Beam, Fulcrum, Torque, Law of the Lever and Mechanical Advantage

Posted by PITHOCRATES - April 30th, 2014

Technology 101

(Originally published May 1st, 2013)

A Lever is a Rigid Beam on a Fulcrum

Archimedes said, “Give me a place to stand, and I shall move the Earth with it.”  At least we think he did.  Archimedes of Syracuse was a Greek genius.  Mathematician.  Physicist.  Engineer.  Inventor.  And astronomer.  One of many of the ancient Greeks who advanced modern civilization.  By using math and science.  He did a lot.  And explained why things worked the way they did using math.  Like the Law of the Lever.

In the days before the twist-off bottle cap we used bottle openers.  Because try as we might we could not pry a bottle cap off with our hands.  Most grown men just didn’t have the strength to do that.  But a child could open a bottle if that child used a bottle opener.  For that bottle opener is a lever.  Giving the child leverage.  The ability to use a little bit of force to do a lot of work.

A lever is a rigid beam on a fulcrum.  Like a seesaw.  A common playground fixture.  If two kids of equal weight are on either end of the seesaw and the fulcrum is in the center these kids can effortless push up and down.  But if a grown adult sits on one end and a child is on the other the weight of the adult will drop his side of the seesaw down.  Leaving the child up in the air on the other side.

As the Lever increases in Length the more it will Amplify the Input Force we Apply

Now that’s no fun.  Having the seesaw permanently tipped in one direction.  However, even two people of different weights can enjoy playing on the seesaw.  All they have to do is move the fulcrum towards the heavier person until the seesaw balances.  So that there is a short length of seesaw between the fulcrum and the heavy person.  And longer length of seesaw between the fulcrum and the lighter person.  This creates the same amount of torque on both side of the fulcrum.

Torque is the turning force created by a force acting about a fulcrum.  The force in this case is the weight of the people on the seesaw.  Which we calculate by multiplying their mass by the force of gravity.  With the force of gravity being constant the greater the mass the greater the weight.  This weight pressing down on the beam creates torque.   And the further away from the fulcrum the greater the turning force.  Such that a lighter weight at a greater distance from the fulcrum can balance a greater weight at a shorter distance from the fulcrum.  Allowing a child to play on a seesaw with someone of far greater mass.  Because the lever amplified the smaller force of the child.  Allowing the child to move a heavier weight.  To illustrate this consider the following table.

Lever

This is just a visual aid.  The numbers don’t represent anything.  It just shows a relationship between force and the length of the lever.  In this example we need 1000 units of force to move something.  If we use a lever that is 10 units from the fulcrum we need to apply 100 units of force.  If we have a lever that is 40 units from the fulcrum we only need to apply 25 units of force.  If we have a lever that is 80 units from the fulcrum we only need to apply 12.5 units of force.  As the lever increases in length the more it will amplify the input force we apply.  Which is why a child can open a bottle with a bottle opener.

A Wheelbarrel combines the Lever with the Wheel and Axle

A lever gives us mechanical advantage.  The amplification of a small input force into a larger output force.  Such as a hand-held bottle opener.  But what about the kind that used to be fastened to pop machines?  When you bought a glass bottle of pop out of a vending machine?  The fulcrum is the fixed bottle opener.  And the lever is the bottle.  A can opener was often on the other end of a bottle opener.  Instead of a grip to latch onto a bottle cap this end had a triangular knife.  When we lifted up on the lever it pressed down and pierced a hole in a can.

A wheelbarrel allows us to move heavy loads.  This device combines two simple machines.  A wheel and axle.  And a lever.  The wheel and axle is the fulcrum.  The lever runs from the fulcrum to the handles of the wheelbarrel.  We place the load on the lever just before the axle.  When we lift the far end of the lever we can tilt up the load and balance it over the axle.  The lever amplifies the force we apply.  And the wheel and axle reduce the friction between this load and the ground.  Allowing us to move a heavy load with little effort.

Today’s pop bottles have screw-top caps.  Some people still use a lever to help open them, though.  A pair of pliers.  We use the pliers because we don’t have the strength to grip the cap tight enough to twist it open.  The pliers are actually two levers connected together at the fulcrum.  The pliers amplify our hand strand-strength to get a very secure grip on the bottle cap.  While our hands compress the two levers together getting a firm grip on the cap we can then use our arm to apply a force on the handles of the pliers.  Providing a torque to turn the bottle cap.  Very simple machines that make everyday life easier.  Thanks to the knowledge Archimedes handed down to us.

