23 April, 2010

An Externally Pivoted Balance


Chandrakant Doshi

email: jignashi@yahoo.co.uk

Abstract: This paper describes a balance in which friction is no hindrance to the accuracy of mass comparison.


The beam balance has a long history as an instrument of mass comparison. The earliest balances were suspended by a cord wrapped round the beam. Later versions had a central hole in the beam for the cord. In modern precision balances a knife-edge is used.

In a conventional balance as an equal mass comparator the point of suspension of the beam should be accurately located at the centre of the beam. In this paper I describe a method of suspending the beam that does not require the location of its centre.

The conventional balance has the point of suspension, or pivot, within the body of the beam and can be thought of as an internally pivoted balance.

The proposed balance has its pivot external to the beam and is simple to construct. In this paper, it is referred to as an externally pivoted balance.

The balance can be constructed using the materials excavated from various Harappan/Indus sites. While the balance was envisaged as a weighing instrument the ancients might have used, it is analysed here as an instrument in its own right.


The balance comprises a rigid beam suspended by means of a string tied to its ends over a rough pin so that it takes the shape of a triangle with the beam as its base (see Fig. A). Friction at the point of suspension is necessary for the arrangement to work as a balance.

The friction permits the beam to be inclined to the horizontal and still serve as a balance. The range of inclination about the horizontal is restricted and too large an inclination will drive the beam to a vertical position. Increasing the friction at the point of suspension will increase the usable range. An arrangement for clamping the string on the pin can also be useful.

So long as the beam inclination stays within the restricted range the arrangement behaves like a rigid body suspended from a frictionless pivot and hangs with its centre of mass directly below the point of suspension.

Pans are easily attached at A and B by tying the harness to the loose ends of the string ACB. The arrangement may not be as good as the knife-edge pivots but is compatible with the suspension arrangement of the balance. Flexibility of the string would keep the pans hanging vertical.

Equilibrium can be detected by suspending a second beam parallel to the main beam to serve as a reference.


A comprehensive analysis of the conventional beam balance is given in Practical Physics1 . A modern treatment is available in Comprehensive Mass Metrology2 . History of the ancient balances can be read in Skinner3 and Kisch4.

AB is a rigid beam and ACB an inextensible string. The arrangement is suspended on a rough pin at C, as shown in Fig. A. For clarity, the pans and harness are not shown in the sketch.

Assume the beam to be uniform and the string to be of negligible mass.

Let the mass of the beam be M, acting at its centre of mass (CoM).

Let pans, each of mass P (including the harness), be attached at A and B, respectively. To calculate the sensitivity of the balance under load, add an appropriate mass T to each pan. Manually adjust the balance so that the beam, AB, is horizontal. The arrangement hangs so that the centre of mass of the assembly, D, is directly below C.

Let AD = l1 and BD = l2 .

When a small mass m is added on the pan at A the beam changes position, without the string slipping about C, so that the centre of mass is again below C.

In the new position the points A, B, and D move to A´, B´ and D´, respectively. A´H is drawn parallel to AB. Let angle(B´A´H) = θ. This is also the angle of inclination of A´B´ with AB. Let the point of intersection of A´B´ and CD be K, the centre of mass of the balance in the new position.

Taking moments about CD, we obtain

where A´K, B´K and KD´ are lengths on the beam, as marked on Fig. A, and g is acceleration due to gravity.

Dividing both sides by g⋅cosθ we obtain

Now, A´K = A´D´ - KD´ = l1 - CDtanϴ
since KD´ = CD´tanϴ and CD´ = CD

Also, B´K = l2 + CDtanϴ

Hence, (1) becomes

which simplifies to


as an expression for added mass in terms of ϴ and the balance constants.
Equation (2) shows that if the CoM of the beam is not in the middle of the beam an error will occur when making a mass comparison.

Putting l1 = l2 = l in equation (2) we obtain

Since m is much smaller than 2P + M it can be ignored in the RHS of the above equation. Also, for small θ, tanθ ≈ θ , and equation (3) simplifies to

Sensitivity of a balance is defined as radians per gram. The above equation can be written as

This is the sensitivity of the balance with a loading of 2T. By putting T = 0 we obtain the sensitivity of the unloaded balance, equation (5).

Equations (4) & (5) show that the sensitivity of the balance decreases with increasing load.


The externally pivoted balance offers some advantages over the conventional, centrally pivoted balance.

Friction at the point of suspension is essential for the operation of the balance but it does not introduce any errors when comparing masses. Also, unlike the conventional balance, knowledge of the location of the centre of the beam is not required.

Errors in measurement will occur if the centre of mass of the balance is not at the centre of the beam, as shown by equation (2). This can be corrected by the addition of weights at an appropriate place on the balance. A method of weighing known as substitution weighing eliminates this source of error. Accuracy of measurement also depends on the accurate detection of equilibrium.

Several Harappan sites have yielded materials, such as pans and beams, with which to construct a balance. Harappan weights have been analysed as more accurate than those of either of the contemporary civilizations in Egypt and Mesopotamia. The externally pivoted balance, with fewer sources of errors, could explain the achievement of such an accuracy. However, no actual example of such a balance has been reported from any Harappan site.


I would like to thank Dr. Stuart Davidson of National Physical Laboratory, Britain, for reading the paper and making valuable suggestions. I also thank my son Samir and his wife Susan for useful discussions and comments.
Many thanks to the librarians at the Keith Axon Centre of the Redbridge Libraries for obtaining the essential books.


1/. Practical Physics by R. T. Glazebrook and W. N. Shaw. Longmans, 1893 – pp. 99.

Copies available on Internet Archive. Link to one such copy below:

2/.Comprehensive Mass Metrology

Edited by Manfred Kochsiek, Michael Glaser, Weinheim ; Chichester, 2000

3/. Weights and Measures:

their ancient origins and their development in Great Britain up to AD 1855 by F. G. Skinner, formerly Deputy Keeper in the Science Museum, London. HMSO 1967

4/. Scales and weights:

a historical outline by B. Z. Kisch, Yale University Press, 1965.

© 2010 Chandrakant Doshi