Machines and Mechanical Advantage

A machine is a device that makes a task easier by changing the magnitude, direction, or distance over which a force acts.

Machines do not create energy. Instead, they allow us to transfer energy in a more useful way.

For example, a machine may allow you to:

• lift a heavy object using a smaller force
• change the direction of a force
• increase the distance through which a force acts
• increase the speed or distance of an output movement
• apply a force more conveniently

Common simple machines include:

• levers
• pulleys
• inclined planes
• wheel and axle systems
• gears
• screws
• wedges

Input Force and Output Force

When using a machine, we can identify two important forces.

Input Force

The input force is the force applied to the machine.

It is sometimes called the effort force.

For example:

• pulling on a rope
• pushing on a lever
• turning a handle
• pedaling a bicycle

Output Force

The output force is the force the machine applies to the load.

It is sometimes called the load force.

For example:

• the force lifting a box
• the force moving a rock
• the force raising a vehicle
• the force pulling a load

A machine can allow a relatively small input force to produce a larger output force.

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Mechanical Advantage

Mechanical advantage (MA) compares the output force produced by a machine with the input force applied to the machine.

MA = output force / input force

or:

MA = Fout / Fin

Mechanical advantage has no unit because it is a ratio of two forces measured in the same units.

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Interpreting Mechanical Advantage

Suppose:

MA = 4

This means:

output force = 4 × input force

If you apply:

100 N

the machine can ideally produce:

400 N

of output force.

The machine has multiplied the force by a factor of four.

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Worked Example: Mechanical Advantage

A person applies an input force of:

150 N

to a machine.

The machine produces an output force of:

600 N

Calculate the mechanical advantage.

MA = Fout / Fin

MA = 600/150

MA = 4

The machine multiplies the input force by a factor of 4.

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Finding Output Force

The equation can be rearranged:

Fout = MA × Fin

Suppose a machine has:

MA = 5

and the input force is:

80 N

Then:

Fout = 5 × 80

Fout = 400 N

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Finding Input Force

We can also rearrange:

Fin = Fout / MA

Suppose a machine must produce:

1200 N

and has:

MA = 6

Then:

Fin = 1200/6

Fin = 200 N

Only 200 N of input force is required.

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Machines Do Not Reduce the Work in an Ideal System

A machine can reduce the force needed, but there is a trade-off.

Recall:

W = Fd

where:

• W = work done
• F = force
• d = displacement in the direction of the force

In an ideal machine:

input work = output work

Therefore:

Fin × din = Fout × dout

If a machine increases the force, the input force must usually act through a greater distance.

This is one of the most important ideas in simple machines:

smaller force → greater distance

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Force-Distance Trade-Off

Suppose we need to perform:

1000 J

of work.

Without a Machine

Force:

500 N

Distance:

2 m

Work:

W = Fd

W = 500 × 2

W = 1000 J

With a Machine

Suppose the machine allows us to use only:

100 N

To perform the same 1000 J of work:

1000 = 100 × d

Therefore:

d = 10 m

The force has been reduced from 500 N to 100 N, but it must act through a much greater distance.

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Machines Trade Force for Distance

This gives us a central rule for ideal machines:

What you gain in force, you give up in distance.

A machine does not provide free energy.

If it multiplies force by five, the input typically moves approximately five times farther than the output.

For example:

Input force = 100 N

Input distance = 5 m

Output force = 500 N

Output distance = 1 m

Input work:

Win = 100 × 5 = 500 J

Output work:

Wout = 500 × 1 = 500 J

Energy is conserved.

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Ideal Mechanical Advantage

For an ideal machine:

Fin × din = Fout × dout

Rearrange:

Fout / Fin = din / dout

Since:

MA = Fout / Fin

we obtain:

MA = din / dout

Therefore, ideal mechanical advantage can also be understood as a distance ratio.

A machine that multiplies force does so because the input moves farther than the output.

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Worked Example: Force and Distance

A machine has an ideal mechanical advantage of:

MA = 8

The output moves:

0.5 m

The input distance is:

din = MA × dout

din = 8 × 0.5

din = 4 m

The input must move 4 m while the load moves only 0.5 m.

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Levers

A lever is a rigid bar that rotates around a fixed point called a fulcrum.

Important parts include:

• effort force
• load force
• fulcrum
• effort arm
• load arm

A lever can:

• multiply force
• change the direction of force
• increase movement or speed

Mechanical Advantage of a Lever

For an ideal lever:

MA = effort arm / load arm

The effort arm is the distance from the fulcrum to the input force.

