The Strange Effect of the Bathroom Scales

Jack Dikian

A few weeks ago I found myself weighing almost 10Kgs more than what I thought I'd weigh. This was after jumping on a new set of bathroom scales and having to almost do a double-take of the rotating dial. Whilst it’s true that some of my otherwise slim-fitting shirts aren’t fitting so well – I still regarded the 10Kg increases as excessive.

A number of theories prevailed – after all, the scales cost less than ten bucks. Engineering? that's something you wouldn't want to bank the farm on. Having said that though, the complication of most scales is by no means a Swiss chronometer. Also, it seemed to be calibrated, at least without a load.

Could using scales on soft flooring alter the scales’ functioning? More importantly, and by far the more interesting question - what exactly is it that scales measure. It turns out that this isn’t as obvious as it might seem, unless of course you are a physicist.

I’m going to return to the saga of the bathroom scales a little later – after exploring shorthand conventions and assumptions relating to the physical quantities, mass and weight. For example, it is often said that bathroom scales is a measuring instrument for determining the weight or mass of an object. Here, both mass and weight are used synonymously.

Mass and Weight

Students of science often confuse mass and weight and many feel that there is no difference between the two. In fact the two are not the same.

Mass is the amount of matter present in a body and is an intrinsic property of the body. The mass of an object remains unaltered regardless of the reference frame it is being measured in and according to special relativity it is related to energy by the famous relationship formula E = mc2. Weight on the other hand is the force which a given mass experiences due to the gravitational force between itself and another mass point (the Earth in our reader’s case).

Simply, we use the word mass to describe how much matter an object posses. On Earth, we weigh objects in order to calculate their mass. The more matter there is, the more the object will weigh. The difference between mass and weight is that weight is determined by the pull of the Earth’s gravity. If we are comparing two objects to each other on Earth, they are pulled by the same gravitational force and so the one with more mass weighs more. In space, where (where there is large distance between the two mass points) the gravitational force or pull of the Earth is smaller, an object may have no weight and yet still posses mass.

More formally, mass refers to any of three properties of matter, which have been shown experimentally to be equivalent. These are:

 Inertial mass,
 Active gravitational mass and
 Passive gravitational mass.

The inertial mass of an object determines its acceleration in the presence of an applied force. According to Newton's second law of motion, if a body of mass m is subjected to a force F, its acceleration a is given by F/m.

A body's mass also determines the degree to which it generates or is affected by a gravitational field. If a first body of mass m1 is placed at a distance r from a second body of mass m2, each body experiences an attractive force F whose magnitude is :

F = G(m1m2)/ r2

where G is the universal constant of gravitation, equal to 6.67×10−11 kg−1 m3 s−2. This is sometimes referred to as gravitational mass (when a distinction is necessary, M is used to denote the active gravitational mass and m the passive gravitational mass). Repeated experiments since the seventeenth century have demonstrated that inertial and gravitational mass are equivalent; this is entailed in the equivalence principle of general relativity.

The Unassuming Kilogram

Since 1889, the International System of Units (SI system) defines the magnitude of the kilogram to be equal to the mass of the International Prototype Kilogram often referred to as the “IPK”. The IPK is made of a platinum alloy and is machined into a cylinder. The IPK, also affectionately known as the ‘Big K’ and its six sister copies are stored at the International Bureau of Weights and Measures in a vault in the outskirts of Paris.

Three independently controlled keys are required to open the vault. Official copies of the IPK were made available to other nations to serve as their national standards. These are compared to the IPK roughly every 50 years.

For Australians, a metal ingot weighing precisely one kilogram is locked in a safe in a government facility on Sydney's North Shore. It is the kilogram against which all other kilograms in Australia are measured. Every precaution is made to ensure the weights are not contaminated. Given this is the reference standard of mass, any contamination with dust or fingerprints or any sort of foreign material, if unchecked, will impact upon and propagate throughout every aspect of everyday life as we know it.

What of the bathroom scales

David MacKay, a physicist at the University of Cambridge, after a chance conversation about bathroom scales measuring a greater weight when on carpet decided to investigate the reasons.

MacKay and his students tried a number of analogue bathroom scales on different surfaces. Sure enough, they found they weighed in at around 10 per cent more on thick carpet than on the hard floor.

To find out why, the studens took several sets of scales apart and measured the movement of the internal mechanisms when loaded on different surfaces. Inside each set of scales, four levers or "fulcrums", each pointing inwards from one of the corners, transmit the weight of the person to a spring-loaded metal plate at the back of the scales. The movement of the plate is then transferred via a metal rod to turn the dial on the scales.

They found that on a hard surface, the base of the scales bows. This makes the fulcrums at each corner of the scales tilt in slightly, shortening the distance between each fulcrum and the point at which the load pushes onto the lever.

Put the scales on a deep carpet, however, and the scales sink into it, so the carpet supports the base, which prevents it from bending. This increases the distance between each fulcrum and the point at which its lever is loaded, so for the same force the lever moves further. Even a small increase in this distance can add several kilograms to the weight registered on the display.

"I've always thought this was an urban myth," says a spokeswoman for Weight Watchers. "But it sounds like it makes a huge difference."