Animated Solution for Physics - Physics and Measurement: The pairs of physical quantities that have the same dimensions is (are )

Dimensional analysis is like uncovering the fundamental DNA of physical quantities. It allows us to strip away the complex units and look purely at how mass, length, and time interact to form a concept. In this problem, we are tasked with finding pairs of physical quantities that share the exact same dimensional formula. Let's break down each option systematically.

The Dimensionless Duo

First, we look at the Reynolds number ($R_e$) and the coefficient of friction ($\mu$).

The Reynolds number is a critical parameter in fluid mechanics, defined as the ratio of inertial forces to viscous forces within a fluid. Because it is a ratio of two forces, the dimensions cancel out entirely.

Similarly, the coefficient of friction is defined as the ratio of the frictional force to the normal force ($f/N$). Again, this is a ratio of two forces.

Both quantities are pure numbers, meaning they are completely dimensionless. Their dimensional formula is $[M^0{L}^0{T}^0]$. Thus, this pair matches perfectly.

The Rhythms of Time

Next, we examine Curie and the frequency of a light wave.

The Curie is a historical unit of radioactivity, defined as the number of atomic decays per second ($3.7\times 10^{10}$ decays/second). Since "decays" is just a count (a pure number), the dimension is simply inverse time, or $[T^{-1}]$.

Frequency, on the other hand, is defined as the number of cycles or oscillations per second. Just like decays, "cycles" is a pure number, giving frequency the dimension of $[T^{-1}]$.

Both quantities represent an event occurring over time, so they share the exact same dimension. This pair is also a match.

Energy per Unit Mass

Moving on to Latent heat and gravitational potential.

Latent heat ($L$) is the amount of heat energy required to change the phase of a unit mass of a substance ($L=Q/m$). Energy has the dimensions $[M{L}^2{T}^{-2}]$, and mass is $[M]$. Dividing the two gives us $[L^2{T}^{-2}]$.

Gravitational potential ($V$) is defined as the gravitational potential energy per unit mass ($V=U/m$). Just like latent heat, we are dividing an energy term by a mass term. This yields the exact same dimensional formula: $[L^2{T}^{-2}]$.

Therefore, this pair is a perfect match as well.

The Mismatch

Finally, let's look at Planck's constant ($h$) and torque ($\tau$).

Planck's constant relates the energy of a photon to its frequency via the equation $E = h u$. Rearranging for $h$, we get $h = E/ u$. Substituting the dimensions, we have $[M{L}^2{T}^{-2}]/[T^{-1}]$, which simplifies to $[M{L}^2{T}^{-1}]$. Interestingly, this is the dimension of angular momentum.

Torque, however, is the cross product of a position vector and a force vector ($\tau =r\times F$). Multiplying a length $[L]$ by a force $[ML{T}^{-2}]$ gives us $[M{L}^2{T}^{-2}]$, which is dimensionally equivalent to energy or work.

Comparing the two, $[M{L}^2{T}^{-1}]$ is clearly not equal to $[M{L}^2{T}^{-2}]$. This pair does not match.

After a thorough dimensional breakdown, we find that options (a), (b), and (c) all contain pairs with identical dimensions. Since this is a multiple-correct question, we confidently select all three!