Simple Machine

Many tools make our work easier: screws, pulleys, the wheel and axle, scissors, the jack used to lift a vehicle, sickles and axes. None of them needs electricity, diesel or petrol: human force is enough. Simple tools like these, which help us do work more easily, more comfortably or faster, are called simple machines.

By their structure there are four basic types of simple machine: the lever, the pulley, the wheel and axle, and the inclined plane. The wedge and the screw are special forms of the inclined plane.

Mechanical advantage

A simple machine can magnify a force. In Figure 8.1, a lever lifts a load of 600 N with an effort of only 300 N: the load is twice the effort.

The ratio of the load lifted to the effort applied is the mechanical advantage (MA) of the machine:

$$\text{Mechanical advantage (MA)} = \frac{\text{load}}{\text{effort}}$$

For the lever, MA = 600 N ÷ 300 N = 2. Mechanical advantage is a ratio of two forces, so it has no unit.

Velocity ratio

Look again at Figure 8.1. The effort end has to move twice as far as the load end. The velocity ratio (VR) of a machine is the ratio of the velocity of the effort to the velocity of the load:

$$\text{Velocity ratio (VR)} = \frac{\text{velocity of effort}}{\text{velocity of load}}$$

The effort and the load move for the same time, so velocity ratio is also the ratio of the distances they move:

$$\text{VR} = \frac{\text{effort distance (ED)} \div t}{\text{load distance (LD)} \div t} = \frac{\text{effort distance (ED)}}{\text{load distance (LD)}}$$

For the lever in Figure 8.1, the effort moves 2 times as far as the load, so VR = 2. Velocity ratio is a ratio of two velocities (or two distances), so it too has no unit. The same two formulas give the MA and VR of every simple machine.

To raise a flag without a pulley you would have to climb the pole and pull it up. The pulley at the top of a flagpole turns the direction of your force round: you pull down, and the flag goes up.

A pulley is a wheel that turns freely on an axle, with a groove round its rim for a rope. It is made of plastic, wood or metal. The effort is applied at one end of the rope and the load is attached to the other. Pulleys are used to lift buckets of water from a well, to raise flags, to lift building materials up tall buildings, in cranes, and in lifts.

There are three kinds of pulley system:

• A fixed pulley is fixed to a rigid support. It doesn’t move up or down, but turns freely. It is used to change the direction of the force.
• A movable pulley moves up and down with the load. It doesn’t change the direction of the force, but it halves the effort needed.
• A combined pulley (a pulley system or block and tackle) has at least one fixed and one movable pulley. The more pulleys it has, the less effort is needed, so the mechanical advantage increases with the number of pulleys.

Velocity ratio of a pulley system

In a single movable pulley, the load hangs in a loop of rope, held up by two segments of rope. To raise the load by 1 m, both segments must get 1 m shorter, so you must pull 2 m of rope. The effort distance is twice the load distance: the velocity ratio of a single movable pulley is 2.

In a single fixed pulley, the rope moves just as far on the effort side as on the load side, and the load is held by one segment: its VR is 1. In general,

$$\text{VR of a pulley system} = \text{number of rope segments supporting the load}$$

Mechanical advantage of a pulley system

As for every machine, $\text{MA} = \dfrac{\text{load}}{\text{effort}}$. For example, if a pulley system lifts a load of 200 N with an effort of 50 N,

$$\text{MA} = \frac{200\ \text{N}}{50\ \text{N}} = 4$$

With this pulley system, an effort of 50 N lifts a load four times as large.

With no friction, a single fixed pulley would need an effort exactly equal to the load (MA = 1). Because of friction, the effort is a little more than the load, and MA is a little less than 1. So why use it? Because it changes the direction of the force: it is much easier to pull down, using your own weight to help, than to lift a load straight up.

A sloping surface used to raise a heavy load to a height with less effort is called an inclined plane. A plank leaning on a truck, a ramp, a sloping path and a staircase are all inclined planes.

When a load is pulled up an inclined plane, the effort moves along the sloping length l of the plane, while the load rises only through its height h. So:

$$\text{VR of an inclined plane} = \frac{\text{effort distance}}{\text{load distance}} = \frac{\text{length of the inclined plane } (l)}{\text{height of the inclined plane } (h)}$$

For the same height, a longer plane makes a smaller angle with the ground: the slope is gentler, and less effort is needed to push or pull a load up it. The effort distance grows, so the velocity ratio increases as the length of the plane increases. That is why mountain roads have so many bends: they make the road longer and gentler, so that vehicles can pull heavy loads up it.

2 (a) Wedge

We cut fruit with a knife, chop vegetables on a chulesi, and split wood with an axe. Each of these tools is a wedge. Look at its edge: it is made of two inclined planes meeting at a sharp edge.

A wedge is a tool with a broad, flat surface at one end and a sharp edge at the other, where two inclined planes meet. It works on the principle of the inclined plane, so it is a special form of the inclined plane. The sharp edge makes it easy to push in, and the sloping faces push the material apart, against the friction and toughness of the material (the load).

