A pulley system with a mechanical advantage of 4. Only 100 pounds of force are needed to lift a 400 pound resistance.

### Simple 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 but the input force and distance do not have to be the same as the output force and distance. A machine can be as simple as a can opener or a screwdriver to as complex as a bicycle or an automobile.

The six common simple machines are the lever, pulley, inclined plane, screw, wedge, and wheel and axle. Wheel and axle calculations are similar to pulley calculations and a wedge is simply a double-sided inclined plane. The wheel and axle and wedge will not be considered here.

####
**Lever**

A lever consists of an inflexible length of material placed over a pivot point called a fulcrum. An object to be lifted is on one end of the lever and is called the **resistance** **force**. The **effort** **force** is exerted on the other end of the level to lift the resistance force. The distance from the fulcrum to the resistance is called the resistance arm or resistance lever arm and the distance from the effort force to the fulcrum is called the effort lever arm.

The effort work is the effort force times the effort lever arm. 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. For ideal machines, effort work always equals resistance work.

Suppose the resistance force is 500. N and the resistance arm is 0.400 m. If the effort arm is 0.800 m, we can calculate exactly how much effort force is required to lift this resistance using this lever.

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

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

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

\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 have the force of the objects weight. 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.

The ratio of the effort arm to the resistance arm is called the **Ideal Mechanical Advantage.**

**\begin{align*}\text{IMA}=\frac{\text{effort arm}}{\text{resistance arm}}\end{align*}**

The ratio of the resistance force to the effort force is called the **Actual Mechanical Advantage.**

**\begin{align*}\text{AMA}=\frac{\text{resistance force}}{\text{effort force}}\end{align*}**

Can openers, nail pullers, and wheelbarrows are all examples of tools that are levers. We can design levers with the fulcrum in the middle or with the fulcrum on the end and the resistance in the middle. On some occasions, we are willing to have the input effort greater than the output effort because we wish to have a large output distance. We exert a large input effort for a short distance and get a small output force but a large output distance. The human elbow is an example of this type of lever.

**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?

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

\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*}

(b) \begin{align*}\text{IMA}=\frac{\text{effort arm}}{\text{resistance arm}}=\frac{2.00 \ \text{m}}{0.100 \ \text{m}}=20\end{align*}

(c) \begin{align*}\text{AMA}=\frac{\text{resistance force}}{\text{effort force}}=\frac{4900 \ \text{N}}{305 \ \text{N}}=16\end{align*}

####
**Pulleys**

A ** pulley** is a wheel on an axle that is designed to support movement of a cable or belt along its circumference. Pulleys are used in a variety of ways to lift loads, apply forces, and to transmit power.

A pulley is a drum that has a groove between two flanges around its circumference. The drive element of a pulley system can be a rope, cable, belt, or chain that runs over the pulley inside the groove. A set of pulleys assembled so they rotate independently on the same axle form a block. Two blocks with a rope attached to one of the blocks and threaded through the two sets of pulleys form a block and tackle. A block and tackle is assembled so one block is attached to fixed mounting point and the other is attached to the moving load.

Consider the pulley systems sketched in the image below. The resistance force is hung on the moving pulley and the effort force is applied at the point of the arrow head in the image above. The distance that the effort force moves is easy to visualize, it is the distance that the arrowhead moves. The distance that the resistance force moves is not so easy to see. Consider the system diagrammed in C above. If the arrowhead in C is pulled 10 cm, the resistance DOES NOT rise by 10 cm. If that were to happen, it would mean that BOTH of the ropes between the two pulleys shortened by 10 cm and in that case, the arrowhead would have moved 20 cm. The distance that the effort force moves is shared equally by all the supporting strands of rope in the pulley system. In system C, if the effort force is pulled 10 cm, each supporting strand will shorten by 5 cm and therefore, the resistance will rise 5 cm.

The IMA of a pulley system can be determined by counting the number of supporting strands of rope in the pulley system. You must be careful, however, about whether or not to count the strand with the arrowhead as a supporting strand or not. If the arrowhead moves in the same direction as the resistance, it is a supporting strand but if it moves in the opposite direction of the resistance, it is NOT a supporting strand.

Let’s count the supporting strands in the five examples pictured above.

A = 1 supporting strand so IMA = 1 (essentially no mechanical advantage)

B = 2 supporting strands so IMA = 2

C = 2 supporting strands so IMA = 2

D = 3 supporting strands so IMA = 3

E = 3 supporting strands so IMA = 3

In the pulley system shown in diagram A, the pulley allows a change in the direction of the effort, that is, you pull down instead of lifting up, but there is no mechanical advantage gain.

**Example Problem: ** 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) \begin{align*}\text{Effort force}=\frac{\text{resistance force}}{\text{IMA}}=\frac{8500 \ \text{N}}{5}=1700 \ \text{N}\end{align*}

(c) 0.20 m

(d) \begin{align*}\text{AMA}=\frac{\text{resistance force}}{\text{effort force}}=\frac{8500 \ \text{N}}{2000 \ \text{N}}=4.25\end{align*}

**Inclined Plane**

An inclined plane is also a simple machine. The resistance is represented as 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, to slide the box up the incline requires a smaller effort force (assuming no friction). 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 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. Therefore, the ratio of the resistance distance to the effort distance is the \begin{align*}\sin \theta\end{align*}. This was also defined as the IMA, the IMA for an inclined plane is \begin{align*}\sin \theta\end{align*}.

**Example Problem:** 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, the parallel force, and the IMA for the box on this incline.

**Solution:** \begin{align*}\text{IMA} = \sin \theta = \sin 35^\circ= 0.57\end{align*}

\begin{align*}\text{Normal force} = (400. \ \text{N})(\cos 35^\circ) = (400. \ \text{N})(0.82) = 330 \ \text{N}\end{align*}

\begin{align*}\text{Parallel force} = (400. \ \text{N})(\sin 35^\circ) = (400. \ \text{N})(0.57) = 230 \ \text{N}\end{align*}

####
**Screw**

**<!-- @@image_id="IntPhysC-08-03-Screw-thread" -->**

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

\begin{align*}\text{IMA}=\frac{\text{effort distance}}{\text{resistance distance}}=\frac{3.0 \ \text{cm}}{0.60 \ \text{cm}}=5.\end{align*}

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 but the input force and distance do not have to be the same as the output force and distance.
- The five common simple machines are the lever, pulley, inclined plane, and screw.
- For a lever, the effort work is the effort force times the effort lever arm. The resistance work is the resistance force times the resistance lever arm.
- For a lever, \begin{align*}\text{IMA}=\frac{\text{effort arm}}{\text{resistance arm}}\end{align*}.
- For a lever, \begin{align*}\text{AMA}=\frac{\text{resistance force}}{\text{effort force}}\end{align*}.
- For an inclined plane, the \begin{align*}\text{IMA} = \sin \theta\end{align*}.
- For a screw, the \begin{align*}\text{IMA}=\frac{\text{head circumference}}{\text{thread width}}\end{align*}.
- 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.

#### Practice

The following is a video on simple machines. Use this resource to answer the questions that follow.

http://www.backpack.tv/video/physics/bozeman-science/paul-andersen/simple-machines

- Potential energy is present in objects that are ___________________.
- Kinetic energy is present in objects that are ________________.
- What formula is given for kinetic energy?

Practice sheet for work and simple machines.

http://www.proprofs.com/quiz-school/story.php?title=physics-chapter-10-energy-work-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?

- 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?