Learning Objectives

By the end of this section, you will be able to do the following:

Describe simple and complex machines
Calculate mechanical advantage and efficiency of simple and complex machines

Section Key Terms
complex machine	efficiency output	ideal mechanical advantage
inclined plane	input work	lever
mechanical advantage	output work	pulley
screw	simple machine	wedge
wheel and axle

Simple Machines

Simple machines make work easier, but they do not decrease the amount of work you have to do. Why can’t simple machines change the amount of work that you do? Recall that in closed systems the total amount of energy is conserved. A machine cannot increase the amount of energy you put into it. So, why is a simple machine useful? Although it cannot change the amount of work you do, a simple machine can change the amount of force you must apply to an object, and the distance over which you apply the force. In most cases, a simple machine is used to reduce the amount of force you must exert to do work. The down side is that you must exert the force over a greater distance, because the product of force and distance, fd, (which equals work) does not change.

Let’s examine how this works in practice. In Figure 9.8(a), the worker uses a type of lever to exert a small force over a large distance, while the pry bar pulls up on the nail with a large force over a small distance. Figure 9.8(b) shows the how a lever works mathematically. The effort force, applied at F_e, lifts the load (the resistance force) which is pushing down at F_r. The triangular pivot is called the fulcrum; the part of the lever between the fulcrum and F_e is the effort arm, L_e; and the part to the left is the resistance arm, L_r. The mechanical advantage is a number that tells us how many times a simple machine multiplies the effort force. The ideal mechanical advantage, IMA, is the mechanical advantage of a perfect machine with no loss of useful work caused by friction between moving parts. The equation for IMA is shown in Figure 9.8(b).

Two diagrams are shown side by side. The diagram on the left shows a drawing of a hand pulling a mail out of wood using a lever. The diagram on the right shows an illustration of a lever and a fulcrum. The fulcrum is closer to the left side of the lever than the right side. The distance of the left side of the lever is labeled L subscript r and the distance of the right side is labeled L subscript e. A force vector is pointing down on the left side and is labeled F subscript r, and a force vector is point

Figure 9.8 (a) A pry bar is a type of lever. (b) The ideal mechanical advantage equals the length of the effort arm divided by the length of the resistance arm of a lever.

In general, the IMA = the resistance force, F_r, divided by the effort force, F_e. IMA also equals the distance over which the effort is applied, d_e, divided by the distance the load travels, d_r.

I M A = \frac{F_{r}}{F_{e}} = \frac{d_{e}}{d_{r}}

Getting back to conservation of energy, for any simple machine, the work put into the machine, W_i, equals the work the machine puts out, W_o. Combining this with the information in the paragraphs above, we can write

\begin{array}{l} W_{i} = W_{o} \\ F_{e} d_{e} = F_{r} d_{r} \\ If F_{e} F_{r}, thend_{e}>d_{r}. \end{array}

The equations show how a simple machine can output the same amount of work while reducing the amount of effort force by increasing the distance over which the effort force is applied.

Watch Physics

Introduction to Mechanical Advantage

This video shows how to calculate the IMA of a lever by three different methods: (1) from effort force and resistance force; (2) from the lengths of the lever arms, and; (3) from the distance over which the force is applied and the distance the load moves.

Click to view content

Grasp Check

Two children of different weights are riding a seesaw. How do they position themselves with respect to the pivot point (the fulcrum) so that they are balanced?

The heavier child sits closer to the fulcrum.
The heavier child sits farther from the fulcrum.
Both children sit at equal distance from the fulcrum.
Since both have different weights, they will never be in balance.

Some levers exert a large force to a short effort arm. This results in a smaller force acting over a greater distance at the end of the resistance arm. Examples of this type of lever are baseball bats, hammers, and golf clubs. In another type of lever, the fulcrum is at the end of the lever and the load is in the middle, as in the design of a wheelbarrow.

The simple machine shown in Figure 9.9 is called a wheel and axle. It is actually a form of lever. The difference is that the effort arm can rotate in a complete circle around the fulcrum, which is the center of the axle. Force applied to the outside of the wheel causes a greater force to be applied to the rope that is wrapped around the axle. As shown in the figure, the ideal mechanical advantage is calculated by dividing the radius of the wheel by the radius of the axle. Any crank-operated device is an example of a wheel and axle.

A diagram of a wheel is shown. The equation IMA equals capital R over lowercase r. A small grey circle is inside a larger blue circle. An arrow labeled lowercase r points from the center of the smaller circle to its edge. An arrow labeled capital R points from the center of the smaller circle to the edge of the larger circle. A mass is attached to the smaller circle and has a force vector, F subscript r, pointing downward from it. Another force vector, F subscript e, points downward and to the right from

Figure 9.9 Force applied to a wheel exerts a force on its axle.

An inclined plane and a wedge are two forms of the same simple machine. A wedge is simply two inclined planes back to back. Figure 9.10 shows the simple formulas for calculating the IMAs of these machines. All sloping, paved surfaces for walking or driving are inclined planes. Knives and axe heads are examples of wedges.

Two diagrams are shown side-by-side. On the left, an inclined plane is shown. Its horizontal surface distance is labeled L and its height is labeled h. A box is shown on the surface with a force vector that points up the plane. The equation IMA equals L over h is shown. On the right, a wedge is shown. Its horizontal surface distance is labeled L. Its height is labeled t. The equation IMA equals L over t is also shown.