When the wedge is driven all the way in:

• the effort moves through the length (depth) of the wedge, H: this is the effort distance;
• the material is pushed apart by the width (thickness) of its broad end, L: this is the load distance.

$$\text{VR of a wedge} = \frac{\text{effort distance}}{\text{load distance}} = \frac{\text{length of the wedge } (H)}{\text{width of the wedge } (L)}$$

So a long, thin wedge has a large VR: that is why a sharp knife with a thin blade cuts more easily than a blunt, thick one.

2 (b) Screw

We join pieces of wood with screws, lift a heavy vehicle with a jackscrew, and fasten sheets of metal with nuts and bolts. Look closely at a screw (Figure 8.5): it has a sharp thread winding round and round it. The thread is an inclined plane wound in a spiral round a cylinder, like a mountain road winding round a hill; so a screw is called a spiral inclined plane, and it is a special form of the inclined plane.

The distance between two neighboring threads is called the pitch (P). Each time the screw is turned once, it goes into the material (or moves up, in a jackscrew) by one pitch.

In one full turn, if the effort is applied at a distance R from the screw’s axis (the radius of the screwdriver’s handle, or the length of a jack’s handle; for a screw turned by its head, the head’s radius):

• effort distance = the circumference of that circle = 2πR;
• load distance = how far the screw goes in = the pitch P.

$$\text{VR of a screw} = \frac{2\pi R}{P}$$

The velocity ratio depends on the pitch: the smaller the pitch, the larger the VR, and the easier it is to turn the screw. A longer handle helps in the same way.

A lattai (latai) winds up kite string; a doorknob turns a latch; bicycle pedals turn the chain wheel. These are all examples of a wheel and axle: two cylinders of different sizes fixed on the same axis, so that they turn together. The bigger one is the wheel and the smaller one the axle. When a load hanging from the axle is lifted by an effort applied to the wheel, the force is magnified.

In one turn, the rope unwinds from the wheel by one circumference of the wheel, and winds on to the axle by one circumference of the axle. So the effort moves through the circumference of the wheel, and the load through the circumference of the axle:

$$\text{VR} = \frac{\text{effort distance}}{\text{load distance}} = \frac{\text{circumference of the wheel}}{\text{circumference of the axle}} = \frac{2\pi R}{2\pi r} = \frac{R}{r}$$

where R is the radius of the wheel and r the radius of the axle. The velocity ratio of a wheel and axle is the ratio of the radius of the wheel to the radius of the axle.

A wheel and axle is also called a continuous lever. Seen from the end (Figure 8.6), the center of the axis O is the fulcrum, the effort acts at A on the wheel, and the load at B on the axle, as on the two arms of a lever. But this lever can turn all the way round, 360°, and as it turns, a new point of the wheel becomes A and a new point of the axle becomes B, over and over: a lever that never runs out of swing.

Magnifying force, or gaining speed

A wheel and axle can be used either way round:

• Effort on the wheel, load on the axle: the effort moves farther than the load, so the machine magnifies force. A driver’s small force on the steering wheel turns the heavy front wheels of a bus. A bigger steering wheel makes the effort distance bigger and the force needed smaller, which is why buses and trucks have bigger steering wheels than cars (power steering now helps them too).
• Effort on the axle, load on the wheel: the load moves farther than the effort in the same time, so the machine increases speed. Flying a kite, you turn the lattai by its thin axle (the stick) with small movements of your hands, and the wide drum (the wheel) winds the string in quickly.

Some work is too complicated for a single simple machine: factories make complicated products, large trucks carry heavy loads, bulldozers dig the ground. For such work, simple machines of different kinds are combined into a complex machine (or compound machine). Complex machines do work with less effort, in less time and with ease.

In Activity 8.2 you worked out the output work and the input work of each pulley system. Their ratio tells us how much of the energy we put into a machine goes into the useful job. If it is 75%, then 75% of the energy we put in lifts the load, and the other 25% is wasted, mostly by friction, which turns it into heat and sound.

• The output work is the work the machine does on the load: output work = load × load distance.
• The input work is the work done by the effort on the machine: input work = effort × effort distance.

The ratio of the output work to the input work, as a percentage, is the efficiency (η, the Greek letter eta) of the machine:

$$\text{Efficiency } (\eta) = \frac{\text{output work}}{\text{input work}} \times 100\%$$

Put in the output and input work:

$$\eta = \frac{\text{load} \times \text{load distance}}{\text{effort} \times \text{effort distance}} \times 100\% = \frac{\text{load} \div \text{effort}}{\text{effort distance} \div \text{load distance}} \times 100\%$$

$$\eta = \frac{\text{MA}}{\text{VR}} \times 100\%$$

In every real machine, the output work is less than the input work. For the efficiency to be 100%, all the input work would have to become output work; but friction between the moving parts (and, in a pulley system, lifting the movable pulleys themselves) always uses some of it. So the efficiency of a real machine is always less than 100%, and its MA is always less than its VR. Oiling and greasing the moving parts reduces friction and raises the efficiency a little.

Perfect machine

A machine in which the output work equals the input work would have an efficiency of 100%. Such an imaginary machine is called a perfect machine (or ideal machine). In a perfect machine, MA = VR. A machine with no friction at all is impossible in practice, so a perfect machine is impossible too, although a good lever or a well-oiled pulley comes close.

1. Choose the best answer

2. Give the differences between

3. Give reasons

4. Answer the following questions

5. Solve these numerical problems