The load arm is the distance from the fulcrum to the load.

A longer effort arm can provide a greater mechanical advantage.

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Worked Example: Lever

A lever has:

Effort arm = 1.5 m

Load arm = 0.30 m

Mechanical advantage:

MA = 1.5/0.30

MA = 5

If the input force is:

120 N

then:

Fout = 5 × 120

Fout = 600 N

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Why a Long Lever Helps

Imagine trying to loosen a tight bolt.

Using a longer wrench makes the task easier because the applied force acts farther from the pivot.

Similarly, a long crowbar can allow a relatively small input force to produce a much larger force on a load.

This is why long handles are common on:

• wrenches
• crowbars
• bolt cutters
• pliers
• bottle openers

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Classes of Levers

Levers can be classified according to the positions of the:

• fulcrum
• effort
• load

First-Class Lever

The fulcrum lies between the effort and load.

Examples:

• seesaw
• crowbar
• scissors

Second-Class Lever

The load lies between the fulcrum and effort.

Examples:

• wheelbarrow
• bottle opener
• nutcracker

Third-Class Lever

The effort lies between the fulcrum and load.

Examples:

• tweezers
• fishing rod
• human forearm

Third-class levers often have mechanical advantage less than 1, but they can increase the speed and distance of movement.

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Mechanical Advantage Less Than 1

A machine does not always multiply force.

Suppose:

Fin = 200 N

and:

Fout = 50 N

Then:

MA = 50/200

MA = 0.25

The machine reduces the force.

Why would this be useful?

Because it may increase:

• movement distance
• speed
• precision

Human limbs often work this way.

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Pulleys

A pulley consists of a wheel with a rope or cable passing around it.

Pulleys can:

• change the direction of a force
• provide mechanical advantage
• make lifting loads easier

Fixed Pulley

A single fixed pulley mainly changes the direction of the force.

For an ideal fixed pulley:

MA = 1

For example, instead of lifting a load upward directly, you can pull downward on the rope.

The force is not multiplied, but the task may be more convenient.

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Movable Pulleys

Movable pulleys can provide mechanical advantage.

In an ideal pulley system, mechanical advantage can often be determined by counting the number of rope sections supporting the moving load.

For example:

2 supporting sections:

MA = 2

4 supporting sections:

MA = 4

6 supporting sections:

MA = 6

The exact arrangement of the pulley system must always be considered.

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Worked Example: Pulley

A pulley system has an ideal mechanical advantage of:

4

A load weighs:

800 N

Required ideal input force:

Fin = Fout / MA

Fin = 800/4

Fin = 200 N

However, the person must pull a greater length of rope than the distance the load rises.

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Pulley Distance Trade-Off

Suppose the pulley has:

MA = 4

and the load rises:

1 m

Ideally, the input rope must move:

4 m

Check the work:

Input:

Win = 200 × 4

Win = 800 J

Output:

Wout = 800 × 1

Wout = 800 J

Again:

input work = output work

for an ideal machine.

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Inclined Planes

An inclined plane is a sloped surface used to raise or lower objects.

Examples include:

• ramps
• loading ramps
• wheelchair ramps
• sloping roads

An inclined plane reduces the force needed to raise an object by increasing the distance over which the force acts.

Mechanical Advantage of an Inclined Plane

For an ideal inclined plane:

MA = length of slope / vertical height

For example:

Ramp length = 6 m

Height = 1.5 m

Then:

MA = 6/1.5

MA = 4

The ramp ideally reduces the required force by a factor of four.

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Worked Example: Ramp

A 1000 N load must be raised vertically by:

1 m

using a:

5 m

ramp.

Ideal mechanical advantage:

MA = 5/1

MA = 5

Ideal input force:

Fin = 1000/5

Fin = 200 N

Check the work:

Input:

Win = 200 × 5

Win = 1000 J

Output:

Wout = 1000 × 1

Wout = 1000 J

The ramp reduces the required force but increases the distance.

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Wheel and Axle

A wheel and axle consists of a large wheel connected to a smaller axle.

Examples include:

• steering wheels
• door handles
• screwdrivers
• winches
• water wheels

When the input force acts on the larger wheel, the system can multiply force at the axle.

Gears

Gears are rotating wheels with teeth that transfer force and motion.

Gears can change:

• force
• rotational speed
• direction of rotation

A small gear driving a larger gear can increase output force but reduce rotational speed.

A large gear driving a smaller gear can increase rotational speed but reduce output force.

Again, there is a trade-off between:

force and movement

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Screws

A screw is essentially an inclined plane wrapped around a cylinder.

