### Making work easier (but there's always a catch)

Simple machines are (simple) devices that either change the direction of a needed force or that reduce the amount of force required to do some work. An example of the latter would be a rope and pulley, so that a pulling force can be used to raise a load. In this case, the amount of downward force required to lift the load is the same as the upward force if we just lifted it; only the direction is different.

As we'll see below, other pulley arrangements not only change the direction of a required force, but also reduce the amount of force required to do the same amount of work, but always at the expense of more distance.

A good example of this idea is the inclined plane.

In the diagram, the crate can be lifted directly to height h, or it can be moved there via the inclined plane. In the first case, the work required is w = mgh. In the second, the force required to push the crate up the ramp is less than mg, but the distance up the ramp to the destination height is larger, in such a way as to keep the required work, w = F·d, the same. This is an example of the principle of conservation of energy.

### The simple machines

Simple machines have been studied since the Greeks. They are fundamental parts of many more complicated mechanical machines, and the thinking was that any machine at all could be represented as a collection of one or more of those listed below. For example, a car uses wheels, pulleys, levers, screws, and so forth.

Simple machines allow us to exert more force than we might be capable of producing by spreading that force out over more distance, or iterations of some smaller distance. A wrench (really a lever) is a good example. Few of us are capable of exerting enough twisting force on a stuck bolt with our hands, but if we use a long-enough wrench, we can do it. The cost of being able to exert enough force comes in the greater distance through which we have to rotate the wrench handle, compared to just turning the bolt with our fingers.

In this section, we'll work through the list of simple machines and see how they allow us to spread force over distance to do useful work.

The most common list of simple machines is

• Lever
• Wheel & axle
• Pulley
• Inclined plane
• Wedge
• Screw

But as you'll see below, it's possible to pare that down to just two. The list isn't really that important. More important are the concepts that each have in common, namely the trade-off of reduced force for greater distance. Notice also that distance is generally proportional to time, so simple machines decrease the power required to do a job, too.

### Conservation of energy

Simple machines obey the principle of conservation of energy, or more specifically, the idea that the amount of mechanical work done is independent of the path taken to change the position.

The change in work,

$$w = F \cdot d,$$

depends only on the initial and final positions of the object moved.

An inclined plane is a good example. Let's see how it works. Let's slide a 100 Kg crate up a ramp angled at 20˚, so that the bottom-left corner of the crate reaches a height of 4 m, as shown below. We'll ignore friction here, something we could add later without loss of any meaning in this example.

We need force and distance. The distance we can get by trigonometry:

\begin{align} sin(20˚) &= \left( \frac{4}{d} \right), \; \text{so} \\ d &= \frac{4}{sin(20˚)} = 11.695 \; m^* \end{align}

*In order to avoid round-off error, I like to keep more digits around during the calculation, then round to significant digits at the end.

To calculate the force needed to push the box that distance up the ramp, we resolve the weight (gravitational force vector) into components into the ramp and down the ramp using a little trigonometry:

$$F_g = 100 Kg \cdot 9.8 \, \frac{m}{s^2} = 980 \; N$$

\begin{align} F_x &= F_g sin(20˚) = 335.18 \; N \\ F_y &= F_g cos(20˚) = 920.90 \; N \\ \end{align}

We only need Fx for our purposes. The work is

\begin{align} w &= F\cdot d \\ &= 335.28\;N (11.695 \; m) \\ &= 3921 \; J = 3.92 \; KJ \end{align}

#### Lifting straight up

Now let's compare that result to the work of just lifting the crate straight up to a height of 4 m.

\begin{align} w &= mgh \\ &= F_g \, h \\ &= 980 \; N \cdot 4.0 \; m \\ &= 3920 \; J = 3.92 \; KJ \end{align}

We were off by a Joule, but that's just due to rounding error in the first calculation. The energies (work) are the same. This isn't proof of conservation of energy, but it does show that work is independent of path for this simple machine.

### Levers

There are three "classes" of levers, first, second and third. We'll go through those and discuss their mechanical advantage properties.

All levers consist of a fulcrum, a bar, a load (Fload) and a place where force is applied (Fapp). The load is the thing we want lifted, moved or turned. The bar is the device (usually something linear) that we use to transmit the force from application to load, and the fulcrum is the point around which the bar rotates. Here we'll call the distance between load and fulcrum a and the distance between fulcrum and applied force b.

#### Class 1 lever

The class 1 lever is probably the most familiar to most people. We often use it when we want to lift something we can't lift with just our muscles. The fulcrum is between the load and the applied force.

The mechanical advantage of such a lever is

$$a_m = \frac{b}{a}$$

Let's say we need to raise a 100 Kg load to a height of 1m using a lever with   $b = 2a.$ The mechanical advantage is 2, which means that the force required to raise the load is half of its weight,

\begin{align} F_{app} &= \frac{1}{2} F_g \\ &= \frac{1}{2}(100 \; Kg)(9.8 \; m/s^2) \\ &= 490 \; N \end{align}

The distance over which the force must be applied, however, is the mechanical advantage multiplied by the lifting distance, or 2·1 m = 2 m. So this lever divides the required force by two (the mechanical advantage), but multiplies the required distance by two as well. In this way the work is the same for either lifting the load straight up or using the lever.

