# Friction of a Wheel on the Axle?

• B

## Main Question or Discussion Point

So say ive got a wheel who's outer radius is 12" and the radius of the axle is 1.5", so the simple machine here is an 8 to 1 ratio.

So im wondering, since 1/2mv^2 is for kinetic energy, and the velocity here is 1/8th, does it mean the friction is 1/8th or 1/64th? does it get 1/8th hotter or 1/64th hotter?

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CWatters
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No.

The friction at the axle bearing has no connection with the friction between wheel and road. Just for info...Car designers try to minimise one and maximise the other.

CWatters
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The _torque_ caused by friction between wheel and ground is usually the same or similar to the torque at the axle.

rcgldr
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The _torque_ caused by friction between wheel and ground is usually the same or similar to the torque at the axle.
One exception would be accelerating and non-driven wheels, such as the front wheels on a rear wheel drive car. Friction would be related to torque which in turn would be related to acceleration and inertia of the non-driven wheel. The torque at the axle in this scenario could be near zero.

jbriggs444
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The frictional torque produced by a bronze tire sliding on a bronze road ought to be about eight times the frictional torque between a bronze axle and the bronze wheel hub. That is one motivation for using wheels instead of sled runners.

right, so a simple machine comparison, lets revise how this is being asked, as briggs seems closer to what I'm asking.

if you had an axle with a radius of 11" and a wheel of radius 12" wrapped around it, (11/12) (no ball bearings), so you have direct contact, and friction as the wheel rolls (such as a cart or chariot, pushed or pulled by a force such as a a cow, human, or gravity down a hill)
As the axle grinds against the surface of the wheel, it has friction, and as the wheel is forcibly rotated (such as by a team of 12 oxen pulling with great force) the friction of the two surfaces causes temperatures at their contact points to increase.
meanwhile, if the radius of the axle is 1" but the wheel radius is 12", you have 1/12. 0.916 inches of surface friction travel per inch of road travel, vs. 0.0833 inches of surface friction per inch of road travel. Or 11 times slower. Friction comes in static friction, and kinetic friction.

Obviously the static friction for the rotating wheel with the larger axle is higher, because of the larger surface, but
what about the kinetic friction? And what about the heat generated? If the kinetic friction is caused by an object moving 11 times slower, does it resist 11 times less? Or is it only the torque that matters? And the heat of the surfaces of the axle and wheel, they should be lower, but is it proportional or is it squared? Like if a friction surface is moved twice as fast does the kinetic energy, being 4 times higher produce 4 times as much heat or twice as much heat? Likewise, if our axle is 11 times smaller, does it produce 11 times less heat or 121 times less heat?

jbriggs444
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Obviously the static friction for the rotating wheel with the larger axle is higher, because of the larger surface, but
what about the kinetic friction? And what about the heat generated? If the kinetic friction is caused by an object moving 11 times slower, does it resist 11 times less? Or is it only the torque that matters? And the heat of the surfaces of the axle and wheel, they should be lower, but is it proportional or is it squared? Like if a friction surface is moved twice as fast does the kinetic energy, being 4 times higher produce 4 times as much heat or twice as much heat? Likewise, if our axle is 11 times smaller, does it produce 11 times less heat or 121 times less heat?
The relevant notion is that of work. Work = force times distance. Or torque times rotation angle.

We typically model kinetic friction as depending only on the normal force and not on the slip rate. So a bronze on bronze bearing supporting the weight of a wagon should have the same linear frictional force as a bronze-shod tire on a bronze rail supporting the same wagon.

If the wheels were locked and the wagon was skidding along the rails, that frictional force is what the horses would need to overcome. The wagon tires skid a long distance against the frictional force, dissipating a large amount of work.

With the wheels released, the moving surfaces are now at the hub. The frictional force is the same, but the hub moves a small distance against the frictional force, dissipating a small amount of work.

It is a direct proportion. A mechanical advantage of 8 to 1 translates to a reduction in work dissipated of 8 to 1.

CWatters
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I'm not really sure you are asking the right questions. What are you trying to understand/design?

Wheels roll so the power required to move a cart depends mainly on the rolling resistance between wheel and road not the friction between wheel and road.

There are several ways to model friction but the simple model is independent of contact area because the larger the area the lower the pressure between the surfaces.

Smaller diameter plan bearings move the friction nearer to the axis of rotation so the torque and power is reduced...but ball bearings are better.

Static friction in a bearing is only significant when the cart starts to move. Once moving there is no static friction only kinetic friction. Kinetic friction in simple models is independent of velocity but thats not always true.

• jbriggs444
From a grinding wheel, I'm able to intuit that as an object increases in speed, the amount of friction inch per inch is constant, while the total friction occurring per second increases, and that increases heat, which can lead to friction welding. So im trying to understand the other mechanical advantages of a wheel in relation to its axle, in terms of surface contact, heat, and friction, and not just in terms of a lever to the radius.

so if the wheel moves at a constant, say 300 rotations per minute, and assuming an axle radius of "1" unit, wouldn't a fatter axle, with a radius of 1.4142 produce twice as much heat? since while the revolutions of the whole wheel are constant, the total amount of surface area moving against another surface increases? Then, without bearings, an optimal axle would have the smallest possible diameter while being strong enough not to break?

CWatters
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so if the wheel moves at a constant, say 300 rotations per minute, and assuming an axle radius of "1" unit, wouldn't a fatter axle, with a radius of 1.4142 produce twice as much heat? since while the revolutions of the whole wheel are constant, the total amount of surface area moving against another surface increases? Then, without bearings, an optimal axle would have the smallest possible diameter while being strong enough not to break?
Increasing the contact area reduces the contact pressure (Newton's per square meter, or lbf per square inch) so many model of friction ignore contact area.

The simplest model uses the equation..

F=uN

Where F is the force needed to overcome friction, u is the coefficient of friction that depends on the material and N is the Normal force between the surfaces. No mention of contact area.

Obviously this model is very simple. Other more complicated models exist but this isn't my field.