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JEE Advanced 1995 · Units and Measurements

The pairs of physical quantities that have the same dimensions is (are) :

Answers: A. Reynolds number and coefficient of friction · B. Curie and frequency of a light wave · C. Latent heat and gravitational potential

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Written solution

Step 1: Analyze Pair (1) — Reynolds number and coefficient of friction.

The Reynolds number Re\text{Re} is defined as:

Re=ρvDμ\text{Re} = \frac{\rho v D}{\mu}

Since all units cancel out, it is dimensionless:

[Re]=M0L0T0[\text{Re}] = M^{0}L^{0}T^{0}

The coefficient of friction μfr\mu_{\text{fr}} is the ratio of two forces:

μfr=FfrN\mu_{\text{fr}} = \frac{F_{\text{fr}}}{N} [μfr]=M0L0T0[\mu_{\text{fr}}] = M^{0}L^{0}T^{0}

Both quantities are dimensionless, so they have the same dimensions.

Step 2: Analyze Pair (2) — Curie and frequency of a light wave.

Curie is a unit of radioactivity representing decay rate per unit time:

Curie=ΔNΔt\text{Curie} = \frac{\Delta N}{\Delta t} [Curie]=T−1[\text{Curie}] = T^{-1}

The frequency ν\nu of a light wave is:

ν=1T\nu = \frac{1}{T} [ν]=T−1[\nu] = T^{-1}

Both quantities have the same dimension of inverse time.

Step 3: Analyze Pair (3) — Latent heat and gravitational potential.

Latent heat LL is heat energy per unit mass:

L=QmL = \frac{Q}{m} [L]=ML2T−2M=M0L2T−2[L] = \frac{ML^{2}T^{-2}}{M} = M^{0}L^{2}T^{-2}

Gravitational potential VV is work done per unit mass:

V=−GMrV = -\frac{GM}{r} [V]=M0L2T−2[V] = M^{0}L^{2}T^{-2}

Both quantities have the same dimensions.

Step 4: Analyze Pair (4) — Planck's constant and torque.

Planck's constant hh is defined via energy and frequency:

h=Eνh = \frac{E}{\nu} [h]=ML2T−2T−1=ML2T−1[h] = \frac{ML^{2}T^{-2}}{T^{-1}} = ML^{2}T^{-1}

Torque τ\tau is defined as force times perpendicular distance:

τ=F×r\tau = F \times r [τ]=ML2T−2[\tau] = ML^{2}T^{-2}

These two quantities have different dimensions.

Answer: The pairs of physical quantities that have the same dimensions are (1) Reynolds number and coefficient of friction, (2) Curie and frequency of a light wave, and (3) Latent heat and gravitational potential.

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