# Tilt balance

Historical inclination scales
Letter scales

The tilt scale is a scale that the measured weight as opposed to the beam balance does not compensate with a different weight, but by the deflection of a mass on a lever , the reading on a scale possible. Since the inclination balance can be viewed as a variant of the beam balance with a bent balance beam, it is also referred to as an articulated lever balance .

The letter scales are probably the most common application of this principle and these scales are currently still produced in this way.

## inventor

Inclination scales were developed in the 18th century by the pastor and inventor Philipp Matthäus Hahn and built several times. Hahn had already built artistic developments in today's Albstadt district of Onstmettingen from 1764 to 1770 and refined them with his own inventions (clocks, scales, calculating machines, astronomy). The construction of the inclination scales was quickly adopted by the manufacturers of scales.

As a result of Hahn's suggestions, at the beginning of industrialization in the middle / end of the 19th century, an economic center was created in what is now the Zollernalb district , made up of factories specializing in precision engineering, which still shapes this area today. In honor of Hahn, the Philipp-Matthäus-Hahn-Museum was built in Onstmettingen , in which his inclination balance and many of his other inventions can be found.

## principle

### construction

Fine scale made of garbage, 0.1-0.8 g; as a test 0.3 ml of water in the weighing pan

In the case of the inclination balance, the load changes a weight that is unchangeable in terms of size, which is usually attached to a lever, from its rest position and thus knocked out. The size of this deflection angle, i.e. the inclination of this weight, serves as a measure of the size of the load, which can then be read off on a scale .

The main difference to the beam balance is the kink of the balance beam in the pivot point. As a result, there is no longer an unstable equilibrium , as with the pure beam balance , but a stable equilibrium. To do this, the balance beam must be bent down on both sides of the pivot point so that an overweight on one side turns the weight closer to the stand and thus reduces its torque, while the other side is turned more upwards, so that its torque increases. In principle, a stable equilibrium can then always be achieved. The sensitivity of the balance is slightly reduced compared to the straight beam balance, but it is sufficient for normal purposes.

### Mathematics of scales

In order to find out which angle corresponds to which mass ratio, an equilibrium of the applied torques is simply established:

${\ displaystyle {\ begin {matrix} m_ {1} g \; \; l \; \ cos (\ omega + \ beta) & = & m_ {2} g \; \; l \; \ cos (\ omega - \ beta) \\ m_ {1} \ cos \ omega \ cos \ beta -m_ {1} \ sin \ omega \ sin \ beta & = & m_ {2} \ cos \ omega \ cos \ beta + m_ {2} \ sin \ omega \ sin \ beta \\ m_ {1} -m_ {1} \ tan \ omega \ tan \ beta & = & m_ {2} + m_ {2} \ tan \ omega \ tan \ beta \\ {\ frac {m_ {1}} {m_ {2}}} & = & {\ frac {1+ \ tan \ omega \ tan \ beta} {1- \ tan \ omega \ tan \ beta}} \ end {matrix}} }$

• The angle is the angle at which the formerly straight beam was bent in the middle to offset the center of gravity. This angle depends on the construction of the balance.${\ displaystyle 2 \ beta}$
• The angle is the angle by which the beam is rotated due to the different masses, i.e. the deviation from the horizontal position.${\ displaystyle \ omega}$

## Areas of application and model variants

Bizerba inclination scales with additional switching weight, up to 2 kg, Deutsche Bundespost

The best-known inclination scales were mechanical letter scales , where you could sometimes even change the weight by folding it down and thus also weigh higher masses, which could then be read on a changed scale. The scales are available with a stand as well as with a hook or handle at the top for hanging or holding in the hand.

There are also high quality bench scales with a large circular pointer measuring head for use in industry, trade and laboratories or kitchens that work with the inclination scale principle. These are usually solid devices in heavy metal housings with a large display behind glass and multiple, usually five-fold pointer circulation. The usual weighing ranges and graduations are z. B. 30 kg / 10 g, 60 kg / 20 g and 100 kg / 50 g, whereby an estimated reading between the scale divisions is usually still possible and sufficiently accurate. These scales were widely used in large kitchens, workshops and laboratories and were or are valued for their uncomplicated handling (no need to switch on). Today (2010) they are only built by very few manufacturers.

Truck and vehicle scales were even manufactured according to this principle (weighing range up to 50 or 60 tons, scale division 20 kg).

Inclination scales were reduced in importance after the development of electronic scales. Production fell sharply, especially after using the strain gauges in scales.