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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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Beam, Fulcrum, Torque, Law of the Lever and Mechanical Advantage

Posted by PITHOCRATES - May 1st, 2013

Technology 101

A Lever is a Rigid Beam on a Fulcrum

Archimedes said, “Give me a place to stand, and I shall move the Earth with it.”  At least we think he did.  Archimedes of Syracuse was a Greek genius.  Mathematician.  Physicist.  Engineer.  Inventor.  And astronomer.  One of many of the ancient Greeks who advanced modern civilization.  By using math and science.  He did a lot.  And explained why things worked the way they did using math.  Like the Law of the Lever.

In the days before the twist-off bottle cap we used bottle openers.  Because try as we might we could not pry a bottle cap off with our hands.  Most grown men just didn’t have the strength to do that.  But a child could open a bottle if that child used a bottle opener.  For that bottle opener is a lever.  Giving the child leverage.  The ability to use a little bit of force to do a lot of work.

A lever is a rigid beam on a fulcrum.  Like a seesaw.  A common playground fixture.  If two kids of equal weight are on either end of the seesaw and the fulcrum is in the center these kids can effortless push up and down.  But if a grown adult sits on one end and a child is on the other the weight of the adult will drop his side of the seesaw down.  Leaving the child up in the air on the other side.

As the Lever increases in Length the more it will Amplify the Input Force we Apply

Now that’s no fun.  Having the seesaw permanently tipped in one direction.  However, even two people of different weights can enjoy playing on the seesaw.  All they have to do is move the fulcrum towards the heavier person until the seesaw balances.  So that there is a short length of seesaw between the fulcrum and the heavy person.  And longer length of seesaw between the fulcrum and the lighter person.  This creates the same amount of torque on both side of the fulcrum.

Torque is the turning force created by a force acting about a fulcrum.  The force in this case is the weight of the people on the seesaw.  Which we calculate by multiplying their mass by the force of gravity.  With the force of gravity being constant the greater the mass the greater the weight.  This weight pressing down on the beam creates torque.   And the further away from the fulcrum the greater the turning force.  Such that a lighter weight at a greater distance from the fulcrum can balance a greater weight at a shorter distance from the fulcrum.  Allowing a child to play on a seesaw with someone of far greater mass.  Because the lever amplified the smaller force of the child.  Allowing the child to move a heavier weight.  To illustrate this consider the following table.

Lever

This is just a visual aid.  The numbers don’t represent anything.  It just shows a relationship between force and the length of the lever.  In this example we need 1000 units of force to move something.  If we use a lever that is 10 units from the fulcrum we need to apply 100 units of force.  If we have a lever that is 40 units from the fulcrum we only need to apply 25 units of force.  If we have a lever that is 80 units from the fulcrum we only need to apply 12.5 units of force.  As the lever increases in length the more it will amplify the input force we apply.  Which is why a child can open a bottle with a bottle opener.

A Wheelbarrel combines the Lever with the Wheel and Axle

A lever gives us mechanical advantage.  The amplification of a small input force into a larger output force.  Such as a hand-held bottle opener.  But what about the kind that used to be fastened to pop machines?  When you bought a glass bottle of pop out of a vending machine?  The fulcrum is the fixed bottle opener.  And the lever is the bottle.  A can opener was often on the other end of a bottle opener.  Instead of a grip to latch onto a bottle cap this end had a triangular knife.  When we lifted up on the lever it pressed down and pierced a hole in a can.

A wheelbarrel allows us to move heavy loads.  This device combines two simple machines.  A wheel and axle.  And a lever.  The wheel and axle is the fulcrum.  The lever runs from the fulcrum to the handles of the wheelbarrel.  We place the load on the lever just before the axle.  When we lift the far end of the lever we can tilt up the load and balance it over the axle.  The lever amplifies the force we apply.  And the wheel and axle reduce the friction between this load and the ground.  Allowing us to move a heavy load with little effort.

Today’s pop bottles have screw-top caps.  Some people still use a lever to help open them, though.  A pair of pliers.  We use the pliers because we don’t have the strength to grip the cap tight enough to twist it open.  The pliers are actually two levers connected together at the fulcrum.  The pliers amplify our hand strand-strength to get a very secure grip on the bottle cap.  While our hands compress the two levers together getting a firm grip on the cap we can then use our arm to apply a force on the handles of the pliers.  Providing a torque to turn the bottle cap.  Very simple machines that make everyday life easier.  Thanks to the knowledge Archimedes handed down to us.

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