Turning the screw requires the input force to move through a relatively large distance.

The screw itself advances only a small distance.

This can produce a large mechanical advantage.

Examples include:

• bolts
• screws
• screw jacks
• vises

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Wedges

A wedge can be thought of as a moving inclined plane.

Examples include:

• knives
• axes
• chisels
• nails

The input force moves the wedge forward, producing forces that push material apart.

A wedge changes both the direction and magnitude of forces.

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Real Machines

So far, many examples have assumed an ideal machine.

An ideal machine has:

input work = output work

Real machines experience:

• friction
• deformation
• vibration
• sound
• heating
• air resistance

Therefore:

useful output energy < input energy

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Efficiency of Machines

Machine efficiency can be calculated using:

efficiency = useful output energy / input energy

or:

efficiency (%) = useful output energy / input energy × 100

Since work is an energy transfer:

efficiency (%) = output work / input work × 100

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Worked Example: Real Machine

A machine receives:

1000 J

of input work.

It produces:

800 J

of useful output work.

Efficiency:

efficiency = 800/1000 × 100

efficiency = 80%

The remaining:

200 J

has been dissipated into other forms.

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Actual and Ideal Mechanical Advantage

In an ideal machine, there is no friction.

In a real machine, friction means that the actual output force may be lower than predicted by the ideal geometry.

For example:

Ideal MA = 5

Input force = 100 N

Ideal output:

500 N

But suppose the real output is:

420 N

Actual mechanical advantage:

MA = 420/100

MA = 4.2

The actual mechanical advantage is lower because some energy is dissipated.

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Worked Example: Real Ramp

A ramp is:

5 m long

and raises a load:

1 m

Ideal mechanical advantage:

IMA = 5/1

IMA = 5

The load weighs:

1000 N

Ideally, the required force would be:

1000/5 = 200 N

But suppose the actual force required is:

250 N

Actual mechanical advantage:

AMA = 1000/250

AMA = 4

The actual mechanical advantage is lower than the ideal value because of friction.

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Efficiency from Mechanical Advantage

For many simple machines:

efficiency = AMA / IMA

As a percentage:

efficiency (%) = AMA / IMA × 100

Using the previous ramp:

AMA = 4

IMA = 5

Therefore:

efficiency = 4/5 × 100

efficiency = 80%

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Work Analysis of the Same Ramp

We can verify this using work.

Input force:

250 N

Input distance:

5 m

Input work:

Win = 250 × 5

Win = 1250 J

Output force:

1000 N

Output distance:

1 m

Output work:

Wout = 1000 × 1

Wout = 1000 J

Efficiency:

efficiency = 1000/1250 × 100

efficiency = 80%

Both methods give the same result.

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Why Machines Make Work Easier

It may seem strange to say machines make work easier if they do not reduce the total work in an ideal system.

Machines make tasks easier because humans and motors often have force limitations.

For example, you might not be able to lift:

1000 N

directly.

But you may be able to push:

200 N

over a longer distance.

The total work may remain approximately the same, but the task becomes physically possible.

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Direction Can Also Make Work Easier

Machines can also make tasks easier without multiplying force.

A fixed pulley is a good example.

Instead of pulling upward on a load, you can pull downward.

This allows you to:

• use your body weight
• work in a more comfortable position
• apply the force in a convenient direction

So machines can make work easier by changing either the magnitude or direction of force.

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Force Multiplication Does Not Mean Energy Multiplication

Suppose a machine has:

MA = 10

Input force:

100 N

Output force:

1000 N

It may appear that the machine has created extra energy.

It has not.

If the output moves:

0.20 m

then ideally the input must move:

2.0 m

Input work:

100 × 2.0 = 200 J

Output work:

1000 × 0.20 = 200 J

The machine multiplies force, not energy.

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Machines and Power

Machines can also affect the relationship between force and speed.

Recall:

P = Fv

For an ideal machine:

input power = output power

Therefore:

Fin vin = Fout vout

If the machine produces a larger output force, the output generally moves more slowly.

For example:

Input:

100 N at 4 m/s

Input power:

P = 100 × 4

P = 400 W

Suppose output force is:

400 N

Then:

400 = 400 × vout

vout = 1 m/s

The machine multiplied the force by four but reduced the speed by four.

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Mechanical Advantage and Speed

This leads to another important trade-off.

A machine can produce:

high force + low speed

or:

low force + high speed

This is why gears are useful.

