Animated Solution for Physics - Physics and Measurement: The physical quantities not having same dimensions are

When faced with a question asking to identify pairs of physical quantities with different dimensions, the most efficient strategy is to quickly recall the fundamental formulas for each quantity. Dimensional analysis is a powerful tool that acts as a sanity check in physics. Let's break down each option systematically.

Let's start with the first pair: torque and work.

Both of these quantities are fundamentally the product of a force and a distance. Work is defined as the dot product of force and displacement ($W=\vec{F}\cdot \vec{d}$), while torque is the cross product of position vector and force ($\vec{\tau }=\vec{r}\times \vec{F}$).

Because they share this core relationship of force multiplied by distance, their dimensional formulas are identical:

They match perfectly, so option (a) is not our answer.

Next, let's examine stress and Young's modulus.

Stress is defined as the restoring force per unit area ($\text{Stress}=\frac{F}{A}$). Young's modulus, on the other hand, is the ratio of stress to longitudinal strain ($Y=\frac{\text{Stress}}{\text{Strain}}$).

Here is the crucial insight: strain is a dimensionless quantity because it is the ratio of change in length to original length ($\frac{\Delta L}{L}$). Therefore, dividing stress by a dimensionless number leaves the dimensions unchanged.

This pair also matches perfectly.

Now, let's look at speed and the expression $({\mu }_0{\epsilon }_0{)}^{-1/2}$.

This expression might look intimidating, but it is one of the most famous results from Maxwell's equations of electromagnetism. The speed of light in a vacuum, $c$, is given exactly by this relation:

Since this expression literally represents a speed, it must have the dimensions of speed:

Another perfect match!

Momentum and Planck's Constant

Finally, we arrive at momentum and Planck's constant.

Momentum ($p$) is the product of mass and velocity ($p=mv$). Its dimensional formula is straightforward:

Planck's constant ($h$), however, relates the energy of a photon to its frequency via the equation $E = h u$. Rearranging for $h$, we get $h = \frac{E}{ u}$. Since energy has dimensions $[{\text{ML}}^2{\text{T}}^{-2}]$ and frequency has dimensions $[{\text{T}}^{-1}]$, we find:

Comparing the two, $[{\text{MLT}}^{-1}]$ is clearly not equal to $[{\text{ML}}^2{\text{T}}^{-1}]$. They differ by a factor of length $[\text{L}]$.

Therefore, momentum and Planck's constant do not share the same dimensions. This makes option (b) the correct answer. Memorizing these standard dimensional pairs can save you precious minutes during competitive exams!