Animated Solution for Physics - Physics and Measurement: Planck's constant $h$, speed of light $c$ and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then, the correct options is/are

The universe is governed by a set of fundamental constants, and sometimes, we can uncover deep relationships between physical quantities just by looking at their dimensions. This problem is a classic example of dimensional analysis, where we express macroscopic quantities like mass and length purely in terms of the fundamental constants of nature: Planck's constant ($h$), the speed of light ($c$), and the gravitational constant ($G$).

Dimensional Formulae

Before we can build mass or length out of $h$, $c$, and $G$, we need to know what these constants are made of in terms of mass (M), length (L), and time (T).

Let's break them down: 1. Planck's Constant ($h$): From the famous equation $E = h u$, we know $h = E/ u$. Energy has dimensions $[{\text{ML}}^2{\text{T}}^{-2}]$ and frequency is $[{\text{T}}^{-1}]$. So, $[h]=[{\text{ML}}^2{\text{T}}^{-1}]$. 2. Speed of Light ($c$): This is simply a velocity, so $[c]=[{\text{LT}}^{-1}]$. 3. Gravitational Constant ($G$): From Newton's law $F=G{m}_1{m}_2/r^2$, we get $G=F{r}^2/(m_1{m}_2)$. Plugging in the dimensions gives $[G]=[{\text{M}}^{-1}{\text{L}}^3{\text{T}}^{-2}]$.

Imagine mass $M$ is a recipe, and $h$, $c$, and $G$ are the ingredients. We don't know the exact amounts, so we assign unknown powers $a$, $b$, and $c$:

Now, we substitute the dimensional formulae into this equation:

By combining the powers of M, L, and T on the right side, we get:

For this equation to hold true, the powers of M, L, and T on both sides must be exactly equal. This gives us a system of three linear equations: 1. $a-c=1$ 2. $2a+b+3c=0$ 3. $-a-b-2c=0$

Solving this system is straightforward algebra. Adding the second and third equations yields $a+c=0$, or $a=-c$. Substituting this into the first equation gives $-2c=1$, so $c=-1/2$ and $a=1/2$. Finally, plugging these into the third equation gives $b=1/2$.

So, our recipe for mass is:

This tells us that $M$ is directly proportional to $\sqrt{h}$ and $\sqrt{c}$, and inversely proportional to $\sqrt{G}$. Looking at the options, Option (a) $M\propto \sqrt{c}$ is correct!

We repeat the exact same process for length $L$. Let's assume:

Substituting the dimensions:

Equating the powers gives us a new set of equations: 1. $x-z=0⟹x=z$ 2. $2x+y+3z=1$ 3. $-x-y-2z=0$

Since $x=z$, the second equation becomes $5x+y=1$, and the third becomes $-3x-y=0$ (or $y=-3x$). Substituting $y=-3x$ into the second equation gives $2x=1$, so $x=1/2$. This means $z=1/2$ as well, and $y=-3/2$.

Our recipe for length is:

This reveals that $L$ is directly proportional to both $\sqrt{h}$ and $\sqrt{G}$. Therefore, Options (c) and (d) are also correct!

What we've just derived are the proportionalities for the Planck Mass and Planck Length! These are fundamental units of the universe, constructed purely from the constants that govern quantum mechanics ($h$), relativity ($c$), and gravity ($G$). It's a beautiful glimpse into how dimensional analysis can reveal the hidden structure of physics.