# 7.2: Machine

- Page ID
- 2820

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A "Rube Goldberg Machine" is a complex construction of many simple machines connected end-to-end in order to accomplish a particular activity. By design, Rube Goldberg Machines are far more intricate than necessary, and may be quite entertaining. Although this kind of construction may be extremely inefficient, simple machines commonly make work easier, and can be found all around us.

## Machines

A machine is an object or mechanical device that receives an input amount of work and transfers the energy to an output amount of work. For an **ideal** **machine*** ,* the input work and output work are always the same. Remember that work is force times distance; even though the

*work*input and output are equal, the input

*force*does not necessarily equal the output

*force*, nor does the input

*distance*necessarily equal the output

*distance*.

Machines can be incredibly complex (think of robots or automobiles), or very simple, such as a can opener. A **simple machine **is a mechanical device that changes the magnitude or direction of the force. There are six simple machines that were first identified by Renaissance scientists: lever, pulley, inclined plane, screw, wedge, and wheel and axle. These six simple machines can be combined together to form **compound machines**.

We use simple machines because they give us a **mechanical advantage**. Mechanical advantage is a measurement of the force amplification of a machine. In ideal machines, where there is no friction and the input work and output work are the same,

(Effort Force)(Effort Distance)=(Resistance Force)(Resistance Distance)

The **effort** is the work that you do. It is the amount of force you use times the distance over which you use it. The **resistance** is the work done on the object you are trying to move. Often, the resistance force is the force of gravity, and the resistance distance is how far you move the object.

The **ideal mechanical advantage** of a simple machine is the ratio between the distances:

IMA=effort distance/resistance distance

Again, the IMA assumes that there is no friction. In reality, the mechanical advantage is limited by friction; you must overcome the frictional forces in addition to the resistance force. Therefore, the **actual mechanical advantage** is the ratio of the forces:

AMA=resistance force/effort force

When simple machines are combined to form compound machines, the product of each simple machine's IMA gives the compound machine's IMA.

## Simple Machines

### Lever

A **lever** consists of an inflexible length of material placed over a pivot point called a **fulcrum**. The resistance is the object to be moved (shown here in red), and is placed to one side of the fulcrum. The resistance distance in a lever is called the resistance arm. The effort is exerted elsewhere on the lever, and the effort distance is called the effort arm or effort lever arm. The lever shown here is the most common type of lever, a Class One Lever, but there are two other types of levers.

The effort work is the effort force times the effort lever arm. Similarly, the resistance work is the resistance force times the resistance lever arm. If we ignore any friction that occurs where the lever pivots over the fulcrum, this is an ideal machine. Suppose the resistance force is 500. N, the resistance arm is 0.400 m, and the effort arm is 0.800m. We can calculate exactly how much effort force is required to lift the resistance in this system:

Output Work=Input Work

(Resistance Force)(Resistance Arm)=(Effort Force)(Effort Arm)

(500. N)(0.400 m)=(x)(0.800 m)

x=250. N

In this case, since the effort arm is twice as long as the resistance arm, the effort force required is only half the resistance force. This machine allows us to lift objects using only half the force required to lift the object directly against the pull of gravity. The distance the effort force is moved is twice as far as the resistance will move. Thus, the input work and the output work are equal.

Example 7.2.1

(a) How much force is required to lift a 500. kg stone using an ideal lever whose resistance arm is 10.0 cm and whose effort arm is 2.00 m?

(b) What is the IMA?

(c) If the actual effort force required to lift the stone was 305 N, what was the AMA?

**Solution**

(a) (resistance force)(resistance arm)=(effort force)(effort arm)

effort force=(resistance force)(resistance arm)/(effort arm)=(4900 N)(0.100 m)/(2.00 m)=245 N

(b) IMA=effort arm/resistance arm=2.00 m/0.100 m=20

(c) AMA=resistance force/effort force=4900 N/305 N=16

## Pulley

A **pulley** is a wheel on an axle that is designed to rotate with movement of a cable along a groove at its circumference. Pulleys are used in a variety of ways to lift loads, apply forces, and to transmit power, but the simplest of pulleys serves only to reverse the direction of the effort force. It consists of a single pulley attached directly to a non-moving surface with a rope or cable through it. As a downward force is applied to one side of the pulley, the other side of the pulley, with the attached resistance force, is pulled upward. This type of pulley is called a **fixed pulley**, and is labeled *A* in the image below.

Another type of pulley is shown above as *B*. This type of pulley is called a **movable pulley. **A set of pulleys assembled so they rotate independently on the same axle form a block. It is shown below in a system called a **block and tackle**. A block and tackle consists of two blocks, in which one block is fixed and the other is movable; the movable block is attached to the load. Play around with the simulation below to further explore the block and tackle:

Interactive Element

The IMA of a pulley system can be determined by counting the number of supporting strands of rope in the system. Be careful though, because in some systems the rope to which the effort force is applied will be a supporting strand, but in others it is not. For example, in the image above with the five pulley systems, the rope to which the effort force is applied (the one with the arrowhead) in *A* is not a supporting strand because it does not hold up any of the weight of the load. The IMA of *A* is 1. In *B*, however, the effort rope is supporting half of the weight of the load and is therefore a supporting strand. *B* has 2 supporting strands and an IMA of 2.

Example 7.2.2

Determine the IMA for *C, D, *and *E* in the image above.

**Solution**

C = 2 supporting strands; IMA = 2

D = 3 supporting strands; IMA = 3

E = 3 supporting strands; IMA = 3

If the direction of the effort force is in the same direction and the movement of the load, the effort strand will be a supporting strand. If the direction of the effort force is in the same direction as the resistance force, the effort strand is not a supporting strand. Look again at the five pulley systems to ensure this is true.

