Simple Machines, Work, Energy, and Simple Machines | Physics

Learning objectives

Teacher Support

The learning objectives in this section will help your students master the following standards:

(6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
- (C) describe simple and complex machines and solve problems involving simple machines;
- (D) define input work, output work, mechanical advantage, and efficiency of machines.

In addition, the High School Physics Laboratory Manual addresses content in this section in the lab titled: Work and Energy, as well as the following standards:

(6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
- (D) demonstrate and apply the laws of conservation of energy and conservation of momentum in one dimension.

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

Teacher Support

In this section you will apply what you have learned about work to find the mechanical advantage and efficiency of simple machines.

[BL][OL] Ask the students what they know about machines and work. Dispel any misconceptions that machines reduce the amount of work. Be sure students do not equate machines and motors by asking for (and, if necessary, providing) examples of machines that are not motorized. Explain that simple machines are often hand-held, and that they reduce force, not work.

[AL] Ask for recall of the formula W = fd. Explain that the product of force and distance is critical to understanding simple machines. Because the amount of work is not changed, the term fd does not change, but force can decrease if distance increases. This is the underlying principle of all simple machines.

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_09_03_lever(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_09_03_lever(b) shows 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_09_03_lever(b).

View original figure (external; image not reproduced in this text edition)

Original figure description: 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 pointing down on the right side and is labeled F subscript e. The equation IMA equals L subscript e over L subscript r is also shown.

(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{matrix} W_{i} = W_{o} \\ F_{e} d_{e} = F_{r} d_{r} \\ If F_{e} < F_{r}, then d_{e} > d_{r} . \end{matrix}

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.

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.

Introduction to mechanical advantage | Work and energy | Physics | Khan Academy

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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_09_03_axle 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.

View original figure (external; image not reproduced in this text edition)

Original figure description: 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 the edge of the larger circle.

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_09_03_inclined 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.

View original figure (external; image not reproduced in this text edition)

Original figure description: 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.

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

The screw shown in Figure_09_03_screw 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.

View original figure (external; image not reproduced in this text edition)

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

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

Figure_09_03_pulley 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.

View original figure (external; image not reproduced in this text edition)

Original figure description: 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. A force vector, F e, points downward from the pulley. N equals four is shown.

Three pulley systems are shown here.

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.

Mechanical advantage (part 3) | Work and energy | Physics | Khan Academy

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_09_03_wire combine two levers and two wedges. Bicycles include wheel and axles, levers, screws, and pulleys. Cars and other vehicles are combinations of many machines.

View original figure (external; image not reproduced in this text edition)

Original figure description: An illustration shows a pair of hand-held wire cutters.

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?

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?

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.

Teacher Support

Teaching tip—When calculating efficiency, it is easy enough to understand what force in and force out are: the force you apply is force in and the weight of the object that is being lifted is force out. The input and output distances are easier to see for the lever, inclined plane and wedge. The other three are not as obvious. For a pulley system, the input distance is how far you pull the rope, and the output distance is the distance the load rises. For a wheel and axle, the input distance is the circumference of the wheel, and the output distance is the circumference of the axle. For a screw, the input distance is the circumference of the circle over which the force is applied, and the output distance is the distance between the screw threads.

Practice Problems

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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

True or falseThe 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

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

Section Summary

The six types of simple machines make work easier by changing the fd term so that force is reduced at the expense of increased distance.
The ratio of output force to input force is a machine’s mechanical advantage.
Combinations of two or more simple machines are called complex machines.
The ratio of output work to input work is a machine’s efficiency.

Key Equations

ideal mechanical advantage (general)	$I M A = \frac{F_{r}}{F_{e}} = \frac{d_{e}}{d_{r}}$
ideal mechanical advantage (lever)	$I M A = \frac{L_{e}}{L_{r}}$
ideal mechanical advantage (wheel and axle)	$I M A = \frac{R}{r}$
ideal mechanical advantage (inclined plane)	$I M A = \frac{L}{h}$
ideal mechanical advantage (wedge)	$I M A = \frac{L}{t}$
ideal mechanical advantage (pulley)	$I M A = N$
ideal mechanical advantage (screw)	$I M A = \frac{2 π L}{P}$
input work	$W_{i} = F_{i} d_{i}$
output work	$W_{o} = F_{o} d_{o}$
efficiency output	$% efficiency = \frac{W_{o}}{W_{i}} \times 100$

Concept Items

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Which type of simple machine is a knife?

