Animated Solution for Physics - Physics and Measurement: Out of the following pairs, which one does not have identical dimensions?

Welcome to a classic exploration of dimensional analysis! This problem might look like a simple memory test, but it is actually a profound journey into the structural DNA of physics. Dimensional analysis is the ultimate lie-detector test in science. If an equation claims that two quantities are equal, their fundamental dimensions must match perfectly. Let's break down each pair and see what their dimensional DNA reveals.

Angular momentum ($L$) is the rotational analog of linear momentum. For a particle, it is defined as $L=mvr$.

Let's extract its dimensions: $[L]=[M]\cdot [L{T}^{-1}]\cdot [L]=[M{L}^2{T}^{-1}]$

Now, consider Planck's constant ($h$). It bridges the quantum and classical worlds through the energy equation $E = h u$, where $ u$ is frequency.

Rearranging for $h$, we get $h = \frac{E}{ u}$.

Substituting the dimensions of energy and frequency: $[h]=\frac{[M{L}^2{T}^{-2}]}{[T^{-1}]}=[M{L}^2{T}^{-1}]$

Fascinatingly, both share the exact same dimensional formula. This is no coincidence; it is the very reason why Niels Bohr could quantize angular momentum in units of $\frac{h}{2\pi }$!

Impulse ($J$) is the total effect of a force acting over a time interval, defined as $J=F\Delta t$.

Its dimensions are: $[J]=[ML{T}^{-2}]\cdot [T]=[ML{T}^{-1}]$

Momentum ($p$) is the quantity of motion, defined as $p=mv$.

Its dimensions are: $[p]=[M]\cdot [L{T}^{-1}]=[ML{T}^{-1}]$

According to the Impulse-Momentum Theorem, impulse equals the change in momentum. Naturally, they must be dimensionally identical.

Here is where the trap lies. The word "moment" appears in both, but they describe entirely different physical realities.

Moment of inertia ($I$) is an object's resistance to rotational acceleration. For a point mass, $I=m{r}^2$.

Its dimensions are simply: $[I]=[M]\cdot [L^2]=[M{L}^2]$

Moment of force, more commonly known as torque ($\tau$), is the rotational equivalent of force. It is defined as the cross product of the position vector and the force vector, $\tau =r\times F$.

Its dimensions are: $[\tau ]=[L]\cdot [ML{T}^{-2}]=[M{L}^2{T}^{-2}]$

Clearly, $[ML^2] eq [ML^2T^{-2}]$. These two quantities do not share the same dimensions. We have found our culprit!

Let's verify the final pair just to be thorough. Work ($W$) is the dot product of force and displacement, $W=F\cdot d$.

Its dimensions are: $[W]=[ML{T}^{-2}]\cdot [L]=[M{L}^2{T}^{-2}]$

As we just calculated, torque ($\tau$) also has the dimensions $[M{L}^2{T}^{-2}]$.

This is a beautiful paradox in physics: Work is a scalar representing energy, while torque is a vector representing rotational effort. They have the exact same dimensions, yet they describe completely different physical phenomena. You can never add work and torque together!

By systematically breaking down the dimensional formulas, we have proven that the moment of inertia and the moment of force are the only pair with non-identical dimensions. This exercise reinforces a golden rule: always trust the fundamental dimensions over the names of the quantities.