Figure 9.10 An inclined plane is shown on the left, and a wedge is shown on the right.

The screw shown in Figure 9.11 is actually a lever attached to a circular inclined plane. Wood screws (of course) are also examples of screws. The lever part of these screws is a screw driver. In the formula for IMA, the distance between screw threads is called pitch and has the symbol P.

An illustration of a screw is shown. It has a long lever, labeled L. P is pitch, the distance between the threads.

Figure 9.11 The screw shown here is used to lift very heavy objects, like the corner of a car or a house a short distance.

Figure 9.12 shows three different pulley systems. Of all simple machines, mechanical advantage is easiest to calculate for pulleys. Simply count the number of ropes supporting the load. That is the IMA. Once again we have to exert force over a longer distance to multiply force. To raise a load 1 meter with a pulley system you have to pull N meters of rope. Pulley systems are often used to raise flags and window blinds and are part of the mechanism of construction cranes.

Three pulley systems are shown in side-by-side diagrams. The first consists of a single pulley and is shown holding a mass. The mass is labeled with a force vector, F r, that also points downward. Another force vector, F e, is shown pointing downward from the pulley. N equals one is also shown. The second pulley system consists of a single pulley holding a mass. A force vector, F e, points upward from the pulley. N equals two is shown. The last pulley system shows four pulleys holding a mass with a hook.

Figure 9.12 Three pulley systems are shown here.

Watch Physics

Mechanical Advantage of Inclined Planes and Pulleys

The first part of this video shows how to calculate the IMA of pulley systems. The last part shows how to calculate the IMA of an inclined plane.

Click to view content

Grasp Check

How could you use a pulley system to lift a light load to great height?

Reduce the radius of the pulley.
Increase the number of pulleys.
Decrease the number of ropes supporting the load.
Increase the number of ropes supporting the load.

A complex machine is a combination of two or more simple machines. The wire cutters in Figure 9.13 combine two levers and two wedges. Bicycles include wheel and axles, levers, screws, and pulleys. Cars and other vehicles are combinations of many machines.

An illustration shows a pair of hand-held wire cutters.

Figure 9.13 Wire cutters are a common complex machine.

Calculating Mechanical Advantage and Efficiency of Simple Machines

In general, the IMA = the resistance force, F_r, divided by the effort force, F_e. IMA also equals the distance over which the effort is applied, d_e, divided by the distance the load travels, d_r.

I M A = \frac{F_{r}}{F_{e}} = \frac{d_{e}}{d_{r}}

Refer back to the discussions of each simple machine for the specific equations for the IMA for each type of machine.

No simple or complex machines have the actual mechanical advantages calculated by the IMA equations. In real life, some of the applied work always ends up as wasted heat due to friction between moving parts. Both the input work (W_i) and output work (W_o) are the result of a force, F, acting over a distance, d.

W_{i} = F_{i} d_{i} and W_{o} = F_{o} d_{o}

The efficiency output of a machine is simply the output work divided by the input work, and is usually multiplied by 100 so that it is expressed as a percent.

% efficiency = \frac{W_{o}}{W_{i}} \times 100

Look back at the pictures of the simple machines and think about which would have the highest efficiency. Efficiency is related to friction, and friction depends on the smoothness of surfaces and on the area of the surfaces in contact. How would lubrication affect the efficiency of a simple machine?

Worked Example

Efficiency of a Lever

The input force of 11 N acting on the effort arm of a lever moves 0.4 m, which lifts a 40 N weight resting on the resistance arm a distance of 0.1 m. What is the efficiency of the machine?

Strategy

State the equation for efficiency of a simple machine, $% efficiency = \frac{W_{o}}{W_{i}} \times 100,$ and calculate W_o and W_i. Both work values are the product Fd.

Solution

$W_{i} = F_{i} d_{i}$ = (11)(0.4) = 4.4 J and $W_{o} = F_{o} d_{o}$ = (40)(0.1) = 4.0 J, then $% efficiency = \frac{W_{o}}{W_{i}} \times 100 = \frac{4.0}{4.4} \times 100 = 91 %$

Discussion

Efficiency in real machines will always be less than 100 percent because of work that is converted to unavailable heat by friction and air resistance. W_o and W_i can always be calculated as a force multiplied by a distance, although these quantities are not always as obvious as they are in the case of a lever.

Practice Problems

What is the IMA of an inclined plane that is

5 m

long and

2 m

high?

$0.4$
$2.5$
$0.4 m$
$2.5 m$

If a pulley system can lift a 200N load with an effort force of 52 N and has an efficiency of almost 100 percent, how many ropes are supporting the load?

1 rope is required because the actual mechanical advantage is 0.26.
1 rope is required because the actual mechanical advantage is 3.80.
4 ropes are required because the actual mechanical advantage is 0.26.
4 ropes are required because the actual mechanical advantage is 3.80.

Check Your Understanding

Exercise 6

True or false—The efficiency of a simple machine is always less than 100 percent because some small fraction of the input work is always converted to heat energy due to friction.

True
False

Exercise 7

The circular handle of a faucet is attached to a rod that opens and closes a valve when the handle is turned. If the rod has a diameter of 1 cm and the IMA of the machine is 6 , what is the radius of the handle?

0.08 cm
0.17 cm
3.0 cm
6.0 cm

9.3 Simple Machines

Learning Objectives