Animated Solution for Physics - Physics and Measurement: A quantity $f$ is given by $f=\sqrt{h{c}^5/G}$, where $c$ is speed of light, $G$ universal gravitational constant and $h$ is the Planck's constant. Dimension of $f$ is that of

Imagine you are a physicist trying to understand the fundamental limits of our universe. You start playing around with the most universal constants known to humanity: Planck's constant ($h$), the speed of light ($c$), and the gravitational constant ($G$). You combine them into a strange-looking formula:

What does this quantity $f$ actually represent? Is it a length? A mass? A momentum? To find out, we don't need to plug in the actual numerical values of these constants. Instead, we can use the elegant tool of Dimensional Analysis.

Before we can find the dimensions of $f$, we need to know the dimensions of its ingredients. Let's recall the basic physics formulas that define these constants.

First, Planck's constant ($h$). We know from quantum mechanics that the energy of a photon is given by $E = h u$, where $ u$ is the frequency. Therefore, $h = \frac{E}{ u}$. The dimensions of energy are $[M{L}^2{T}^{-2}]$ and frequency is $[T^{-1}]$. So, $[h]=\frac{[M{L}^2{T}^{-2}]}{[T^{-1}]}=[M{L}^2{T}^{-1}]$.

Next, the speed of light ($c$). Speed is simply distance over time. So, $[c]=[L{T}^{-1}]$.

Finally, the gravitational constant ($G$). Newton's law of universal gravitation states that $F=G\frac{m_1{m}_2}{r^2}$. Rearranging for $G$, we get $G=\frac{F{r}^2}{m_1{m}_2}$. The dimensions of force are $[ML{T}^{-2}]$, distance squared is $[L^2]$, and mass squared is $[M^2]$. So, $[G]=\frac{[ML{T}^{-2}][L^2]}{[M^2]}=[M^{-1}{L}^3{T}^{-2}]$.

Now, let's carefully substitute these dimensional formulas into our expression for $f$. Remember that a square root is equivalent to raising the entire expression to the power of $1/2$.

Plugging in the dimensions we found:

This looks like a mess of fractions, but don't get intimidated! We just need to patiently group the powers of $M$, $L$, and $T$ one by one.

Let's start with Mass ($M$): In the numerator, we have $M^{1/2}$. In the denominator, we have $M^{-1/2}$. When we bring the denominator up, we subtract the exponent:

Next, Length ($L$): From $h$, we get $L^{2\times 1/2}=L^1$. From $c$, we get $L^{5/2}$. From $G$ in the denominator, we get $L^{3/2}$. Adding them up:

Finally, Time ($T$): From $h$, we get $T^{-1/2}$. From $c$, we get $T^{-5/2}$. From $G$ in the denominator, we get $T^{-2\times 1/2}=T^{-1}$. Adding them up:

Putting it all together, the dimensional formula for $f$ is:

Does this look familiar? It should! This is the exact dimensional formula for Energy (like kinetic energy $\frac{1}{2}m{v}^2$ or work done $F\cdot d$).

So, our mysterious quantity $f$ represents energy. In fact, in theoretical physics, a very similar expression $\sqrt{\frac{\hslash c^5}{G}}$ is known as the Planck Energy, which is the energy scale at which quantum gravity effects become significant. You just derived a fundamental scale of the universe!