For example:

• low gears on a bicycle provide greater force at the wheel but lower speed
• high gears allow greater speed but provide less force multiplication
• vehicle transmissions adjust force and speed according to driving conditions

Comparing Simple Machines

All obey the same basic physical principles:

energy is conserved

and:

force advantage requires a trade-off

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Choosing a Machine

Suppose a worker needs to raise a heavy object.

Possible solutions might include:

• a long lever
• a ramp
• a pulley system
• a hydraulic or mechanical jack

The best machine depends on factors such as:

• available input force
• available distance
• available space
• desired lifting height
• speed
• efficiency
• safety
• friction

There is no single best machine for every situation.

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Worked Multi-Step Problem

A worker uses a 6 m ramp to raise a 1200 N load by 1.5 m.

The worker applies:

400 N

along the ramp.

Ideal Mechanical Advantage

IMA = ramp length / height

IMA = 6/1.5

IMA = 4

Actual Mechanical Advantage

AMA = output force / input force

AMA = 1200/400

AMA = 3

Input Work

Win = Fin × din

Win = 400 × 6

Win = 2400 J

Useful Output Work

Wout = Fout × dout

Wout = 1200 × 1.5

Wout = 1800 J

Efficiency

efficiency = 1800/2400 × 100

efficiency = 75%

Or:

efficiency = AMA/IMA × 100

efficiency = 3/4 × 100

efficiency = 75%

Energy dissipated:

2400 − 1800 = 600 J

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Energy Flow in a Real Machine

Consider a machine receiving:

2000 J

of mechanical input energy.

Suppose:

1600 J → useful mechanical output

300 J → thermal energy

100 J → sound and vibration

Energy conservation:

2000 = 1600 + 300 + 100

Efficiency:

efficiency = 1600/2000 × 100

efficiency = 80%

The machine does not destroy the other 400 J.

That energy has been transferred into less useful forms.

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Improving Machine Performance

Engineers can improve machine efficiency by reducing unwanted energy transfers.

Methods include:

• lubrication
• ball bearings
• smoother surfaces
• better alignment
• reducing unnecessary vibration
• reducing air resistance
• using appropriate materials
• maintaining moving components

Reducing friction is particularly important in machines with many moving parts.

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Machines in the Human Body

The human body contains biological lever systems.

Bones act as:

levers

Joints act as:

fulcrums

Muscles provide:

input forces

Loads may include:

• body parts
• objects being carried
• external forces

Many human-body levers have mechanical advantages less than 1.

This may seem inefficient in terms of force, but it provides advantages in:

• speed
• range of movement
• precision

Common Mistakes

Thinking Machines Create Energy

Machines cannot create energy.

They redistribute force, distance, speed, and direction while obeying conservation of energy.

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Thinking Mechanical Advantage Has Units

Mechanical advantage is:

force / force

Therefore, the units cancel.

Mechanical advantage has no unit.

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Reversing the Mechanical Advantage Equation

Correct:

MA = output force / input force

Not:

input force / output force

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Thinking MA Must Be Greater Than 1

Mechanical advantage can be:

greater than 1

equal to 1

or:

less than 1

A machine with MA < 1 may provide advantages in speed, distance, or precision.

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Thinking Smaller Force Means Less Work

In an ideal machine:

input work = output work

A smaller input force usually acts through a greater distance.

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Ignoring Friction

Real machines require additional input energy because some energy is dissipated through friction, heating, sound, and vibration.

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Confusing Force Advantage with Energy Advantage

A machine can multiply force.

It cannot multiply energy.

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Confusing Ideal and Actual Mechanical Advantage

Ideal mechanical advantage is based on machine geometry.

Actual mechanical advantage uses measured input and output forces.

Friction usually causes:

AMA < IMA

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Assuming a Machine Must Multiply Force

Some machines mainly:

• change force direction
• increase speed
• increase movement distance

Force multiplication is only one possible function.

