# Why do wheels spin?

I'm pretty sure this is probably a stupid question but it is annoying me a lot. In my physics textbook it says, cars move because the wheels exert a backward force on the ground and the ground exerts an equal and opposite forward force on the car. If the wheel spins clockwise, then there is a resultant clockwise moment on the wheel due to the force from the engine. My question is, why doesn't the equal and opposite force from the ground create an anticlockwise moment on the wheel which cancels out the clockwise moment and hence stop the wheel from spinning, instead of making the car move forward?

The engine provides a force to turn the wheel ( in physics lingo torque ) this causes the wheel to turn counter clockwise. Now look at the point of the tire that touches the ground. This point on the ground experiences one force as the wheel tries to turn - static friction (not kinetic, but that's an entire discussion in itself). This force applies a push that is proportional to the weight of the car (assuming the car is on a flat surface). This pull on the wheel creates traction (a difference in forces) which allows the wheel to turn and the car to move. Using Newton's third law which you cited isn't the correct line of thought as it means that the force friction exerts on the wheel will be equal to the force that the earth experiences from the wheel. I hope my explanation helps clarify this subject.

• Hi user287069, Welcome to Physics S.E. It is indeed a nice answer. It just needs a little corrections and more clarification. – Benjamin Apr 1 '16 at 4:30
• This is my first post, so I aim to get better at explaining myself. Thank you for the feedback – user287069 Apr 1 '16 at 11:31
• I don't think this answer is correct, but let me ask you, when you say, "Using Newton's third law ... isn't the correct line of thought..." what do you mean? Is Newton's third law wrong here? In what sense is it not the "correct line of though"? – tom10 Apr 1 '16 at 17:34
• Newton's third law is that for every action there is an equal and opposite reaction. This doesn't mean that the torque and the friction are equal. Both are different forces. It means that the axel of the wheel will experience a stress force, and that friction will act on both the earth and tire with equal magnitude. It doesn't relate the two forces. @tom10 – user287069 Apr 1 '16 at 23:56
• @tom10 and user287069 Which is the right answer? Can you two please agree with each other and give me an answer that is correct? – Nazmus Saadat Apr 2 '16 at 19:20

...why doesn't the equal and opposite force from the ground create an anticlockwise moment on the wheel which cancels out the clockwise moment and hence stop the wheel from spinning, instead of making the car move forward?

It does! That is, the moment on the wheel from the road does (almost) stop the angular acceleration of the wheel. The approach you describe is standard practice in understanding components of more complicated systems, as I explain below, so this is a good example to work that out.

I'm sure you've seen a car wheel without the friction of the road, for example on a lift, on ice, or when a car is stuck in the mud. There you can see the angular speed of the wheel ramps up extremely fast. That's what a wheel looks like without the torque from the road. But on the road it doesn't do this. That is, the force from the road balances almost the entire torque on the wheel bringing it close to zero. There is a little bit of torque that doesn't balance which is why the wheel spins (and has angular acceleration) when the car is moving, but, by far, most of the torque is cancelled out by the mechanism you suggest.

edit:
A more detailed understanding is easy to see if one writes down the equation of motion for the car and wheel. Here

• $M$ is the mass of the car
• $m$ is the mass of the wheel
• $R$ is the radius of the wheel
• $I$ is the moment of inertia of the wheel, which I'll assume to be a disk, so $I = {1\over2}mR^2$
• $a$ is the acceleration of the car
• $\alpha$ is the angular acceleration of the wheel, so $\alpha = a/R$
• $\tau_A$ is the torque supplied by the engine at the axle

Keep in mind that the mass of the wheel is much less than the mass of the car, something like $M = 100m$. Then, the torque at the axle needs to both accelerate the car and make the wheel spin: $$\tau_A = I\alpha + MaR$$ or $$\tau_A = ({m\over2} +M)Ra$$

So one can look at the above equation in several ways. It's true to say that part of the axle torque $\tau_A$ goes to spinning the wheel and part to pushing the car forward. But it's equally correct to describe a net torque on the wheel, $\tau_{NET}$, that's the axle torque less the torque caused by the reaction force, $MaR$, or, $$\tau_{NET} = \tau_A - MaR = {m\over2}Ra$$ That is, if the wheel is going to move in a certain way (here have acceleration $a$), then the net torque on the wheel must be consistent with that motion, so it's good to see here that it is. But it requires the understanding that you described where the reaction force counters the drive torque.

Since $m\ll{M}$, we have $\tau_{NET}\ll{\tau_A}$. That is, the reaction force almost completely cancels out the input torque, leaving just a small net torque on the wheel. That's expected since the wheel is much lighter than the car. The result of this is that the wheel moves appropriately to roll with the acceleration of the car, but it spins much much slower than it would if all of the torque $\tau_A$ went purely to making the wheel spin (eg, on ice, etc). So you're right that the reaction force creates a counter torque which cancels the drive torque and stops the wheel from spinning... it's not a complete cancellation, but almost complete, i.e, to within about $m/M$.

By the way, this chain of reasoning is actually fairly common. For example, consider a lever on a fulcrum with a mass on one end. When the lever lifts the mass you could ask, "doesn't the mass provide a reaction force that makes the lever not move?"; and you could analyze the whole system (lever + mass), or you could just look at the lever and find that the mass does provide a reaction force but when you work it out fully, there's just enough net force on the lever to allow it to move.

In general, this approach is important when comparing components as parts of a system vs in isolation. Specifically, forces that were action forces (where, say, part A pushes into part B) become reaction forces when the part is isolated (where now the force on A from B are considered to hold it in place -- as B would have but now it's been removed), etc. (Also, Newton's third law is the key to understanding these isolated situations, and is standard practice and understanding, which is why I don't like the other answer which says that the 3rd law is not the correct line of thinking. It is exactly the correct line of thinking, and one just needs to work it through consistently.)

Imagine that the car is moving to the right and the wheels are rotating clockwise.
The no slip condition is $v=r \omega$ where $v$ is the linear speed of wheel axle to the right, $r$ is the radius of the wheel and $\omega$ is the clockwise angular speed of the wheel.

Now suppose that the wheels are made to rotate faster (spin) which means that the angular speed of the wheels is too large or the linear speed of the axle is too small for the no slip condition.

So to get to the no slip condition again the angular speed of the wheels must decrease and/or the linear speed of the axle must increase.

A forward frictional force on the wheel will exert an anticlockwise torque on the wheels thus trying to reducing the angular speed of the wheels whilst at the same time exerting a forward force on the axle thus trying to increase the linear speed of the axle.

This continues until the new rolling without slipping condition is achieved.

This is my understanding:

The car and the Earth apply equal and opposite forces on eachother by the torque of the engine trying to turn the wheel. This causes the Earth to move backwards a little bit, and more significantly, causes the whole car to move forwards which in turn makes the wheels move.

I think what makes this case with cars confusing is that it is the wheels themselves that apply the force in order to move the vehicle forward. If you think of a car that accelerated with the help a giant fan on the back, it is easier to see that the wheels turn simply because the car is being accelerated into motion by the fan.

In summary: The torque from the wheels creates a force between the Earth and the whole car which makes the car move and it is this movement that turns the wheels.

• I will mention that the case where a car moves because torque is applied to the wheels and the wheels move the car is almost completely opposite (in a sense) to the case where an external force (the fan) moves the car and the wheels roll in response. To see this you should consider the direction of the friction force on the wheels in each case. – Todd R Apr 1 '16 at 14:17

## protected by Qmechanic♦Apr 1 '16 at 3:02

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