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.

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.

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:

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 (F_load) and a place where force is applied (F_app). 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.

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.

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.

The screw

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

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.