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Six simple force-multiplying devices - the simple machines. Lever, pulley, wheel and axle, inclined plane, screw, wedge.

 

There are six simple force-multiplying devices that are employed in various forms in many ways in our mechanized society and are important components of many machines. They are


(1) the lever

(2) the pulley system (i.e. block and tackle)

(3) the wheel and axle

(4) the inclined plane

(5) the screw

(6) the wedge


Many machines are either modifications of one of these devices or combinations of two or more of them. In physics these six devices are called the simple machines.


In all of the above machines there is a force F1 applied to the machine which moves through some distance s1 and a force F2 exerted by the machine on the load which moves through some distance s2. The product F1s1 represents work put into the machine. The product F2s2 represents useful work done by the machine.


Input and output of a machine. The work W1 = F1s1 representing work input into the machine is called the input. The work W2 = F2s2 representing work output by the machine is called the output.


In a frictionless machine W2 = W1 and F2s2 = F1s1. In this ideal case, where no friction is assumed, we have


             ole.gif                          



Mechanical advantage. The actual mechanical advantage (AMA) of a machine is

 

AMA = F2 /F1


Thus if an applied force of 1 lb generates a multiplied force of 5 lb, the actual mechanical advantage of the machine is 5.


The ideal mechanical advantage (IMA) of a machine is the mechanical advantage it would have if there were no friction and is given by

 

IMA = s1/s2



However, there is usually friction. Consequently, of the work input into the machine, some is lost to friction. Thus


            W1 = Wf + W2


where Wf is work done against friction, or


            F1s1 = Wf + F2s2



Efficiency of a machine. The efficiency E of a machine is given by


             ole1.gif  




ole2.gif

The lever. A lever consists of a rigid bar which is free to turn about a fixed pivot point called a fulcrum. The lever can be used to multiply force and lift weights. The part of the lever between the load and fulcrum is called the load arm. The part between the effort and fulcrum is called the effort arm. See Fig. 1.


The effort needed to lift a load is given by the law of the lever:


Law of the lever: The effort times the length of the effort arm is equal to the load times the length of the load arm.


This law corresponds to the rule that for equilibrium (i.e. balance) the sum of the counterclockwise moments about the pivot point must be equal to the sum of the clockwise moments.


Example. What force F is needed to lift a load of 100 lb if the length of the load arm is 2 ft and the length of the effort arm is 10 ft?


Solution. 10F = 2×100 so F = 20 lb



Mechanical advantage of a lever. The mechanical advantage of a lever is given by


ole3.gif

ole4.gif


---------------------------------------------

Proof. In Fig. 2 an applied force F1 at point A causes a multiplied up-acting force at point B, lifting the load. In doing this, point A moves through a distance s1 along arc AC to point C while point B moves a distance s2 along arc BD to point D. The ideal mechanical advantage of this machine is then

 

IMA = s1/s2

 

ole5.gif

where s1 is the length of arc AC and s2 is the length of arc BD. However, s1/s2 = AO/OB. Why? Because arcs of circles subtended by equal central angles are directly proportional to the radii of the circles. Thus IMA = AO/OB. The efficiency of levers is often nearly 100%.                

---------------------------------------------


                                                            

Three classes of levers. There are three classes of levers.


1. First class lever. In a first class lever the fulcrum is located between the effort and the load.


2. Second class lever. In a second class lever the load is located between the fulcrum and the effort. Example: wheelbarrow.


3. Third class lever. In a third class lever the effort is located between the fulcrum and the load. Example: forearm. A third class lever multiplies speed rather than force.


See Fig. 3.



ole6.gif

The pulley system (block and tackle). There are a variety of pulley systems that can be used for lifting loads. Pulleys are mounted in frames called blocks. The rope is called tackle.


The ideal mechanical advantage of a pulley system can be computed from the formula

 

IMA = s1/s2


ole7.gif

where s1 is the distance through which the applied force travels and s2 is the distance through which the load travels. In multiple block and tackle the IMA is equal to the number of ropes supporting the load. Because of friction in the blocks and rigidity of the ropes the efficiency of block and tackle is usually less than 60%.                                                                              


The wheel and axle. The wheel and axle consists of a wheel or crank that is rigidly attached to an axle. See Fig. 5. In Fig. 6 we see that a wheel and axle is similar to a lever with unequal arms.

 

The ideal mechanical advantage of the wheel and axle is


             ole8.gif


where


            R = radius of the wheel

            r = radius of the axle

 

The wheel and axle is often used to multiply speed instead of force. Example: wheels on bicycles and motor vehicles.


ole9.gif

The inclined plane. When it is desired to raise something that is too heavy to lift, an inclined plane or ramp is sometimes used. One may load heavy barrels on a truck by rolling them up an inclined plane constructed of planks.

                                                                                                 

The ideal mechanical advantage of an inclined plane is         

             ole10.gif                                                                                      


where


            l = length of the plane (length of incline)

            h = height of plane

ole11.gif

See Fig. 7.



The screw. A screw is really an inclined plane wound on a cylinder. The distance between the threads is called the pitch of the screw. One complete revolution of the screw will move it into an object the distance of the pitch. Wood screws, bolts, and screw jacks represent applications of the screw.                                                                          

ole12.gif

The mechanical advantage of a screw depends on the length of the lever arm used in turning the screw. See Fig. 8. While the effort force completes a full circle, the head and axis of the screw make one complete turn and the load moves a distance equal to the pitch of the screw. If r is the length of the lever arm, then in one complete revolution, the distance s1 through which F moves is 2πr. As F moves this distance, the weight w moves the distance d, which is the pitch of the screw. Thus the ideal mechanical advantage is given by


             ole13.gif


where


            r = length of lever arm

            d = pitch of screw



The wedge. The wedge is really a double inclined plane. There is so much friction in using a wedge that a theoretical mechanical advantage has no significance. A long thin wedge is easier to drive than a short thick one so one can say that the mechanical advantage of a wedge depends on the ratio of its length to its thickness. Examples of wedges: axes, nails, pins.




References

 Dull, Metcalfe, Brooks. Modern Physics.

 Schaum. College Physics.

 Sears, Zemansky. University Physics.


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