Mechanical Transmission of Energy

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MECHANICAL TRANSMISSION OF ENERGY

Mechanical devices are used to magnify the applied force (mechanical advantage), to magnify the distance moved, or to change the direction of the applied force. They of course cannot decrease the amount of work (force × distance) necessary to do a job; they only make it more convenient to do it. In many cases, without a machine, the job would be impossible.

There is generally considered to be five distinct simple machines: lever, wedge, wheel and axle, pulley, and screw. The transmission of energy by these simple machines is so basic that people use them with little understanding of the physical principles involved. Most learn their use intuitively, through experience, and consider their application just plain common sense.

THE BASICS

The history of the origin of simple machines is largely conjectural, but there also exists documentation of the ancient Egyptians using simple machines to build pyramids nearly 5,000 years ago. An inscription in a 4,000-year-old tomb tells of 2,000 men pulling a statue estimated at 132 tons into place. The mass of the 2,000 men would be about the same as the mass of the statue, and it would probably take that many because they moved it on sledges without wheels.

The use of simple machines has sometimes been taken as a definition of what separates humans from animals; however, some primates have been observed fashioning probes out of sticks to pry out or to reach food. One of the most powerful images depicting the use of tools as defining humanity is the opening scene in the movie 2001. An ape has discovered the club and is bashing some bones. One of the bones flies upward and in slow motion morphs into a spaceship. The club or the hammer is such a basic tool that it does not even make it into the classical listing of the five simple machines. However, it is also a mechanical device that multiplies force and transmits energy.

In the transmission of energy by these simple machines, the conservation law always applies: The work input equals the work output. When work is done by a system, energy is transferred out of it; and when work is done on a system, energy is transferred into it. When two objects interact by way of a machine (e.g. a lever), the work out of one object equals the work into the other. The work done by a person forcing one end of a lever downward equals the work done lifting a load at the other end as the lever moves upward. In any practical situation, the frictional forces resisting motion will always increase the amount of force (and work) required to do a job.

The amount of work done on an object is determined by the force exerted on it multiplied by the distance it moves in the direction of the force. Therefore the key to figuring out how much the force is magnified by a simple machine is to compare distances moved. For example, if the end of a lever under a stone weighing 2,000 newtons moves upward 1 meter, the amount of work done lifting the stone is 1 meter × 2,000 newtons = 2,000 joules. An equal amount of work must be done by a person on the other end of the lever. If that end of the lever is pushed downward 2 meters, then the person needs to apply a force of only 1,000 newtons to do the same amount of work (2,000 joules) and lift the 2,000 newton stone.

In brief, the mechanical advantage of a lever or any machine equals the ratio of the distance the applied force moves to the distance the load moves.

THE LEVER

The lever is such a part of everyday activity that its application usually requires no conscious thought: the pop top on a soda can, a doorknob, a wrench, pliers, a fishing pole, a faucet with a handle that lifts, a wheelbarrow, fingernail clippers, and so on. The crank and winch used to pull a heavy boat out of the water up onto a trailer can be thought of as a lever in circular form.

Basically, a lever is a solid object with an axis about which it rotates (fulcrum). As the lever rotates about its fulcrum, a point on the lever farther from the fulcrum moves a greater distance. The conservation of energy applied to the lever results in the fact that the output force times its distance from the fulcrum equals the input force times its distance from the fulcrum. A little experience lifting heavy loads with a lever soon teaches one that to maximize the output force, the load should be placed as close to the fulcrum as possible and the input force as far from the fulcrum as possible. To dramatize the nearly infinite possibility of the lever to magnify force, Archimedes said that if he had a lever long enough and somewhere to stand, he could move Earth.

As an illustration of the lever, consider the current design of the pop top on a popular brand of cola (Figure 1). The built-in opener is a piece of aluminum 25 mm long, made rigid by crimping its edges. The fulcrum is 5 mm from the end which presses into the top of the can to "pop" it open. This leaves 20 mm from the fulcrum to the end, which the user lifts up on with a force of about 5 pounds to open the can. (Since ratios of quantities are involved, there is no problem with mixing English and metric units.) The ratio of distances is 20 mm/5 mm = 4, which means that the opener presses into the top of the can with a force of 4 × 5 pounds = 20 pounds. If the built-in opener is missing, it is necessary to open the can by pressing a small object such as a key into the top with a force of 20 pounds. It is not impossible but it is awkward, and a slip could be messy.

THE WEDGE, THE INCLINED PLANE, AND THE SCREW

These three simple machines change the direction of the applied force as well as magnify it. Each one's operation can be understood by nearly the same physical principles.

A wedge is fairly easy to understand. One side of a heavy rock can be lifted a small amount by pounding a wedge under it, as illustrated in Figure 2. If friction is neglected, the force pushing the wedge under the rock is magnified by the ratio of the distance the wedge moves to the amount the rock is lifted. This follows from requiring the work done by the driving force to equal the work done in lifting the rock. This magnification is the ratio of the length of the wedge to its width, and is obviously greater the smaller the angle of the wedge. The friction force of the wedge against the rock decreases the available lifting force, but it is to some extent an advantage. It holds the wedge in place even when no driving force is applied. A wedge is usually used in combination with a hammer.

