Animated Solution for Physics - Physics and Measurement: If speed $V$, area $A$ and force $F$ are chosen as fundamental units, then the dimensional formula of Young's modulus will be

Imagine for a moment that we decide to completely rewrite the rules of physics. Instead of measuring the universe using Mass, Length, and Time, we decide that Force, Area, and Speed are the true fundamental building blocks of reality.

In this new universe, how would we describe a complex property like Young's Modulus? This is the core challenge of dimensional analysis problems where fundamental units are swapped. It might sound like science fiction, but it is actually a beautiful exercise in algebraic logic and physical intuition.

Whenever we are asked to express a physical quantity in terms of a new set of fundamental units, we start by assuming a general power-law relationship. We don't know exactly how Young's modulus ($Y$) depends on Force ($F$), Area ($A$), and Speed ($V$), so we assign them unknown powers: $x$, $y$, and $z$.

Our master equation becomes:

Our mission is now clear: we must find the exact values of $x$, $y$, and $z$ that make this equation dimensionally consistent.

To solve for these unknown powers, we need a translator—a bridge between our standard $MLT$ system and this new $FAV$ system. We achieve this by writing out the standard dimensional formulas for all the quantities involved.

We know that Young's modulus is defined as Stress divided by Strain. Since Strain is just a ratio of lengths (and therefore dimensionless), Young's modulus has the exact same dimensions as Stress (Force/Area).

Similarly, we write down the dimensions for our new base quantities:

Now, we substitute these standard $MLT$ dimensions back into our master equation. This is where the magic of the Principle of Dimensional Homogeneity comes into play.

For this equation to be true, the total power of Mass ($M$), Length ($L$), and Time ($T$) must be perfectly equal on both the left and right sides. Let's break it down dimension by dimension.

Equating powers of M: On the left side, $M$ has a power of $1$. On the right side, $M$ only appears inside the Force term, raised to the power of $x$. This gives us our first breakthrough instantly:

Equating powers of T: Next, we look at Time. On the left, we have $-2$. On the right, $T$ appears in the Force term as $-2x$ and in the Speed term as $-z$.

Since we already know that $x=1$, we can substitute it in:

This is a fascinating result! A power of zero means that in this new system, Young's modulus is completely independent of Speed.

Equating powers of L: Finally, we balance Length to find our last variable, $y$. On the left, we have $-1$. On the right, we gather the powers from all three terms:

Substituting our known values ($x=1$ and $z=0$):

We have found our powers: $x=1$, $y=-1$, and $z=0$. Plugging these back into our original master equation yields the final dimensional formula:

While the algebraic method is foolproof, there is a beautiful, intuitive shortcut hidden in the physics.

Remember how we established earlier that Young's modulus has the same dimensions as Stress? And what is the definition of Stress? It is simply Force divided by Area!

Because we already had Force ($F$) and Area ($A$) available as our new fundamental units, we didn't even need Speed ($V$). The relationship was staring us right in the face. Recognizing these physical definitions can save you immense time in competitive exams!