# Beam balance Small beam scales (so-called “seed scales”), southern Germany, first third of the 19th century

A beam balance is a weighing device . It consists of a horizontal beam, which is movably mounted on a horizontal axis, and two weighing pans, which are suspended from the ends of the beam. The theoretical-physical basis is provided by the law of levers . In contrast to scales with springs, beam balances are not dependent on local differences in gravity. They allow a direct determination of the mass if the measured object has approximately the same density as the weight . If the density of the object differs significantly from the density of the weight piece (e.g. the mass of a body made of plastic or wood versus a weight made of metal), the measured weight value deviates more or less clearly from the actual mass.

The accuracy of a beam balance depends on the one hand on the accuracy of the weights used and on the other hand on the construction parameters of the balance (beam length, position of the center of gravity).

## construction

If the beam were supported in its center of gravity and all three bearings, i.e. that of the beam and the two bearings of the weighing pans, were on a straight line, this would have the following effects:

• In the case of equilibrium (both weighing pans are filled with the same weight), there is a state of equilibrium in any position of the bar, since the two torques are always the same ( indifferent equilibrium ).
• In the event of an imbalance, the bar would rotate vertically so that the heavier weighing pan would reach the lowest point ( stable equilibrium ).

This means that weighing is possible in principle, but impractical. Therefore, the beam is placed above its center of gravity and the bearings of the weighing pans are placed a little lower. That is why the balance leans to the heavier side, but it does not necessarily come through. The distance created by such a construction determines the sensitivity of the balance (deflection per mass difference).

Beam scales are usually symmetrical: the two weighing pans are suspended at the same distance from the pivot point - the two lever arms are the same length. If the masses in the scales are the same, the torques acting on the beam are also the same and the beam remains horizontal. If the masses in the weighing pans are unequal, the torques in the starting position are unequal. The shell with the greater weight lowers, whereby the effective lever arm on the heavier side is shortened and the lighter side is extended until the torques are balanced again.

Beam scales usually have a pointer in the middle that points vertically up or down in equilibrium and deviates from the vertical in the event of an imbalance. The pointer makes it easier to see whether the weights are balanced, especially when the pointer is combined with a scale on which the deflection of the pointer can be read.

## Weighing process

### principle

For weighing, the object to be weighed is placed in one of the bowls. Now mass in the form of pieces of weight is placed in the other bowl until the balance is in equilibrium. The mass of the object to be weighed is then as great as the mass of all weights in the other bowl.

If the mass of the object to be weighed is slightly smaller than a large piece of weight, it can make sense to compensate for this by using weights in the same bowl. If, during the first comparison, you find, for example, that the object to be weighed weighs slightly less than a 1 kg piece, you will weigh down the bowl with the weighing object with small pieces of weight until an equilibrium is achieved, instead of the 1 kg piece many smaller pieces to replace. The difference in the sums of the masses of all weights in each of the bowls gives the mass of the object.

### Coarse and precise weighing Laboratory balance with knurled screws for taring and long pointer pointing downwards. The rotary knob under the scale is used to raise the bearing of the beam for weighing. In the photo it is lowered, the bowls are on the table, the bearings have play and are spared when handling.

In the trade, the beam balance is or was mostly used to slightly exceed a minimum amount. If a customer requests 2 kilograms of apples, first two 1 kg pieces are placed on the scales and then the other bowl is filled with apples until the 2 kg is slightly exceeded. The customer only pays for 2 kg.

An error by a factor of L / L ' by differently long lever arms L and L' may be covered by a second measurement with reversed position of the sample and the weights is performed in error by a factor L '/ L . In the geometric mean of the two results, these factors cut off. This assumes that the balance is tared, i.e. that it swings around the middle position when empty.

## accuracy

### Factors

The sensitivity of the beam balance depends on various factors.

• Length of the balance beam: the longer, the more sensitive.
• The higher the center of gravity of the unloaded system (the closer it is to the pivot point), the more sensitive the balance.
• Dimensions of the weighing pans and the balance beam: the smaller, the more sensitive the balance.
• The heavier the goods to be weighed and the counterweights, the lower the sensitivity of the balance.
• Quality of the bearings : The lower the static friction in the bearings, the smaller the differences in weight, the lower the hysteresis and the better the repeatability of the balance.

### The role of the pointer

Pharmacist scales are often designed in such a way that a long pointer (the proverbial tip on the scales ) is attached above the pivot point , which reaches to the base of the scales and oscillates there in front of a small scale . It is very light, so that it only makes a very small contribution to the torque compared to the weighing pans.

Weighing can now be carried out with two different weights on the scales, which differ, for example, by one milligram . After reading off the pointer positions from the scale, the weight that corresponds to the pointer zero position can be calculated by interpolation . An additional decimal place (valid place) of the weighing result can be determined, which in the example enables an accuracy of tenths of a milligram.

## Similar scales

As a rule, only scales with two hanging scales are referred to as beam scales. However, the following types of scales also have a bar as a central element:

• The table scales . The platforms for supporting the weights - it can also be bowls - are located above the balance beam.
• The sliding weight scale has a bar with a sliding counterweight. Moving it changes its leverage .
• The tilt balance . The bar is kinked with her. The counterweight is constant - the decisive factor is the variable leverage.
• The decimal scale was designed for larger loads, for example for weighing sacks of flour. Only part of the weight of the weighed item is transferred to the lever arm in such a way that a tenth of the weight on the other side of the bar is sufficient to compensate.

These scales are also independent of local gravitational differences.

## Individual evidence

1. See Duden online: beam balance
2. Lexicon of Physics: Beam balance. Spektrum.de, accessed on February 9, 2020 .
3. ^ The beam balance, Virtual Physics Museum. Laurentianum grammar school in Warendorf, archived from the original ; accessed on February 9, 2020 .