A ramp
A wedge
A pulley
A screw

Critical Thinking

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Problems

A man is using a wedge to split a block of wood by hitting the wedge with a hammer. This drives the wedge into the wood creating a crack in the wood. When he hits the wedge with a force of 400 N it travels 4 cm into the wood. This caused the wedge to exert a force of 1,400 N sideways increasing the width of the crack by 1 cm . What is the efficiency of the wedge?

0.875 percent
0.14
0.751
87.5 percent

An electrician grips the handles of a wire cutter, like the one shown, 10 cm from the pivot and places a wire between the jaws 2 cm from the pivot. If the cutter blades are 2 cm wide and 0.3 cm thick, what is the overall IMA of this complex machine?

View original figure (external; image not reproduced in this text edition)

Original figure description: The image is a picture of wire cutters with red handles. The pinchers are closed.

1.34
1.53
33.3
33.5

Performance Task

Conservation of Energy and Energy Transfer; Cause and Effect; and S&EP, Planning and Carrying Out Investigations

Plan an investigation to measure the mechanical advantage of simple machines and compare to the IMA of the machine. Also measure the efficiency of each machine studied. Design an investigation to make these measurements for these simple machines: lever, inclined plane, wheel and axle and a pulley system. In addition to these machines, include a spring scale, a tape measure, and a weight with a loop on top that can be attached to the hook on the spring scale. A spring scale is shown in the image.

View original figure (external; image not reproduced in this text edition)

Original figure description: An illustration of a spring scale is shown. The left side shows grams and the right side shows Newtons.

A spring scale measures weight, not mass.

LEVER: Beginning with the lever, explain how you would measure input force, output force, effort arm, and resistance arm. Also explain how you would find the distance the load travels and the distance over which the effort force is applied. Explain how you would use this data to determine IMA and efficiency.

INCLINED PLANE: Make measurements to determine IMA and efficiency of an inclined plane. Explain how you would use the data to calculate these values. Which property do you already know? Note that there are no effort and resistance arm measurements, but there are height and length measurements.

WHEEL AND AXLE: Again, you will need two force measurements and four distance measurements. Explain how you would use these to calculate IMA and efficiency.

SCREW: You will need two force measurements, two distance traveled measurements, and two length measurements. You may describe a screw like the one shown in Figure_09_03_screw or you could use a screw and screw driver. (Measurements would be easier for the former). Explain how you would use these to calculate IMA and efficiency.

PULLEY SYSTEM: Explain how you would determine the IMA and efficiency of the four-pulley system shown in Figure_09_03_pulley. Why do you only need two distance measurements for this machine?

Design a table that compares the efficiency of the five simple machines. Make predictions as to the most and least efficient machines.

Test Prep Multiple Choice

A large radius divided by a small radius is the expression used to calculate the IMA of what?

A screw
A pulley
A wheel and axle
An inclined plane.

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Which statement correctly describes the simple machines, like the crank in the image, that make up an Archimedes screw and the forces it applies?

View original figure (external; image not reproduced in this text edition)

Original figure description: A classic illustration of an Archimedes' screw, a historical water-lifting device, showing its helical design within a tube, used to transport water upwards from a lower to a higher level.

The crank is a wedge in which the IMA is the length of the tube divided by the radius of the tube. The applied force is the effort force and the weight of the water is the resistance force.
The crank is an inclined plane in which the IMA is the length of the tube divided by the radius of the tube. The applied force is the effort force and the weight of the water is the resistance force.
The crank is a wheel and axle. The effort force of the crank becomes the resistance force of the screw.
The crank is a wheel and axle. The resistance force of the crank becomes the effort force of the screw.

Refer to the pulley system on right in the image. Assume this pulley system is an ideal machine.
How hard would you have to pull on the rope to lift a 120 N load?
How many meters of rope would you have to pull out of the system to lift the load 1 m?

View original figure (external; image not reproduced in this text edition)

Original figure description: A series of 3 pulley systems are shown in this image. At the top of the image there is a formula I M A equals N. The first pulley on the left is shown. A red arrow is following diagonally down and to the right from the top of the pulley labeled F sub e. A weight is shown on the left side of the pulley with a red arrow from the weight denoted F sub r. A dotted line goes across the line from the weight to the pulley that reads N equals 1. The pulley in the center has a red arrow on the right side of the pulley going directly up labeled F sub e. A dotted line goes between the pulley and the fixed portion labeled N equals 2. A weight is suspended straight down from the pulley. The pulley system on the right has four pulleys in vertical succession. The top pulley and bottom pulley are larger than the pulleys in the middle. A red arrow is following diagonally down and to the right from the top of the pulley labeled F sub e. A dotted line goes between the two smaller pulleys in the middle labeled N equals 4. A hook on the bottom of the second pulley from the top starts a line that winds under the second smaller pulley, over the first smaller pulley, under the bottom larger pulley and over the top larger pulley. A weight is suspended from the bottom of the larger pulley.