The hammer itself is an important simple machine used to magnify force. The ape in 2001 was shown inventing the hammer or the club. The user applies a force on the hammer through a relatively large distance, giving it kinetic energy. The hammer comes into contact with an object, compressing it or moving it some short distance. The average delivered force depends on how much compression or motion of the object there is and on whether the hammer rebounds off the object. The mechanical advantage is approximately the ratio of the distance the hammer moves to the distance the wedge, for instance, moves. A person might apply a relatively small force to a hammer for a 1-meter swing. If it drives the wedge 1 millimeter, then the mechanical advantage is 1,000. No wonder that striking a wedge with a hammer can lift an object weighing several tons.

The inclined plane is stationary, and the load to be lifted is moved; whereas it is the other way around for the wedge. The force is magnified by the ratio of the distance the load is pushed to the height through which it is lifted. Again friction will increase the amount of force that has to be applied. If the coefficient of friction is large enough (at least as great as the tangent of the angle of incline), then the load will not slip back down the incline when the pushing force is released. For most materials (coefficient of friction greater than, 0.6 say), an angle less than 30 degrees will keep the load from slipping back on its own. This allows the pushers to take a break to regain their strength before finishing the job. Or they can push the load up the incline a little at a time by lunging at it. In this way they play the role of the hammer relative to the wedge as they collide with the heavy load.

The screw is an inclined plane that is conveniently wrapped around a circular cylinder. The incline of the screw is in the form of a helix similar to a spiral staircase. The mechanical advantage is the ratio of the distance the driving force moves in a circle as it rotates the screw to the distance the load is lifted. A screwdriver with a large handle will provide a larger mechanical advantage, as will a screw with threads closer together. By using a screw jack with a long enough handle, a person can easily lift an automobile or a house. Again, friction conveniently keeps the screw from backing up on its own so that the user can just leave the jack supporting the load, assured that the jack will stay in place.

THE PULLEY

A fixed pulley is a device for changing the direction of an applied force. A common form is a mounted wheel with a rim around which a rope passes. In a very primitive form it could be a vine looped over a tree branch. A pull downward on the rope (vine) results in lifting a load on the other end. Neglecting friction, the mechanical advantage of the single fixed pulley is 1; the load moves the same distance as the applied force.

A combination of fixed and movable pulleys can achieve larger mechanical advantages. The diagrams in Figure 3 show systems of pulleys with mechanical advantages of up to 4. The mechanical advantage is always the ratio of the distance the applied force moves to the distance the load moves. Another, and perhaps easier, way to determine the mechanical advantage of a system of pulleys is to count the number of rope segments that support the load. That number is the mechanical advantage.

Friction is a very important factor in the actual mechanical advantage of a system of pulleys because the frictional losses are compounded each time the rope passes over a pulley. For a typical coefficient of friction of 0.03 (greased shaft with no ball bearings), the system with a theoretical mechanical advantage of 4 would be decreased by friction to an actual mechanical advantage of about 3.5. The largest mechanical advantage that could be obtained with pulleys having a coefficient of friction of 0.03 is about 16 no matter how many pulleys are used. A coefficient of friction of more than 0.333 (such as a vine over a tree branch) results in an actual mechanical advantage of less than 1 for any system of pulleys.

The friction coefficient of 0.03 used above is so low due to the advantage of a wheel on an axle. The friction force at the axle of the pulley wheel is overcome by a smaller force applied at the rim in accord with the principle of the lever.

The friction forces in a pulley system will never hold the load by themselves, but the effort required can be quite small. In the case of a block and tackle used to lift an engine out of an automobile, the weight of the chain hanging from the last pulley may be enough to keep the engine in place. By wrapping a rope once around a post, a cowboy can hold a raging bull in check.

GEARS

Gears are used almost entirely in rotary motion applications, and as such it is easier to discuss the mechanical advantage as a multiplication of torque rather than as a multiplication of force. The work involved in rotary motion is torque times angle; whereas for the linear motion discussed above, it is force times distance.

Torque arises when a force is applied so that it tends to rotate an object about an axis. The force must have a component at right angles to the axis and at some distance from the axis. The torque produced is the product of the force component and its perpendicular distance from the axis. The units of torque turn out to be the same as the units of energy: force × distance. However, torque is not energy. An angle is the ratio of two lengths (arc length/radius for the angle in radians) and has no units; thus the work in rotary motion (the product of torque and angle) has the appropriate units for work.

