# Why does friction not accelerate a wheel?

It seems like a silly question because this defies common sense, but it appears that friction is supposed to accelerate a wheel (not attached to anything).

We can derive from Newton's laws that $$\mathbf{F} = m\mathbf{a}$$ works for an extended object just as it does for a point particle---we just need to treat the center of mass of the object as the object's position. A rolling wheel has three forces acting on it: the force due to gravity, the normal force and friction. The net force on the body is the friction---which is nonzero---and so $$\mathbf{F} = m\mathbf{a}$$ tells us that the center of mass of the wheel must be accelerating.

I doubt this is the correct conclusion, but why am I wrong? The argument appears to be indisputable.

• Why is friction in this case non-zero? – BowlOfRed Mar 11 '19 at 17:19
• @BowlOfRed If the friction were zero then the wheel wouldn't be able to roll. – PiKindOfGuy Mar 11 '19 at 18:03
• Friction DOES cause the wheel to accelerate. Try spinning a wheel above the ground to some rotational speed. When you drop it, it slides against the ground, and kinetic friction will accelerate the wheel until it stops slipping. – David White Mar 11 '19 at 19:00
• If there is friction, then the wheel accelerates, yes (assuming no other influences). But ideally, on flat, horizontal ground there is no friction during rolling. The rolling doesn't accelerate or slow down. It just continues with constant rotational speed and constant translational speed of the centre of mass. – Steeven Mar 11 '19 at 19:20

We know two things about the friction force; firstly it is in the direction of motion: otherwise it would accelerate the wheel up or down. Secondly it works from the point of contact. For a force to do any work it has to act in the direction of motion: $$W=\vec F\cdot\Delta \vec x$$. If we consider a perfect wheel there will be only one point that has contact with the ground, in the case of perfect rolling (no slipping) the force of friction is zero. The contact point is actually stationary for a very short amount of time. In the picture you can see that the cusp is actually vertical at the point of contact. The contact point moves purely up and down so the direction of motion is normal to the friction force so no energy is transmitted. In fact, you could remove the ground and gravity and the wheel would keep moving since momentum and angular momentum is conserved.

It is possible for the friction force to do work and accelerate the wheel though. Consider the case of a burnout. Spin the wheel really fast but keep it locked in place at $$x=0$$. When $$t=0$$ you release the wheel and it will accelerate: the point of contact moves with respect to the ground so friction can perform work. The rotation of the wheel slows because rotational energy is converted to regular kinetic energy until you have perfect rolling again.

Finally I would like to mention that in rotation momentum can be 'hidden' quite easily. Imagine you strap a couple of rockets to a wheel in vacuum in a clockwise direction. If you fire the rockets all the rockets will lose momentum, each of the rockets exerts a force. But if you align the rockets well enough the wheel will not accelerate. In the case the total momentum stays zero but if you consider a small part of the wheel it will be moving very fast. The individual parts have momentum but the total sum has zero net momentum.

• Okay, you've shown that the frictional force does no work when there's rolling without slipping, but you have not addressed directly why my argument is wrong. – PiKindOfGuy Mar 12 '19 at 22:53

Consider the left hand wheel: it is either stationary or rolling at constant speed, without slipping.

For rolling without slipping the condition: $$v=\omega R\tag{1}$$ holds, where $$\omega$$ is the angular velocity about the CoG of the wheel and $$R$$ its radius.

As both $$v$$ and $$\omega$$ are either $$0$$ or constant, this means no net forces or torques act on the wheel.

Friction isn't needed here. The left hand scenario can be imagined for instance where a wheel moves like this on a perfectly frictionless surface. But it works only on a surface capable of providing friction.

But on the right we introduce a net force, $$F_{Net}$$ which will cause acceleration $$a>0$$.

To maintain the rolling w/o slipping condition of $$v=\omega R$$, a friction force $$F_f$$ is needed. The torque about the CoG this force causes is needed for the angular acceleration $$\dot{\omega}$$:

$$F_f\times R=I\frac{\mathrm{d}\omega}{\mathrm{d}t}=I\dot{\omega}\tag{2}$$

where I is the inertial moment of the wheel.

We can quantify $$F_f$$ as follows.

Derive $$(1)$$ with respect to time:

$$a=\dot{\omega}R\tag{3}$$

The acceleration $$a$$ respects $$F_{Net}=ma$$ ($$m$$ is the mass of the wheel) and with $$(2)$$ and $$(3)$$:

$$\frac{F_{Net}}{m}=\frac{F_f R^2}{I}$$

If the friction were zero then the wheel wouldn't be able to roll.

This is incorrect. Friction may be a reason that affects the rolling of a wheel, but it is not necessary.

An a thought experiment, let's imagine a wheel on an icy surface that provides zero friction. We can still move the wheel left and right, and we can still spin the wheel. It's just that without friction, these two motions become independent.

What friction does is oppose relative motion between the wheel surface and the ground surface. When this relative motion is zero, there is no frictional force at all.

This means that friction accelerates the wheel (forward) in cases where the motion of the wheel would tend to move the edge of the wheel backward with respect to the ground. This is true when an automobile wheel is driven by the engine and the axle. When you accelerate the car, you are using friction to drive the car forward. If you were to attempt this on a frictionless surface, the wheel would rotate, but would not accelerate forward.

In the opposite case, a car under braking, the rotation of the wheel is slowed by the brakes and friction with the ground creates a rearward force that decelerates the car. The wheel is moving forward with the same speed as before, but the frictional force is in the opposite direction.

You can't determine the magnitude or the direction of the frictional force simply from the velocity of the wheel. Additional information is required.