Animated Solution for Physics - Physics and Measurement: If velocity $[v]$, time $[T]$ and force $[F]$ are choosen as the base quantities, the dimensions of the mass will be

Have you ever stopped to think about the measuring tapes we use to define the universe?

Since the dawn of classical mechanics, we have comfortably relied on Mass, Length, and Time as our absolute, unshakable foundation. We weigh things in kilograms, measure distances in meters, and track the flow of reality in seconds. Everything else—velocity, acceleration, force, energy—is just a derivative, a combination of these three fundamental pillars.

But what if we woke up in a bizarre, parallel universe? What if, in this new reality, the fundamental measuring tapes were completely different?

Imagine a world where Velocity ($v$), Time ($T$), and Force ($F$) are the absolute base quantities. In this universe, you don't measure mass directly. Instead, mass is a derived concept, a strange combination of how fast things move, how hard they are pushed, and how long it takes.

This isn't just a wild sci-fi thought experiment; it is a classic and incredibly powerful application of Dimensional Analysis. By mastering this, you gain the ability to translate the laws of physics into any language, any system of units you desire.

Let's embark on this journey and find out exactly what "mass" looks like when viewed through the lens of velocity, time, and force.

Before we start crunching numbers, we need a guiding philosophy. In physics, we rely on the Principle of Dimensional Homogeneity.

This principle states a very simple but profound truth: you cannot equate apples to oranges. If you have an equation like $A=B+C$, then the fundamental physical nature (the dimensions) of $A$, $B$, and $C$ must be absolutely identical. You cannot add a force to a velocity, and you cannot say a mass equals a length.

We will use this principle as our ultimate truth-detector. If mass can be expressed as a combination of velocity, time, and force, then the dimensions of that specific combination must perfectly match the dimensions of mass.

The Master Equation

We want to find out how mass ($m$) depends on velocity ($v$), time ($T$), and force ($F$). Since we don't know the exact relationship yet, we assume that mass is proportional to some unknown powers of these new base quantities.

Let's call these unknown powers $x$, $y$, and $z$. We can write our master equation as:

To turn this proportionality into an equation, we introduce a dimensionless constant $k$:

Our entire mission now boils down to one objective: Find the exact values of $x$, $y$, and $z$. Once we have those numbers, we have cracked the code of this new universe.

To solve for these unknown powers, we need a common ground. We need to translate both sides of our master equation back into the familiar, classical language of Mass ($M$), Length ($L$), and Time ($T$).

Let's write down the standard dimensional formulas for all the players involved:

1. Mass ($m$): This is simply mass. It has no length and no time component. $[m]=[M^1{L}^0{T}^0]$

2. Velocity ($v$): Velocity is distance divided by time. $[v]=[L^1{T}^{-1}]$

3. Time ($T$): This is just time. $[T]=[T^1]$

4. Force ($F$): From Newton's Second Law ($F=ma$), force is mass times acceleration. $[F]=[M^1{L}^1{T}^{-2}]$

Now, let's substitute these classical dimensions back into our master equation:

Equating the Powers

The equation looks a bit messy right now. Let's clean it up by using the basic laws of exponents. When you raise a power to a power, you multiply them. When you multiply terms with the same base, you add their exponents.

Let's group all the $M$, $L$, and $T$ terms on the right side together.

For Mass ($M$): The only contribution comes from the force term, which is raised to the power of $z$. So, we have $M^z$.

For Length ($L$): We get an $L^x$ from the velocity term and an $L^z$ from the force term. Multiplying them gives $L^{x+z}$.

For Time ($T$): We get $T^{-x}$ from velocity, $T^y$ from time, and $T^{-2z}$ from force. Combining them gives $T^{-x+y-2z}$.

Putting it all together, our equation becomes:

Now, we invoke the Principle of Dimensional Homogeneity. For this equation to be true, the power of $M$ on the left must equal the power of $M$ on the right. The same goes for $L$ and $T$. This gives us a system of three simple linear equations.

1. Comparing the powers of $M$: $1=z$ Wow, that was fast! We already have our first piece of the puzzle. $z=1$.

2. Comparing the powers of $L$: $0=x+z$ Since we just found out that $z=1$, we can substitute it in: $0=x+1$ $⟹x=-1$ Two down, one to go!

3. Comparing the powers of $T$: $0=-x+y-2z$ This is where you need to be careful with your minus signs. Let's substitute our known values of $x=-1$ and $z=1$: $0=-(-1)+y-2(1)$ $0=1+y-2$ $0=y-1$ $⟹y=1$

We did it! We have successfully hunted down all three unknown powers: $x=-1$ $y=1$ $z=1$

Now, let's bring these numbers back to our original master equation:

Rearranging this to look a bit more elegant, we get the final dimensional formula for mass in this new universe:

Looking back at our multiple-choice options, this perfectly matches option (b).

Your Turn to Play God

What we just did is nothing short of mathematical magic. We took the fundamental concept of mass and completely redefined it using a different set of physical rules.

This technique of dimensional analysis is not just a trick for solving textbook problems; it is a tool used by theoretical physicists to check complex derivations, predict relationships in fluid dynamics, and even explore the extreme physics of black holes and quantum mechanics.

Now that you understand the mechanics of this process, you can play God with the universe's constants. What if you chose Energy, Momentum, and Density as your base quantities? How would you express Force in that reality?

Grab a piece of paper, set up your master equation, and start equating those powers. The universe is yours to redefine!