480 N
4 m
480 N
$\frac{1}{4}$ m
30 N
4 m
30 N
$\frac{1}{4}$ m

Test Prep Short Answer

Why would you expect the lever shown in the top image to have a greater efficiency than the inclined plane shown in the bottom image?

View original figure (external; image not reproduced in this text edition)

The resistance arm is shorter in case of the inclined plane.
The effort arm is shorter in case of the inclined plane.
The area of contact is greater in case of the inclined plane.

Why is the wheel on a wheelbarrow not a simple machine in the same sense as the simple machine in the image?

View original figure (external; image not reproduced in this text edition)

The wheel on the wheelbarrow has no fulcrum.
The center of the axle is not the fulcrum for the wheels of a wheelbarrow.
The wheelbarrow differs in the way in which load is attached to the axle.
The wheelbarrow has less resistance force than a wheel and axle design.

A worker pulls down on one end of the rope of a pulley system with a force of 75 N to raise a hay bale tied to the other end of the rope. If she pulls the rope down 2.0 m and the bale raises 1.0 m, what else would you have to know to calculate the efficiency of the pulley system?

the weight of the worker
the weight of the hay bale
the radius of the pulley
the height of the pulley from ground

True or falseA boy pushed a box with a weight of 300 N up a ramp. He said that, because the ramp was 1.0 m high and 3.0 m long, he must have been pushing with force of exactly 100 N.

True
False

Test Prep Extended Response

To dig a hole, one holds the handles together and thrusts the blades of a posthole digger, like the one in the image, into the ground. Next, the handles are pulled apart, which squeezes the dirt between them, making it possible to remove the dirt from the hole. This complex machine is composed of two pairs of two different simple machines. Identify and describe the parts that are simple machines and explain how you would find the IMA of each type of simple machine.

View original figure (external; image not reproduced in this text edition)

Original figure description: A post hole digger is shown with yellow and black handles. The digging apparatus is to the right and the post hole digger is closed.

Each handle and its attached blade is a lever with the fulcrum at the hinge. Each blade is a wedge.
The IMA of a lever would be the length of the handle divided by the length of the blade. The IMA of the wedges would be the length of the blade divided by its width.
Each handle and its attached to blade is a lever with the fulcrum at the end. Each blade is a wedge.
The IMA of a lever would be the length of the handle divided by the length of the blade. The IMA of the wedges would be the length of the blade divided by its width.
Each handle and its attached blade is a lever with the fulcrum at the hinge. Each blade is a wedge.
The IMA of a lever would be the length of the handle multiplied by the length of the blade. The IMA of the wedges would be the length of the blade multiplied by its width.
Each handle and its attached blade is a lever with the fulcrum at the end. Each blade is a wedge.
The IMA of a lever would be the length of the handle multiplied by the length of the blade. The IMA of the wedges would be the length of the blade multiplied by its width.

A wooden crate is pulled up a ramp that is 1.0 m high and 6.0 m long. The crate is attached to a rope that is wound around an axle with a radius of 0.020 m . The axle is turned by a 0.20 m long handle. What is the overall IMA of the complex machine?

Glossary

complex machine: a machine that combines two or more simple machines
efficiency: output work divided by input work
ideal mechanical advantage: the mechanical advantage of an idealized machine that loses no energy to friction
inclined plane: a simple machine consisting of a slope
input work: effort force multiplied by the distance over which it is applied
lever: a simple machine consisting of a rigid arm that pivots on a fulcrum
mechanical advantage: the number of times the input force is multiplied
output work: output force multiplied by the distance over which it acts
pulley: a simple machine consisting of a rope that passes over one or more grooved wheels
screw: a simple machine consisting of a spiral inclined plane
simple machine: a machine that makes work easier by changing the amount or direction of force required to move an object
wedge: a simple machine consisting of two back-to-back inclined planes
wheel and axle: a simple machine consisting of a rod fixed to the center of a wheel

Attribution. Physics, OpenStax, Rice University, and the contributing authors credited in the book. Licensed under Creative Commons Attribution 4.0 International. Source revision 56e71824e7 of openstax/osbooks-physics (2026-10-13). CC BY 4.0. Original source ↗. Text edition of a pinned CC BY 4.0 source revision of the OpenStax book, converted from CNXML by the EvidenceHelix importer. Original prose, learning objectives, tables, exercises, glossary and captions are preserved; images are not copied (figures link to the original source and keep their credits). No new content was added. Later revisions of this book can carry different licence terms and are not used.