Upon comparing the work input with the work output for a gear system similar to the one shown in Figure 4, the mechanical advantage is found from the ratio of the angles turned by the respective shafts as the gears engage. This ratio in turn is equal to the ratio of the numbers of teeth on each gear. For example, if a gear (pinion) with 10 teeth drives a gear with 40 teeth, the mechanical advantage is 4—that is, the torque imparted to the large gear is 4 times the torque input by the small gear. There is a commensurate speed reduction; the large gear will rotate once for each four revolutions of the small gear.

Gears may be used to increase the available torque, as in most applications of electric or internal-combustion engines, or to increase the amount of motion, as in a bicycle.

In an automobile, except for the highest gear, the transmission reduces the rotation rate of the drive shaft relative to the engine speed. The differential gears further reduce the rate of rotation. For a typical automobile cruising at 60 mph, the engine runs at about 2,000 rpm while the wheels turn at about 800 rpm giving a mechanical advantage of 2,000/800 = 2.5.

For a typical 21-speed bicycle, on the other hand, the lowest gear ratio is 1.0 and the highest is nearly 3.5. In the highest gear the wheels rotate 3.5 times for each rotation of the pedals. The torque with which the rear wheel propels the bicycle is less than the torque exerted on the pedals by a factor of 3.5. The propulsion force is further reduced by the fact that the radius of the pedals is less than the radius of the rear wheel. The advantage of a bicycle is basically that the wheels move faster than the pedals, coupled with the fact that it takes very little force to overcome the rolling friction of the wheels on a hard, smooth surface. This allows the rider to move faster than a pedestrian using the same energy. However, there is no advantage in force even in the lowest gear. To climb a really steep incline with a bicycle, it is better to get off and walk. From a standing start the greatest acceleration is achieved by pushing the bicycle.

BELTS AND CHAINS

Another function of the transmission of energy by mechanical means is to transfer the motion of a rotating shaft to another shaft at a distant location. A historical application was to get the rotation of a water-wheel coupled to a mill located safely away from the stream in order to grind grain. Another was to drive several rotating machines in a factory from a single large steam engine. Before the invention of the electric generator and the electric motor this type of problem was solved by the use of belts. Today it would be ridiculous to use belts in this way. Think how complicated it would be to connect the power company's spinning steam turbine to a refrigerator in a home by means of belts. Electricity makes it easy. The power is distributed electrically through relatively small wires to drive individual small electric motors.

Today belts are used in automobiles to drive auxiliary devices such as air conditioning, power brakes, power steering, the alternator, and the cooling fluid pump. Belts also can be found in household appliances such as vacuum sweepers, on lathes in machine shops, or inside copying machines.

A belt drive is an inexpensive solution for transmitting mechanical energy over short distances. Compared with machining precision gears, fashioning two pulleys to be linked by a belt is not technologically demanding. The mechanical advantage is the ratio of the diameter of the pulley on the load to the pulley on the driver. The length of the belt makes no difference. By adjusting the sizes of the two pulleys, the mechanical advantage can be conveniently changed. Also, one twist in the belt will reverse the direction of rotation. One disadvantage of a belt is that the amount of torque it can deliver is limited to the product of three factors: the coefficient of friction between the belt and the pulley, the force producing tension in the belt, and the radius of the load pulley.

A V-belt greatly increases the deliverable torque, since the wedging of the belt in the sheave groove increases the force of contact between the surfaces (N) far above the tension force (P). The driving action takes place through the sides of the belt rather than the bottom, which normally is not in contact with the sheave at all. This is yet another example of the application of the wedge as a force multiplier.

For a situation where large torques are involved, such as a bicycle drive, a chain linkage is superior to a belt. A person putting all his or her weight on a pedal probably would make most belt systems slip. Another advantage of a chain over a belt is that a chain is more efficient, mainly because it does not require any ambient tension. The return side of a chain drive has only enough tension to support itself. Furthermore, the chain links are equipped with rollers, which can rotate as they contact the teeth, reducing the frictional forces and wear.

The mechanical advantage of a chain linkage can be calculated by counting the teeth on the load sprocket and the drive sprocket. The output torque is found by multiplying the input torque by the ratio of the number of load teeth to the number of drive teeth. A chain drive is also compact compared to a belt. Imagine trying to arrange 21 speeds on a bicycle derailleur using belts.

A serious disavantage of belts and chains for transmitting energy is that they can be quite dangerous. Whereas the low torque of a bicycle chain rarely results in bad injuries when trouser cuffs or shoestrings get caught in the chain, high-torque industrial and agricultural machinery (such as mechannical reapers with numerous belts and chain drives) is another matter. They have caused many grave inuries and loss of limbs because of the tremendous torque that engines and motors transmit to belts and chains. Therefore, as a safety measure, almost all new equipment comes equipped with guards to prevent accidental injuries either from broken belts and chains whipping out with tremendous force, or the careless actions of workers.

Don C. Hopkins

See also: Bicycling; Drivetrains; Electric Power, Generation of; Electric Power Transmission and Distributive Systems; Engines; Flywheels; Kinetic Energy; Propellers; Steam Engines.