For a lever with mechanical advantage am, the required force (Fapp) is   $F_g / a_m$   and the distance over which that force must be applied is am times the lifting distance.

##### Biomechanical example

One of many class-1 levers in the human musculoskeletal system is the system that rotates the skull up and down. The fulcrum is the point where the skull connects to the spine. The applied force is the muscles on the back of the neck (the trapezius is one of many) and the load is the weight of the front of the skull. Normal muscle tension in the bundle of muscles at the back of the neck holds the head level.

#### Class 2 lever

In a class-2 lever system, the load is between the fulcrum and the applied force, as shown below.

The mechanical advantage in this system is the same as a class-1 lever. One common example is a door. The fulcrum is at one edge, the load is the weight of the door, which we can concentrate at its center of mass, and the applied force is at the edge opposite the hinges.

##### Biomechanical example

The muscles that stand you on tip-toe are the calves (soleus & gastrocnemius). They attach to the bottom (distal end) of the femur and to the heel bone. The load of the weight of a human is transferred through the tibia, which is between the fulcrum (the ball of the foot) and the applied force (the constricting calves).

#### Class 3 lever

In a class-3 lever, the applied force is between the fulcrum and the load. In this configuration, the bar must be attached to the fulcrum, or else it would lift off. One example of a 3rd-class lever is a pair of tweezers, with the fulcrum at the closed end, the load at the open end, and the force in between.

In a 3rd-class lever, the maximum mechanical advantage is 1. Any position of the applied force other than directly opposite the load, results in an amplification of the force needed to move the load.

##### Biomechanical example

In order to raise a load held in your hand, your biceps, which attach to the strong bone of the forearm between the fulcrum (the elbow) and the hand, shorten and pull on the forearm bone (ulna), raising the load.

### Pulleys

Pulleys are used to either re-direct the force needed to do work, to reduce it (at the expense of extending the distance over which the applied force is applied), or both.

Pulleys can either be fixed (stationary) or moving. The mechanical advantage in a pulley system is equal to the number of ropes leading to or coming from moving pulleys. In the simplest case, a single pulley is used to re-direct the force needed to move a load:

The upward force is converted to a downward force (which can be a great benefit), but there is no mechanical advantage to this system. The applied force is exactly equal to the load, as long as we're ignoring friction in the system.

#### 2:1 pulley system

In a 2:1 pulley system, shown below, a second pulley is attached to the load, and thus moves along with it. In this system, there are two ropes attached to the moving pulley, giving a mechanical advantage of 2. Thus the force required to lift the crate will be half of the weight of the load, but for every meter of lift, we'll have to pull 2 m of rope through the system.

#### 4:1 pulley system

A 4:1 pulley system is shown below. It contains two moving pulleys, each with two ropes going to or coming from them. The four ropes attached to moving pulleys gives this system a 4:1 mechanical advantage.

The force needed to lift the load is ¼ of the load, but four times as much rope will have to be pulled through the system for each unit of lift of the load.

Pulley systems come in many shapes and sizes. A typical block-and-tackle setup is shown below.

This particular setup can be rigged as a 4:1 or 8:1 system for hauling against the kinds of heavy resistance encountered in larger sailing boats. With an 8:1 system, in order to move a sail or spar by 10 cm, we'd have to haul out 80 cm or 0.8 m of rope.

### The screw

U.S. Patent office excerpt from the 1933 patent of the Phillips-head screw, J.P. Thompson

Screws and bolts of all kinds work on the same basic principle. They are essentially nails — narrow cylinders of metal — wrapped with helical inclined plane — the "threads" of the screw.

Twisting a screw — in a hole in wood, for example — causes the threads to cut into the wood. The force needed to drive a same-sized cylinder straight into the wood is divided by the pitch (slope = rise/run). The trade-off for that reduced force is a greater distance over which that force must be applied, in this case, many circles have to be turned.

A screw is just a specialized version of the inclined plane. A bolt is a screw that is meant to slide into a pre-grooved receptacle with matching threads, such as a nut or a tapped (threaded) hole.

### The wedge

The wedge is closely related to the inclined plane. It is used to translate a force at one angle to a force at right angles to it. The most common example is the wood-splitting wedge used to split logs along the grain of the wood.

A wedge is like two inclined planes set back-to-back. The applied force can be resolved into components along the slanted face of the wedge and perpendicular to it. It is the perpendicular force that does the useful work of pushing outward.

Using a wedge (or a maul or an axe, which are just wedges on a stick), one can prepare enough wood to heat a small cabin for the winter.

Photo: J. Cruzan, 2017

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### spiral/helical

A spiral is a two-dimensional curve emanating from a central point, but with each new point on the curve lying an increasing distance from that central point. A spiral is a curve in a plane.

A helix is a three-dimensional figure, more closely related to a circle than a spiral. A helix is characterized by a radius, the nearest distance between any point on the helix and a central line about which it twists, and the pitch or angle of those twists with respect to a plane normal to the central axis.

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