# Wheel and Axle

Posted by PITHOCRATES - May 8th, 2013

# Technology 101

## The Key to the Wheel and Axle is the different Angular Velocities of the Outer Surfaces of the Axle and Wheel

Have you ever tried to turn a screw using only your fingers?  You might be able to get it started and spin it a few rotations.  But eventually you’ll be unable to turn the screw any further.  If you use a screw driver, though, you’ll be able to turn the screw all the way in.  Why?  For the same reason you can turn the handle on the spigot when you want to water the grass.  And why you can open the door when you enter your home.  Because of a wheel and axle.

The wheel and axle is one of six simple machines.  The others being the lever, the inclined plane, the pulley, the wedge and the screw.  The wheel and axle are two circular parts whose outer surfaces rotate at different speeds.  Think of a large wagon wheel.  Wooden spokes connect the outer rim of the wheel (the felloes) to the hub.  Imagine the wheel turning one quarter turn.  The end of the spoke at the felloes has to cover more distance than the end of the spoke at the hub.  Therefore the spoke end at the felloes travels faster than the spoke end at the hub.

In the ideal machine power in equals power out.  And power equals the torque (twisting force) multiplied by the angular velocity (how fast something spins around).  The key to the wheel and axle is the different angular velocities of the outer surfaces of the axle and wheel.  If power remains the same while the angular velocity changes then the torque must change.  Let’s use some meaningless numbers to illustrate this point.  The angular velocity is 4 and the torque is 2 on a wheel’s surface and the angular velocity is 2 and the torque is 4 on an axle.  Power in equals 8 while power out also equals 8.  But the torque increases.  So using the wheel and axle gives us mechanical advantage.  The ability to amplify force to do useful work for us.

## Mechanical Advantage amplifies our Input Force to do Useful Work for Us

What makes a screwdriver work is the handle on it that we grip.  Which represents the outer surface of the wheel.  While the metal shaft the handle fastens to is the axle.  The handle provides a larger surface for our hand to grip.  Allowing us to apply a greater turning force (torque) to the handle than we could to the metal shaft.  The angular velocity of the surface of the handle is greater than the metal shaft.  So the torque of the metal shaft is greater than the torque we apply to the handle of the screwdriver.

The mechanical advantage amplifies our input force to do useful work for us.  To turn a screw that our fingers aren’t strong enough to turn.  Just as the handle on the water spigot allows us to twist it open.  And the door knob allows us to twist open the latching mechanism to open a door.  Things we couldn’t do without a large handle to grasp and twist.  To amplify our limited force.  To do useful work.

The old-fashioned water well is another example.  Across the top of the well is an axle.  A length of rope long enough to reach the water below is attached to a bucket.  The other end is attached to the axle.  Also attached to the axle is a wheel that we can turn by hand.  Or a hand crank.  As we turn the wheel or crank the rope wraps around the axle.  Pulling up the bucket full of water.  The speed of our hand spinning the wheel or the crank is greater than the speed of the spinning axle.  That is, our input angular velocity is reduced.  Which increases the torque on the axle.  Allowing it to pull up a heavy bucket of water that we couldn’t do as easily without the wheel and axle.

## Using more Gears in a Gear Train can greatly Reduce the Angular Velocity which Greatly Increases the Output Force

We can amplify our input force more by adding some additional wheels.  And some gears.  For example, when we started harvesting sugarcane we used a mechanical press to squeeze the juice out of the cane.  And we did this by running the sugarcane through a couple of rollers with a narrow gap between them.  Crushing and pulling this cane through these rollers, though, required a lot of force.  Which we produced with a couple of wheels and axles.  One axle was the roller.  Attached to this axle was a large wheel.  Only we didn’t turn this wheel.  This wheel was a large gear.  Its teeth meshed with the teeth of a smaller gear on another axle.  Attached to this second axle was another wheel.  With a hand crank attached to it.

When we turned this wheel we rotated the small gear on the hand-crank axle.  This gear turned the larger gear attached to the roller axle.   Which pulled and crushed the cane through the press.  This reduced the angular velocity twice.  Thus increasing the torque twice.  Which twice amplified our input force.  Using more gears in a gear train can greatly reduce the angular velocity from the input axle to the output axle.  Greatly increasing the output force.  Like in a motor vehicle.  The engine spins at a high angular velocity.  The power output of the engine spins a gear train inside a transmission.  Greatly reducing the output angular velocity.  While greatly increasing the turning force sent to the drive wheels.

High-spinning electric motors have replaced the hand-crank on modern sugarcane presses.  These use a gear train or a belt and pulley system (or both) to reduce the spinning speed of the electric motor.  So when the force turns the rollers it doesn’t pull the cane through dangerously fast.  It pulls it through slow but with great force.  Which will flatten the cane and squeeze every last drop of fluid from it.  Or someone’s hand if it gets caught in the rollers.  Which usually have hand-guards around them to prevent that from happening.  But some people still operate machines that have no such guards as they hand-feed the cane into the press.  This is a disadvantage of using mechanical advantage.  For it can cause great harm just as easily as it can do useful work for us.

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