance the forward frictional force of the road when a real sized car is traveling at highway speeds). Could it be a backward frictional force acting on the non driven wheels?

Thanks to a conversation with a few other physics teachers, we devised this ingenious experiment to test to see if there are any horizontal forces acting on the cart when it moves at constant velocity. We started by setting a board atop two metal cylinders to create a platform that felt no frictional force from the table, and would respond to any horizontal forces applied to the board. We also placed a whiteboard just above the board so that it wasn’t touching the board and the cart could roll at constant velocity from the whiteboard to the platform without a significant change in velocity.

I then made an alignment mark on the board and a stationary piece of paper, and allowed the cart to roll onto the board. If there is a frictional force, then we would see the board move in the direction of the frictional force acting on the board (and opposite the frictional force acting on the car, via Newton’s 3rd law).

Here is the video:

As you can see, there is no movement of the board when the cart travels onto the board from the whiteboard. Since the board stays at rest, we can conclude the cart does not exert any horizontal forces on the board when it is moving at constant velocity, and by Newton’s 3rd Law, the board isn’t exerting any horizontal forces on the cart. The cart’s free body diagram is just:

One other possibility exists that perhaps the rollers beneath the board are able to counteract a frictional force of the cart, which might be possible if the frictional force is small enough. However, we can rule this out by seeing that when the car comes to a stop on the board, the board moves forward, indicating a board experienced a forward frictional force during the time when the car was changing velocity.

This seems somewhat counterintuitive at first—aren’t the wheels pushing the car and making it go? If the frictional force is zero, then why is the motor necessary at all?

Here’s one more video that shows when the cart is accelerating forward, there is indeed a forward frictional force of the board acting on the cart. Both of these videos also make great examples of momentum conservation

Finally, all of this connects back to some of my previous wonderings about the mysteries of wheels. But as I read my post about wheels again, I come to this quote:

When the car is going at constant velocity, the net torque and force on the wheel must be zero. this means that the traction force (F in the diagram) must equal the retarding force (f) in the diagram. The road can exert 2 forces on the tire since the the tire isn’t contacting the road in a single point, there is an area of contact (and this helps to understand some of Noah’s question about how the placement of a banana under one’s foot might cause you to fall forward or backward). The key idea is these are point objects, or points of contact, so you can have multiple forces at a single surface.

Maybe my first diagram was correct and the board exerts two forces (a traction force and a retarding force) on the back wheel of the cart that are exerted from two slightly different point on the wheel due to its deformation, but these two forces exactly cancel each other out and so we see no motion in the board. Or maybe instead this wheel really is ideal, the deformation is negligible and the the traction and retarding forces are zero.

Now I’m just confused. What do you think?

7 Comments leave one →
1. Andy "SuperFly" Rundquist permalink
June 30, 2013 12:39 am

A quick thought: We’ve been modeling spinning billiard balls a lot lately, and our model always ends with the ball “rolling without slipping.” At that point, the contact frictional forces don’t do any more work and the system stops losing energy. I think that’s related to this. If you figure the drag forces are pretty small, then I think your wheels are rolling without slipping. However, the motor is needed because of the friction in the bearings, i would think. Hmmm, but if that were the case, the motor would run the same speed whether on the ground or held in your hand, which I would guess is not the case. Interesting!

• June 30, 2013 10:07 am

Andy,
Thanks—I tried to measure the speed of the motor in the air vs ground. I made a chalk mark on the wheels and made a video recording. It seems that they are very close to the same—about 15 frames/rotation in both cases.

June 30, 2013 9:16 pm

If the motor did run faster when the wheels were in the air compared to on the ground, that may be due to the bearings (or bushings) having greater friction when under load.

Great question and analysis!

June 30, 2013 11:26 am

I expect that the answer to this question is that the car is not a point object, and there are a several internal forces. I think the motor is necessary to maintain the wheels’ constant angular velocity, which otherwise would be reduced by internal friction forces. So long as the wheels maintain constant angular velocity, they will be turning at just the right speed to “keep up” with the car without any frictional force being exerted between the tire and the road. If the wheels slow down because of internal friction, linear friction between the tires and road will slow the cart (this is essentially how brakes work, isn’t it?). If this echos your “previous musings”, I apologize for not having read them.

June 30, 2013 12:47 pm

Andy said what I was going to say – the next experiment would be to run the car without it being on the track.

Another cool experiment would be to change the wheel to something harder or something softer and see what that does.

4. June 30, 2013 9:30 pm

Interesting stuff.
In your second video, we see the platform move to the right as the buggy accelerates, but when the buggy deccelerates, the platform doesn’t move much at all. I thought that was a bit curious.

5. Michael Mannix permalink
July 18, 2013 7:31 am

John,
This was something I was thinking about, recently, as I re-tool my lessons for the upcoming year and anticipating discussions. My musing had me thinking about the EMF and back EMF in the motor. My background in circuits is not strong, but my guess is that the EMF and back EMF both fluctuate slightly at a high frequency – the time-averaged result of this fluctuating current is the constant-velocity cart. This high-frequency interplay between EMF and back EMF makes me wonder if the traction and retarding forces vary in a similar fashion that time-averages out to zero net force,

Moving forward, I think the explanation of equal traction and retarding forces works best for the purposes of my classroom.