SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

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 F_{1} applied to the machine which moves through
some distance s_{1} and a force F_{2} exerted by the machine on the load which moves through some
distance s_{2}. The product F_{1}s_{1} represents work put into the machine. The product F_{2}s_{2} represents
useful work done by the machine.

Input and output of a machine. The work W_{1} = F_{1}s_{1} representing work input into the
machine is called the input. The work W_{2} = F_{2}s_{2} representing work output by the machine is
called the output.

In a frictionless machine W_{2} = W_{1} and F_{2}s_{2} = F_{1}s_{1}. In this ideal case, where no friction is
assumed, we have

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

AMA = F_{2} /F_{1}

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 = s_{1}/s_{2}

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

W_{1} = W_{f} + W_{2}

where W_{f} is work done against friction, or

F_{1}s_{1} = W_{f} + F_{2}s_{2}

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

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

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

Proof. In Fig. 2 an applied force F_{1} at
point A causes a multiplied up-acting
force at point B, lifting the load. In doing this, point A moves through a distance s_{1} along arc
AC to point C while point B moves a distance s_{2} along arc BD to point D. The ideal mechanical
advantage of this machine is then

IMA = s_{1}/s_{2}

where s_{1} is the length of arc AC and s_{2} is the length of arc BD. However, s_{1}/s_{2} = 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.

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 = s_{1}/s_{2}

where s_{1} is the distance through which the applied force travels and s_{2} 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

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.

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

where

*l* = length of the plane (length of incline)

*h* = height of plane

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.

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 s_{1} 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

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.

[ Home ] [ Up ] [ Info ] [ Mail ]