Example 7.2.3

Consider the pulley system sketched above. Given that the resistance force is 8500. N, find

(a) the IMA.

(b) the ideal effort force required to list this weight.

(c) the distance the weight will rise if the effort force moves 1.0 m.

(d) the AMA if the actual effort force is 2000. N.

**Solution**

(a) Since the effort strand moves in the opposite direction of the resistance, it is not a supporting strand. Therefore, there are 5 supporting strands and that makes the IMA = 5.

(b) Effort force=resistance force/IMA=8500 N/5=1700 N

(c) Since the IMA is 5, the resistance distace will be 1/5 of the effort distance: the resistance distance is 1.0m/5 = 0.20 m

(d) AMA=resistance force/effort force=8500 N/2000 N=4.25

## Wheel and Axle

Just like it sounds, a **wheel and axle** is composed of two connected cylinders of different diameters. Since the wheel has a larger radius (distance) than the axle, the axle will always have a larger force than the wheel. The ideal mechanical advantage of a wheel and axle is dependent on the ratio between the radii:

IMA=Radius_{wheel}/Radius_{axle}

## Inclined Plane

An inclined plane is also a simple machine. The resistance is the weight of the box resting on the inclined plane. In order to lift this box straight up, the effort force would need to be equal to its weight. However, assuming no friction, less effort (a smaller effort force) is required to slide the box up the incline. We know this intuitively; when movng boxes into a truck or onto a platform, we use angled platforms instead of lifting it straight up.

The red triangle that hangs below the yellow box is a similar triangle to the inclined plane. The vector perpendicular to the inclined surface is the normal force and this normal force is equal to the portion of the weight of the box that is supported by the surface of the plane. The parallel force is the portion of the weight pushing the box down the plane and is, therefore, the effort force necessary to push the box up the plane.

The effort distance, in the case of an inclined plane, is the length of the incline and the resistance distance is the vertical height the box would rise when it is pushed completely up the incline. The mechanical advantages for an inclined plane are

IMA=effort distance/resistance distance=length/vertical height=1/sinθ

IMA=resistance force/effort force=weight/applied force

Example 7.2.4

Suppose, in the sketch above, the weight of the box is 400. N, the angle of the incline is 35°, and the surface is frictionless. Find the normal force (by finding the portion of the weight acting perpendicular to the plane), the parallel force, and the IMA for the box on this incline.

**Solution**

Normal force=(400. N)(cos35∘)=(400. N)(0.82)=330 N

Parallel force=(400. N)(sin35∘)=(400. N)(0.57)=230 N

IMA=1/sinθ=1/sin35∘=1/0.57=1.74

## Wedge

A wedge is essentially two inclined planes back to back. Like an inclined plane, the IMA of a wedge is the ratio between the length of the wedge and the width of the wedge. Unlike an inclined plane, a wedge does not have a right angle; the IMA of a wedge cannot be found with sines.

## Screw

A **screw** is an inclined plane wrapped around a cylinder. When on a screw, inclined planes are called threads, which can be seen in the image above. The mechanical advantage of a screw increases with the density of the threads. The calculations to determine the IMA for a screw involve the circumference of the head of the screw and the thread width. When the screw is turned completely around one time, the screw penetrates by one thread width. So, if the circumference of the head of a screw is 3.0 cm and the thread width is 0.60 cm, then the IMA would be calculated by

IMA=effort distance/resistance distance=3.0 cm/0.60 cm=5.

When **simple** **machines** are joined together to make **compound** **machines**, the ideal mechanical advantage of the compound machine is found by multiplying the IMA’s of the simple machines.

## Summary

- A machine is an object or mechanical device that receives an input amount of work and transfers the energy to an output amount of work.
- For an ideal machine, the input work and output work are always the same.
- The six common simple machines are the lever, wheel and axle, pulley, inclined plane, wedge, and screw.
- For all simple machines, the ideal mechanical advantage is effort distance/resistance distance.
- For all simple machines, the actual mechanical advantage is resistance force/effort force.
- When simple machines are joined together to make compound machines, the ideal mechanical advantage of the compound machine is found by multiplying the IMA’s of the simple machines.

## Review

- Is it possible to get more work out of a machine than you put in?
- A worker uses a pulley system to raise a 225 N carton 16.5 m. A force of 129 N is exerted and the rope is pulled 33.0 m.
- What is the IMA of the system?
- What is the AMA of the system?

- A boy exerts a force of 225 N on a lever to raise a 1250 N rock a distance of 0.13 m. If the lever is frictionless, how far did the boy have to move his end of the lever?
- How can you increase the ideal mechanical advantage of an inclined plane?
- Diana raises a 1000. N piano a distance of 5.00 m using a set of pulleys. She pulls in 20.0 m of rope.
- How much effort force did Diana apply if this was an ideal machine?
- What force was used to overcome friction if the actual effort force was 300. N?
- What was the work output?
- What was the ideal mechanical advantage?
- What was the actual mechanical advantage, if the input force was 300N?

- A mover’s dolly is used to pull a 115 kg refrigerator up a ramp into a house. The ramp is 2.10 m long and rises 0.850 m. The mover exerts a force of 496 N up the ramp.
- How much work does the mover do?
- How much work is spent overcoming friction?

- What is the ideal mechanical advantage of a screw whose head has a diameter of 0.812 cm and whose thread width is 0.318 cm?

## Additional Resources

Interactive: Seesaw

PLIX: Play, Learn, Interact, eXplore: Pulleys, Lever

Videos: