Teaching Environmental Modelling with Hooks Law


We all know that environmental modelling plays an important role in environmental management. Impact assessment is one example where we make an attempt to predict or anticipate environmental impacts due to actions we propose to take. We use models to help us in this endeavor.

Models tell us about what if? and caution or guide us in taking required preventive and mitigative measures. Mitigations include not just physical measures but also required policy reforms. That makes both construct and application of models very interesting, useful and an exciting experience.

Models improve our understanding of the “system”, make us realize the limitations and but at the same time provide clues on how to interpret the modelling results to our advantage.

Environmental modelling deserves a full course in a post-graduate program. Unfortunately, this course is not generally offered today in most universities. There is a clear shortage of faculty. The subject is however actively used in the practice – influencing major decisions.

Modelling is generally taught only over few lectures in the course on Environmental Impact Assessment (EIA). That is not enough. Again some teach modelling by training students on some of the canned software e.g. Aermod, Qual etc. Both students and professors feel that this is how modelling should be “taught” or “learned”!

Apart from the postgraduate students, modelling needs to be taught to other key stakeholders like the government officials, Environmental NGOs etc. Its not that we want these stakeholders to become modelling experts but we want them to know enough to appreciate, question and get involved when models are influencing important decisions.

I asked my Professor Friend about my worries about teaching environmental modelling. He has been a Professor of environmental modelling for long. How do you teach this subject? I asked the Professor.

“Come to my first modelling lecture next week at IIT. I am running a course on environmental modelling for environmental NGOs and government officials”. He said – casually.

I was fascinated. Teaching a heavy subject like environmental modelling to NGOs and Government officials can be quite a challenge.

I reached the class early and took a back seat to look inconspicuous. The “students” walked in shortly and occupied the seats.

Professor entered the classroom with a large bag. Everybody was curious.

Professor asked everyone whether they had studied Hooks Law in their school/college days. The answer was yes. “Great” – Professor said.

I  failed to understand why this question was asked. Must be something naughty as usual, I said to myself.

Professor pulled out a metal spring from his large bag and hung it on a C-Frame.  He took out a weight with a hook from the bag and announced

“I am going to attach this weight on this spring. What would you expect to happen?

There was a quick show of hands.

“Sir, the spring is going to expand because of the weight”A  “Student”, who was junior officer at the Pollution Control Board said.

Professor smiled. “You are right. Let me then do so”

When the weight was attached, indeed the spring elongated.  “This weight was 1 kg by the way.” Professor said.

Professor now pulled out another similar metal spring and placed it on the C frame.

“I am going to put on this second spring a 2 kg of weight”. He placed the weight.

The “students” now saw two springs under different weights. One with 2 kg that elongated more than spring that was loaded with 1 kg.

“Do you see any difference?” Professor asked.

Student, who was an activist in the struggle on stopping encroachment at the Mangroves in Mumbai, said that the spring with 2 kg elongated more.

Professor said “That’s absolutely correct”

He then measured the extent of elongation. For spring 1, it was 0.5 cm and for spring 2 it was 1 cm.

“So more is the weight on the spring, more is the elongation. This proportional relationship is essentially the Hook’s law”. Professor said

If we plot force or weight against extension for a material which obeys Hooke’s law then the relationship will look like the graph below.


The gradient of this graph is the spring constant (k) or Young’s modulus (Y) which is measured in Nm-1. For Copper Y is around 117, for aluminum around 69 and for nylon close to 3. The material matters.

Professor then asked the students to come up to the C-frame and make experiments with springs of different materials and various weights and calculate Y. This was some fun.

Adarak (Ginger) Tea was now served.

“Do you see this experiment has any environmental connection?” Professor asked.

Most faces showed negative

Professor prodded  “say the case of untreated sewage entering the lake?

There was a pause.

“Are you trying to imply that more is the pollution load getting into a lake, more is the deterioration in water quality” One of the smart “students” responded.  He was working on the project “save Powai Lake”. “This is what Hook’s law will tell us”

Another “student”, an assistant engineer from Municipal Corporation added “and it will depend on the lake’s Young’s Modulus, more is the depth, mixing and dilution, less will be the deterioration in water quality”

Professor smiled “Both of you are absolutely right. You got that! We will call the Young’s modulus here as the Assimilative capacity”

Professor then explained the term assimilative capacity. He talked about mixing height, horizontal winds and turbulence in the air sheds; depth, velocity and cross sections influencing waste assimilation in the rivers etc.

Then there was a  discussion that we must not exceed the “allowable” assimilative capacity. We must have policies, regulations, required institutional capacities and the finance.

While the discussion moved to this point, Professor took out a 20 kg weight from his bag. He attached this “heavy weight” to one of the springs. The spring almost “sank”. There was a disproportionate elongation. It did not follow the “linear model” of the Hooks law as shown in the graph above.

Everybody was stunned.

Professor explained that the weight of 20 kg exceeded the limit of proportionality. Hooks law was no more valid as we stretching the material too far.

The linearity ended after Point “a” as shown in the figure below.



See source

There was one more interesting thing Professor asked the students to notice. When he removed the 20 kg weight, the spring did not come back to its original length. Clearly the elastic limit of the spring was crossed and a permanent deformation had occurred indicating a non-linear behavior.

The message to the students was clear. They understood that if the assimilative capacity is exceeded, then the impacts could be non-linear and even irreversible.

If too much wastewater was discharged in to the lake, then the water quality changes could be major, catastrophic or permanent. So one would need interventions on both mitigative and restorative side i.e. treating the wastewater prior to lake discharge, or discharge with treatment at the center of the lake where depths are greater or introduce aeration, carry out dredging or apply biotechnology to improve the restorative capacity (or lakes Young’s modulus). Professor discussed the risks to be understood for each of the options. There were pros and cons and no free lunch.

This experiment led to the understanding that we need to be precautionary and protective to the assimilative capacities available in the nature. We must compute or model in advance the possible consequences.

Professor was ending his first lecture on “modelling” now. He went to the white board, drew a lake in plan and placed three springs connected with each other. MFS_sp_1_c

See source

He then summarized – don’t look at the lake as ”one” spring but made out of several different springs – and all interconnected. A load on any of the springs will not only elongate that spring, but will “transfer” the load to the “connecting” springs leading to elongation in them. Since the Young’s modulus of the springs will be different (as depths vary, so the currents and the extent of reaeration), the net elongations (or water quality changes) are going to be complex. If wastewater is discharged at the bank or in the mid of the lake for instance, it will make a difference as the springs in “action” will be different.

He looked at the Powai Lake outside the window of the class and slowly said.

“I am requesting Dr Prasad Modak who is sitting in the last row, to come up with a “physical model of springs and weights” for the Powai Lake when we meet next. We will do various experiments on this model by hanging weights, measuring the elongations and changing the springs. In the third lecture, I will explain how we will use the principle of Hooks law to develop a simple lake water quality model”

“Oh this is now getting really tricky and difficult” I said to myself.

I decided to bunk the next lecture and jump straight to the third.


Cover image taken from https://en.wikipedia.org/wiki/Hooke%27s_law



  1. This piece is simply superb!! Modelling otherwise sounds so abstract for the learners. Pedagogy like this is much needed for the teaching of environmental science so as to get professionals who understand modelling, not the application of the modelling software.

    And this is true for so many other concepts of environmental science. It’s so frustrating when even the practitioners in field get confused on such basic things like organic pollutants and COD, using the terms interchangeably.

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