# Simple Machines

## Calculate ideal and actual mechanical advantage of the six simple machines

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Simple Machines

Credit: Phil Manker
Source: http://www.flickr.com/photos/philmanker/3654636770/

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 (we will not cover this). These six simple machines can be combined together to form compound machines.

### 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.

Credit: Richard Parsons
Source: CK-12 Foundation

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\begin{align*}\text{Output Work} = \text{Input Work}\end{align*}

(Resistance Force)(Resistance Arm)=(Effort Force)(Effort Arm)\begin{align*}(\text{Resistance Force})(\text{Resistance Arm}) = (\text{Effort Force})(\text{Effort Arm})\end{align*}

(500. N)(0.400 m)=(x)(0.800 m)\begin{align*}(500. \ \text{N})(0.400 \ \text{m}) = (x)(0.800 \ \text{m})\end{align*}

x=250. N\begin{align*}x = 250. \ \text{N}\end{align*}

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 Problem:

(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?

Solution:

(resistance force)(resistance arm)=(effort force)(effort arm)\begin{align*}(\text{resistance force})(\text{resistance arm}) = (\text{effort force})(\text{effort arm})\end{align*}

effort force=(resistance force)(resistance arm)(effort arm)=(4900 N)(0.100 m)(2.00 m)=245 N\begin{align*}\text{effort force}=\frac{(\text{resistance force})(\text{resistance arm})}{(\text{effort arm})}=\frac{(4900 \ \text{N})(0.100 \ \text{m})}{(2.00 \ \text{m})}=245 \ \text{N}\end{align*}

#### Pulley

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.

Credit: Laura Guerin
Source: CK-12 Foundation

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.

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.

Credit: Samantha Bacic
Source: CK-12 Foundation

#### 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.

Credit: Samantha Bacic
Source: CK-12 Foundation

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.

#### Wedge

A wedge is essentially an inclined plane

#### Screw

Credit: Flickr: LawPrieR
Source: CC-BY 2.0

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.

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.
• When simple machines are joined together to make compound machines.

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