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Check Your Understanding

1. Define a machine.
2. Explain two ways a machine can make a task easier.
3. Define input force.
4. Define output force.
5. Define mechanical advantage.
6. A machine receives an input force of 100 N and produces an output force of 500 N. Calculate its mechanical advantage.
7. A machine has MA = 6 and receives 80 N. Calculate the output force.
8. A machine must produce 1200 N and has MA = 4. Calculate the required input force.
9. Explain why mechanical advantage has no unit.
10. What does MA = 5 mean physically?
11. Can mechanical advantage be less than 1? Explain why this might be useful.
12. A machine uses 200 N acting through 5 m. Calculate the input work.
13. If the machine is ideal and produces 1000 N, how far does the output move?
14. Explain the relationship between force and distance in an ideal machine.
15. A lever has an effort arm of 2 m and a load arm of 0.5 m. Calculate its ideal mechanical advantage.
16. If 150 N is applied to the lever in Question 15, calculate the ideal output force.
17. A pulley system has four supporting rope sections. What is its ideal mechanical advantage?
18. A 1200 N load is lifted by an ideal pulley system with MA = 4. Calculate the required input force.
19. If the load rises 2 m, how much rope must ideally be pulled?
20. A ramp is 8 m long and 2 m high. Calculate its ideal mechanical advantage.
21. A 1600 N load is pushed up this ideal ramp. Calculate the ideal input force.
22. Explain why a longer, less steep ramp requires less force.
23. A machine receives 1500 J and provides 1200 J of useful output. Calculate its efficiency.
24. How much energy is dissipated in Question 23?
25. A ramp has IMA = 5 and AMA = 4. Calculate its efficiency.
26. Explain why actual mechanical advantage is usually less than ideal mechanical advantage.
27. A machine has an input force of 300 N moving at 4 m/s. Calculate its input power.
28. If its output force is 1200 N and the machine is ideal, calculate the output speed.
29. Explain why increasing output force generally decreases output speed in an ideal machine.
30. Explain why machines make tasks easier even though an ideal machine does not reduce the total work required.

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Key Terms

Machine — Device that changes the magnitude, direction, distance, or speed associated with a force.

Simple machine — Basic mechanical device used to change how a force is applied.

Input force — Force applied to a machine.

Effort force — Another name for input force.

Output force — Force produced by a machine on a load.

Load — Object or resistance upon which a machine performs work.

Mechanical advantage (MA) — Ratio of output force to input force.

Ideal mechanical advantage (IMA) — Mechanical advantage predicted from the geometry of an ideal machine.

Actual mechanical advantage (AMA) — Mechanical advantage calculated from actual measured forces.

Input work — Energy transferred to a machine by the input force.

Output work — Useful mechanical energy transferred by a machine to its load.

Fulcrum — Fixed pivot point around which a lever rotates.

Effort arm — Distance from the fulcrum to the input force on a lever.

Load arm — Distance from the fulcrum to the load force.

Lever — Rigid bar that rotates around a fulcrum.

Pulley — Wheel and rope system used to change force direction or provide mechanical advantage.

Inclined plane — Sloping surface that reduces required force by increasing the distance over which the force acts.

Wheel and axle — Simple machine consisting of connected rotating components of different radii.

Gear — Toothed wheel used to transfer rotational motion and change force, speed, or direction.

Screw — Inclined plane wrapped around a cylinder.

Wedge — Moving inclined plane used to apply forces to materials.

Efficiency — Fraction of input energy transferred into useful output energy.

Energy dissipation — Transfer of energy into less useful forms such as thermal energy and sound.

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Key Takeaways

• Machines make tasks easier by changing the magnitude, direction, distance, or speed associated with forces.
• The input force is the force applied to the machine.
• The output force is the force the machine applies to the load.
• Mechanical advantage compares output force with input force.
• MA = Fout / Fin.
• Mechanical advantage has no unit.
• An MA greater than 1 means the machine multiplies force.
• An MA of 1 means the output and input forces are equal.
• An MA less than 1 can provide advantages in speed, distance, or precision.
• Machines cannot create energy.
• In an ideal machine, input work = output work.
• Since W = Fd, reducing force requires increasing the distance over which the force acts.
• A machine therefore trades force for distance.
• Ideal mechanical advantage can often be determined from the geometry of the machine.
• Levers can multiply force depending on the relative lengths of their effort and load arms.
• Pulley systems can multiply force by distributing the load across supporting rope sections.
• Inclined planes reduce the force required to raise an object by increasing the distance traveled.
• Gears and wheel-and-axle systems trade force for rotational speed.
• Real machines are less than 100% efficient because of friction, heating, vibration, sound, and deformation.
• Actual mechanical advantage is generally lower than ideal mechanical advantage.
• Machine efficiency can be determined by comparing useful output work with input work.
• For many simple machines, efficiency = AMA/IMA × 100%.
• Force multiplication does not mean energy multiplication.
• Increasing output force generally means decreasing output speed or increasing input distance.
• Energy conservation provides the underlying explanation for how all simple machines operate.

The central trade-off is:

smaller input force → greater input distance

or:

greater output force → smaller output distance

A machine does not allow us to get more energy than we put in. It allows us to use the available energy and force in a way that makes the task more practical, controllable, or convenient.