Animated Solution for Physics - Physics and Measurement: If momentum $P$, area $A$ and time $T$ are taken to be the fundamental quantities, then the dimensional formula for energy is

The universe of physics is traditionally built upon a foundation of fundamental quantities: Mass ($M$), Length ($L$), and Time ($T$). These are the ABCs of our physical reality. Every other mechanical quantity—velocity, force, energy, pressure—is simply a word spelled using these three letters.

But what if we decided to change the alphabet? What if, instead of Mass, Length, and Time, we chose a completely different set of physical quantities to be our fundamental building blocks?

This is not just a mathematical game; it is a profound exercise in understanding the interconnectedness of physical laws. In this problem, we are challenged to express Energy ($E$) not in terms of $M$, $L$, and $T$, but in terms of Momentum ($P$), Area ($A$), and Time ($T$).

Let's embark on this journey of dimensional translation.

Before we dive into the algebra, it is worth pausing to appreciate the elegance of dimensional analysis. Developed formally in the 19th century by giants like Joseph Fourier and Lord Rayleigh, dimensional analysis is based on a simple but profound idea: physical laws must be independent of the units we use to measure them.

Whether you measure length in meters, miles, or lightyears, the fundamental relationship between distance, velocity, and time remains exactly the same. This invariance implies that every physical quantity has an intrinsic "dimension" that dictates how it scales when units change.

By manipulating these dimensions algebraically, we can uncover hidden relationships between seemingly disparate physical phenomena. In our current problem, we are essentially asking: "How is the concept of Energy woven from the threads of Momentum, Area, and Time?"

When we want to express one physical quantity in terms of others, we start with a reasonable assumption: the derived quantity is proportional to the product of the new fundamental quantities, each raised to some unknown power.

We can write this hypothesis mathematically as:

Here, $x$, $y$, and $z$ are the unknown exponents that we need to discover. They represent the "recipe" for cooking up Energy using Momentum, Area, and Time. Our mission is to find the exact values of these exponents.

Why do we use a multiplicative power-law? Because physical quantities are generally derived through multiplication and division (like speed = distance / time), not addition. You cannot add a mass to a length, but you can multiply them.

Translating to M, L, T

To find $x$, $y$, and $z$, we need a common language. We must translate both the left-hand side (Energy) and the right-hand side (Momentum, Area, Time) back into our standard $M$, $L$, $T$ system. This standard system acts as our Rosetta Stone, allowing us to compare the two sides of the equation.

Let's recall the standard dimensional formulas for each quantity involved:

1. Energy ($E$): Energy is the capacity to do work. Work is Force multiplied by displacement. Force is mass times acceleration ($ML{T}^{-2}$). Therefore, multiplying by another length dimension for displacement, Energy is:

2. Momentum ($P$): Momentum is the product of mass and velocity. It represents the "quantity of motion" an object possesses. Velocity is displacement over time ($L{T}^{-1}$). Therefore, Momentum is:

3. Area ($A$): Area is a measure of a two-dimensional surface. It is simply length multiplied by length. Therefore, Area is:

4. Time ($T$): Time is already a fundamental quantity in our standard system. It is the progression of events from the past to the present into the future. Therefore:

Equating the Exponents

Now, we substitute these standard dimensional formulas back into our initial hypothesis. This is where the physics translates purely into algebra:

Next, we need to group the terms on the right-hand side. We apply the fundamental laws of exponents to distribute the powers $x$, $y$, and $z$ to the inner dimensions of $M$, $L$, and $T$:

Combining the like bases by adding their exponents, we get a single, unified expression on the right side of the equation:

This equation is a beautiful algebraic symphony. It tells us exactly how the dimensions of Mass, Length, and Time are distributed on both sides of our relationship. The left side is the target, and the right side is our parameterized attempt to hit that target.

We now invoke one of the most powerful tools in physics: the Principle of Dimensional Homogeneity. This principle states that for any valid physical equation, the dimensions on the left-hand side must be exactly identical to the dimensions on the right-hand side. You cannot equate apples to oranges, and you cannot equate a dimension of $L^2$ to a dimension of $L^3$.

By applying this principle, we can equate the exponents of $M$, $L$, and $T$ from both sides of our equation. This gives us a system of three linear equations.

For Mass ($M$): Looking at the exponent of $M$ on the left (which is 1) and the right (which is $x$), we get:

For Length ($L$): Looking at the exponent of $L$ on the left (which is 2) and the right (which is $x+2y$), we get:

For Time ($T$): Looking at the exponent of $T$ on the left (which is -2) and the right (which is $-x+z$), we get:

Solving the System

We now have a simple system of linear equations to solve. The beauty of this particular problem is that the system is already partially decoupled, making it very easy to solve sequentially.

The first equation immediately gives us our first exponent without any further work:

We substitute this known value of $x$ into the second equation to find $y$: Subtracting 1 from both sides: Dividing by 2:

Finally, we substitute the value of $x$ into the third equation to find $z$: Adding 1 to both sides:

We have successfully cracked the code! The exponents required to build Energy out of Momentum, Area, and Time are $x=1$, $y=1/2$, and $z=-1$.

The last step is to substitute these exponents back into our original hypothesis to reveal the final relationship:

Cleaning this up by removing the explicit power of 1, we arrive at our final, elegant result:

This tells us that if we lived in a universe where Momentum, Area, and Time were the fundamental quantities, Energy would be defined as Momentum multiplied by the square root of Area, divided by Time.

This exercise demonstrates the incredible flexibility and logical consistency of dimensional analysis. It is not just a tool for checking equations; it is a framework for understanding the deep structural relationships that govern the physical world. By mastering this technique, you gain the ability to translate between any set of physical variables, a skill that is invaluable in advanced physics and engineering.