JEE PYQ: Units & Measurements - Question ID e370c787c56c (JEE Main 2024)

To determine whether two physical quantities have the same dimensions, we rely on dimensional analysis. Every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), time ($T$), electric current ($I$), temperature ($\Theta$), amount of substance ($N$), and luminous intensity ($J$). For this question, we focus on $M$, $L$, and $T$.

Key formulas and their dimensions:

• Planck’s constant ($h$): $E = h \nu \implies h = \frac{E}{\nu}$. Energy $E$ has dimensions $ML^2T^{-2}$, and frequency $\nu$ has dimensions $T^{-1}$. Thus, $h$ has dimensions $ML^2T^{-1}$.
• Angular momentum ($L$): $L = r \times p$, where $r$ is displacement ($L$) and $p$ is linear momentum ($MLT^{-1}$). Thus, $L$ has dimensions $ML^2T^{-1}$.
• Linear momentum ($p$): $p = mv$, where $m$ is mass ($M$) and $v$ is velocity ($LT^{-1}$). Thus, $p$ has dimensions $MLT^{-1}$.
• Moment of force ($\tau$): $\tau = r \times F$, where $F$ is force ($MLT^{-2}$). Thus, $\tau$ has dimensions $ML^2T^{-2}$.

Analyzing Statement (I):

Compare the dimensions of Planck’s constant ($h$) and angular momentum ($L$):

• Dimensions of $h$: $ML^2T^{-1}$.
• Dimensions of $L$: $ML^2T^{-1}$.

Since both have the same dimensions, Statement (I) is true.

Analyzing Statement (II):

Compare the dimensions of linear momentum ($p$) and moment of force ($\tau$):

• Dimensions of $p$: $MLT^{-1}$.
• Dimensions of $\tau$: $ML^2T^{-2}$.

The dimensions are not the same. Thus, Statement (II) is false.

Conclusion:

From the above analysis:

• Statement (I) is true.
• Statement (II) is false.

The correct option is A.

Students often confuse the dimensions of moment of force ($\tau$) with linear momentum ($p$) because both involve "momentum" in their names. However, moment of force is the cross product of displacement and force, giving it an extra dimension of length ($L$) compared to linear momentum. Always derive dimensions from first principles rather than relying on names or intuition.

Another common mistake is assuming that Planck’s constant and angular momentum are unrelated. While their physical interpretations differ, their dimensions are identical, which is a key insight for this question.