Unveiling the Power of Dimensional Analysis

A Mathematical Tool for Probing Scientific Phenomena

Author: Nishant Soni

Introduction
In the realm of science, understanding and predicting the behavior of natural phenomena is a fundamental pursuit. This pursuit often involves employing a variety of problem-solving techniques to derive insights and verify the accuracy of results. One such indispensable tool in the scientist’s toolkit is dimensional analysis. This technique plays a pivotal role in ensuring the validity of formulas, estimating quantities, and unraveling hidden relationships among various physical parameters. In this blog post, we delve into the significance of dimensional analysis, exemplified by the work of G.I. Taylor, who harnessed this method to estimate the explosive energy of the Trinity Test, marking a remarkable application of this mathematical approach.

Dimensional Consistency
Physics relies on the formulation of equations that accurately represent the underlying physical phenomena. To ensure the reliability of these equations, they must be dimensionally consistent, aligning with the fundamental dimensions of the quantities involved. Length, mass, and time serve as the building blocks for other dimensions. For instance, area corresponds to length squared, while volume corresponds to length cubed. Furthermore, quantities like velocity (length per unit time) and acceleration (length over time squared) provide insight into the rates of change of other quantities.

Dimensional Analysis in Action
Dimensional analysis is not merely about maintaining consistency in formulas; it also offers a powerful approach to estimating quantities. Consider a brick dropped from a towering building. How long does it take for the brick to reach the ground? By analyzing the relevant parameters – brick mass, building height, and gravitational acceleration- one can form a quantity with the dimensions of time. Dividing the height by the acceleration of gravity and taking the square root yields an estimation of the fall time:

    \[      t \propto \sqrt{\frac{h}{g}}\]


Though slightly off by a factor of root 2, this estimation holds true. Remarkably, the analysis reveals that the fall time is independent of the brick’s mass, a principle previously recognized by Galileo.

G.I. Taylor’s Ingenious Application
G.I. Taylor, a luminary of the twentieth century, demonstrated the prowess of dimensional analysis in an unparalleled manner. Capitalizing on the secrecy shrouding the energy of the Trinity Test explosion-equivalent to detonating 20,000 tons of TNT—Taylor utilized photographs of the event released by the US Army in 1947. These images, taken at precise intervals after the initial blast, provided crucial data. Taylor harnessed dimensional analysis, assuming that the blast radius is determined by the energy, time since detonation, and air density.
Taylor ingeniously combined these factors to derive an expression with energy dimensions, discovering that the energy was proportional to the fifth power of the blast radius. Taking the fifth root, he derived the following expression for the radius:

    \[      R = k \sqrt[5]{\frac{Et^2}{\rho}}\]


where E is the energy, \rho represents density, and k is a constant that Taylor set to unity.


Taylor’s meticulous calculations were aligned with the images from the Trinity Test, confirming the accuracy of this scaling law, as shown in the below figure.

Figure: Logarithmic plot showing that R^{5/2} is proportional to t.

The Impact of Taylor’s Work
Taylor’s estimations yielded an energy value of 17 kilotons, closely resembling the subsequent announcement by President Truman. While his findings were met with some resistance due to the classified nature of atomic weapons, they stood as a testament to the power of mathematical deductions. Taylor’s groundbreaking application of dimensional analysis exemplified its potential for exploring the hidden intricacies of scientific phenomena. In Taylor’s own words, his estimation surpassed expectations considering the limitations of the measurements involved.

Conclusion
The story of G.I. Taylor’s application of dimensional analysis underscores its crucial role as a fundamental tool for investigating and understanding complex scientific phenomena. Beyond maintaining dimensional consistency, this technique provides a means to estimate quantities, validate formulas, and reveal intricate relationships among physical parameters. As scientists continue to unravel the mysteries of the universe,
Dimensional analysis remains an indispensable asset, guiding us toward deeper insights and discoveries.

Appendix
Estimate of the energy released in the Trinity explosion:-
Let us begin with the following assumptions:

  • The energy E was discharged within a confined space.
  • The shock wave propagated spherically.
    Given that the dimensions of the fireball (R, which is a function of t) are observable at various time intervals,
    we can explore the relationship between the radius (R) and the following factors:
  • Energy E
  • Time t
  • Density of the encompassing medium (\rho – initial air density)
    Now, let us perform a dimensional analysis.
  • [R] = L, radius has dimensions of distance.
  • [E] = ML^{2}T^{-2}, energy has dimensions of mass times distance squared divided by time squared.
  • [t] = T, time and
  • [\rho] = ML^{-3}, density is determined by mass divided by distance cubed.

    The ansatz,

        \[      [R] = M^{0}L^{1}T^{0} = [E]^{a} [\rho]^{b} [t]^{c}\]


    Substituting the dimensions for energy, time and density in the above equation we have:

        \[      [R] = M^{0}L^{1}T^{0} = M^{(a+b)} L^{(2a-3b)} T^{(-2a+c)}\]


    Equating the power of M,L and T provides three simultaneous equations:
    a + b = 0,
    2a - 3b = 1 ,
    -2a + c = 0.

    Solving the three equations yields the results:

        \[   a = 1/5, b = (-1/5) , c = 2/5.\]


    Therefore, the radius of the shock wave can be expressed as:
    R = k * E^{1/5} \rho^{(-1/5)} t^{2/5}

    Let us assume the constant of proportionality k is approximately 1.
    Rearranging the terms we obtain the following equation for energy released during the explosion:

        \[   E = \frac{R^{5}\rho}{t^2}\]

Figure: Radius R of blast wave at time t after the explosion

At t = 0.015 seconds the radius of the shock wave was approximately 106.5 meters. Taking the density of air to be, \rho \approx 1.2 kg/m^{3}. Substituting these values into the energy equation gives:

E = (106.5)^{5} \times 1.2 / (0.015)^{2} kgm^{2}/s^{2}

= 7.307 \times 10^{13} kgm^{2}/s^{2}

= 7.307 \times 10^{20} ergs

Now,
\text{1 g of TNT} = 4 \times 10^{10} erg, and hence,

    \[   E \approx 18.267 \text{kilo-tons of TNT}.\]


References
1. Batchelor, George, 1996: The Life and Legacy of G. I. Taylor. Camb. Univ. Press., 285pp. ISBN: 9-780-521-46121-4
2. dimensional.pdf (utoronto.ca)
3. How Big was the Bomb? – ThatsMaths
4. Trinity explosion images taken from www.atlantic.com
5. Blast wave radius data is taken from chegg.com

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