JEE PYQ: Units & Measurements - Question ID 6e50a1698195 (JEE Main 2005)

In the chapter Units & Measurements, the key concept is dimensional analysis. Every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($I$), thermodynamic temperature ($\Theta$), amount of substance ($N$), and luminous intensity ($J$). For this question, we focus on the first three: $M$, $L$, and $T$.

The dimension of a physical quantity is the power to which these fundamental units are raised. For example:

• Force: $F = ma \implies [F] = MLT^{-2}$
• Work: $W = F \cdot s \implies [W] = ML^2T^{-2}$
• Angular momentum: $L = mvr \implies [L] = ML^2T^{-1}$

The question asks us to identify the pair of quantities that do NOT have identical dimensions. To solve this, we derive the dimensions of each quantity in the options and compare them.

Option A: Angular Momentum and Planck's Constant

Angular Momentum ($L$): Angular momentum is defined as $L = mvr$, where $m$ is mass, $v$ is velocity, and $r$ is radius.
Dimensions: $[L] = [m][v][r] = M \cdot (LT^{-1}) \cdot L = ML^2T^{-1}$

Planck's Constant ($h$): Planck's constant relates the energy of a photon to its frequency: $E = h\nu$. Energy has dimensions $ML^2T^{-2}$, and frequency ($\nu$) has dimensions $T^{-1}$. Thus, $[h] = \frac{[E]}{[\nu]} = \frac{ML^2T^{-2}}{T^{-1}} = ML^2T^{-1}$

Conclusion for Option A: Both angular momentum and Planck's constant have dimensions $ML^2T^{-1}$. Hence, they have identical dimensions.

Option B: Impulse and Momentum

Impulse ($J$): Impulse is the change in momentum, defined as $J = F \cdot \Delta t$, where $F$ is force and $\Delta t$ is time.
Dimensions: $[J] = [F][\Delta t] = (MLT^{-2}) \cdot T = MLT^{-1}$

Momentum ($p$): Momentum is defined as $p = mv$.
Dimensions: $[p] = [m][v] = M \cdot (LT^{-1}) = MLT^{-1}$

Conclusion for Option B: Both impulse and momentum have dimensions $MLT^{-1}$. Hence, they have identical dimensions.

Option C: Moment of Inertia and Moment of a Force

Moment of Inertia ($I$): Moment of inertia is defined as $I = mr^2$, where $m$ is mass and $r$ is radius.
Dimensions: $[I] = [m][r]^2 = M \cdot L^2 = ML^2$

Moment of a Force (Torque, $\tau$): Torque is defined as $\tau = r \times F$, where $r$ is radius and $F$ is force.
Dimensions: $[\tau] = [r][F] = L \cdot (MLT^{-2}) = ML^2T^{-2}$

Conclusion for Option C: Moment of inertia has dimensions $ML^2$, while moment of a force (torque) has dimensions $ML^2T^{-2}$. These are not identical. This is the correct answer.

Option D: Work and Torque

Work ($W$): Work is defined as $W = F \cdot s$, where $F$ is force and $s$ is displacement.
Dimensions: $[W] = [F][s] = (MLT^{-2}) \cdot L = ML^2T^{-2}$

Torque ($\tau$): As derived in Option C, torque has dimensions $ML^2T^{-2}$.

Conclusion for Option D: Both work and torque have dimensions $ML^2T^{-2}$. Hence, they have identical dimensions.

Students often confuse the following:

1. Moment of inertia vs. moment of force: Many assume both have the same dimensions because they share the word "moment." However, moment of inertia is purely $ML^2$, while torque involves force ($MLT^{-2}$), introducing a $T^{-2}$ term.
2. Angular momentum and Planck's constant: Students may overlook that both have the same dimensions ($ML^2T^{-1}$) because they appear in different contexts (classical mechanics vs. quantum mechanics).
3. Impulse and momentum: Some students think impulse has dimensions of force ($MLT^{-2}$) because it is $F \cdot \Delta t$, forgetting that the time dimension cancels out to give $MLT^{-1}$.

Exam Tip: Always derive dimensions from first principles (using definitions like $F = ma$, $W = F \cdot s$, etc.) rather than relying on memory. This avoids confusion between similar-sounding quantities.