Home / Opinion / Online-views /  Cauvery dispute: The importance of dimensional analysis

Here is a simple question: If Karnataka releases 6,000 cusec of water to Tamil Nadu for 10 days, what is the flow rate of the water released?

a. 60,000 cusec

b. 6,000 cusec

c. 5.184 tmc

d. None of the above

If you have not answered (b), read on, for we will explore a concept which is critical for science and everyday decision-making, including the ongoing Cauvery issue.

Most physics syllabi begin with a concept called dimensional analysis, which states that there are seven fundamental physical quantities, or dimensions—mass (M), length (L), time (T), temperature (K), electric current (I), luminous intensity (C) and amount of substance (mol), and two further stipulations:

1. Any physical quantity may be represented as a product of various powers of these seven basic quantities.

2. Only quantities which are dimensionally alike, i.e. products of the same dimensions, can be compared, added or subtracted.

Elaborating on the first point, speed is defined as the rate at which distances are traversed with respect to time. Thus, its dimensional form is L/T. Then there is acceleration, defined as the rate of change of speed over time, having the dimensions of speed/time, or L/T2. The second point indicates that speed and acceleration are dimensionally different and, thus, cannot be compared.

Taking a more complicated example, force is the product of mass and acceleration, and of the dimensional form ML/T2, while energy and work are both the product of force and length, of the dimensional form ML2/T2. Thus, work and energy can be compared, added and subtracted (energy is convertible to work and vice versa), but they cannot be compared to force.

Similarly, power, the rate of change of energy over time has the dimensional form ML2/T3.

Dimensional forms of all physical quantities can, thus, be derived from the seven basic dimensions, including the so-called dimensionless quantities such as the Reynolds number, whose dimensional form is simply 1.

However, any physical quantity independent of these seven dimensions is yet to be discovered, and such a discovery would represent a breakthrough in physics.

Measurement units are closely related to dimensions. While ancient people used their own body parts for measurements (foot size, hand size, etc.— we still do such rough measurements occasionally), there arose a need for standardization. The British invented units such as the foot (ft) for length, pound (lbm) for mass, pound-force (lbf) for force, horsepower (hp) for power, etc. However, such units were often arbitrarily specified, causing difficulties in both recall and computation. This led to the development of a standardized set of units known as the Système International D’unités (SI), where every dimension was assigned its own unique standard unit.

Thus, in most of the world except the US and the UK, length is measured in metres (m), mass in kilograms (kg), time in seconds (s), temperature in Kelvin (K), electric current in amperes (A), luminous intensity in candela (Cd) and amount of substance in moles (mol).

With SI units, a lot of the arbitrariness associated with British units vanished, and units of complicated quantities were now easy to both remember and compute. Thus, the SI unit of power is the Watt (W), and 1 W = 1 kgm2/s3, compared with the arbitrary British formulation of 1 hp = 17,696 lbm ft2/s3 (the dimension of power is ML2/T3).

Note that, consistent with dimensional analysis, like quantities can be converted from British units to SI and vice versa, but unlike quantities cannot be. For example, 1 hp = 745.7 W, but horsepower measurements can never be converted to acceleration units of m/s2 or even ft/s2.

With this foundation in dimensional analysis, we move on to the Cauvery judgement, which directed Karnataka to release 6,000 cusec/day of water to Tamil Nadu. This has been faithfully reported by most media outlets, including Mint. Additionally, The Hindu reported D.G. Parameshwara, home minister of Karnataka, as saying, “Karnataka had complied with the earlier order and released 1.68 lakh cusecs until today." Do these numbers make any sense?

Cusec is short for cu. ft per second (ft3/s), a British unit of volume flow rate still in currency today among India’s civil engineers. It is the volumetric analogue of speed, and has the dimensional form L3/T. Cusec/day has the dimensional form L3/T2, and is a meaningless quantity in this context, the volumetric analogue of acceleration. It would only make sense if the court ordered Karnataka to start at 6,000 cusec on day 1, and increase it by 6,000 cusec every day to 12,000 cusec on day 2, 18,000 cusec on day 3, etc.

Clearly, this is not the case and, thus, a fundamentally flawed judgement, reflecting a basic ignorance of a crucial concept, has been accepted without question, by our finest legal minds, reporters and other stakeholders. Gentle attempts to point out these errors online have largely been ignored, while at least one outraged activist’s misinformed tweet has been retweeted over 50 times.

A technically sound analysis would account for Karnataka’s total water holding, and then consider how much water must be released to Tamil Nadu. Tmc (thousand million cu. ft) is commonly used as a volume unit here. If a flow rate of 6,000 cusec (L3/T) were maintained for 10 days, i.e. 864,000 seconds (T), the total volume (L3) of water released would be 5184,000,000 ft3, or 5.184 tmc. But cusec and tmc are not interchangeable units, as they are dimensionally different. Thus, it may be important to pedal back to basic physics, compare apples with apples, and hope that our crucial decisions are taken after a proper understanding of dimensions.

PS: Salman Khan of Khan Academy provides an excellent beginners’ tutorial on dimensional analysis.

Prithwiraj Mukherjee is an assistant professor of marketing at IIM-Bangalore. Views expressed here are personal, and do not reflect those of his employer. This piece is just illustrative, and not intended to endorse any state’s claim to water. He can be contacted at pmnitk@gmail.com

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